Hamiltonian mechanics is a reformulation of classical mechanics that was introduced in 1833 by Irish mathematician William Rowan Hamilton.

It arose from Lagrangian mechanics, a previous reformulation of classical mechanics introduced by Joseph Louis Lagrange in 1788, but can be formulated without recourse to Lagrangian mechanics using symplectic spaces (see Mathematical formalism, below). The Hamiltonian method differs from the Lagrangian method in that instead of expressing second-order differential constraints on an n-dimensional coordinate space (where n is the number of degrees of freedom of the system), it expresses first-order constraints on a 2n-dimensional phase space.^[1]

As with Lagrangian mechanics, Hamilton's equations provide a new and equivalent way of looking at Newtonian physics. Generally, these equations do not provide a more convenient way of solving a particular problem in classical mechanics. Rather, they provide deeper insights into both the general structure of classical mechanics and its connection to quantum mechanics as understood through Hamiltonian mechanics, as well as its connection to other areas of science.

Simplified overview of uses[link]

Illustration of a generalized coordinate q for one degree of freedom, of a particle moving in a complicated path. Four possibilies of q for the particle's path are shown. For more particles each with their own degrees of freedom, there are more coordinates.

The value of the Hamiltonian is the total energy of the system being described. For a closed system, it is the sum of the kinetic and potential energy in the system. There is a set of differential equations known as the Hamilton equations which give the time evolution of the system. Hamiltonians can be used to describe such simple systems as a bouncing ball, a pendulum or an oscillating spring in which energy changes from kinetic to potential and back again over time. Hamiltonians can also be employed to model the energy of other more complex dynamic systems such as planetary orbits in celestial mechanics and also in quantum mechanics.^[2]

The Hamilton equations are generally written as follows:^[3][4]

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} & \dot p_j = -\frac{\partial \mathcal{H}}{\partial q_j}\\ & \dot q_j = +\frac{\partial \mathcal{H}}{\partial p_j} \end{align}

where the dot denotes the ordinary derivative with respect to time of the generalized coordinates q_j = q_j(t) and generalized momenta p_j = p_j(t), where j = 1,2...n.

More explicitly, one can equivalently write

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm d}{\mathrm dt}\mathbf{p}(t) = -\frac{\partial}{\partial \mathbf{q}}\mathcal{H}(\mathbf{q}(t),\mathbf{p}(t),t)

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm d}{\mathrm dt}\mathbf{q}(t) = +\frac{\partial}{\partial \mathbf{p}}\mathcal{H}(\mathbf{q}(t),\mathbf{p}(t),t)

where the functions q and p take values in a vector space, and function Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H} = \mathcal{H}(\mathbf{q},\mathbf{p},t)

is the (scalar valued) Hamiltonian function, and specify the domain of values in which the parameter t (time) varies.

Hamilton's equations are symmetric in the generalized coordinates and momenta, meaning the interchange Failed to parse (Missing texvc executable; please see math/README to configure.): \pm q \rightleftharpoons \mp p

and hence Failed to parse (Missing texvc executable; please see math/README to configure.): \pm \dot{q} \rightleftharpoons \mp\dot{p}
leaves the equations unchanged. Naturally, the more degrees of freedom the system has, the more complicated its behavior (predicted by the solutions), since the degrees of freedom correspond to the configuration of the system i.e. (generalized) positions, momenta and the rates at which these change (time derivatives). As such, for more than two massive particles the solutions cannot be found exactly - the many-body problem. It is still possible to obtain qualitative knowledge about the system by approximate analysis of the differential equations. 

For a detailed derivation of these equations from Lagrangian mechanics, see below.

Basic physical interpretation[link]

The simplest interpretation of the Hamilton equations is as follows, applying them to a one-dimensional system consisting of one particle of mass m under time-independent boundary conditions: The Hamiltonian Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H} represents the energy of the system (provided that there are NO external forces, or additional energy added to the system), which is the sum of kinetic and potential energy, traditionally denoted T and V, respectively. Here q is the x coordinate and p is the momentum, mv. Then

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H} = T + V , \quad T = \frac{p^2}{2m} , \quad V = V(q) = V(x).

Note that T is a function of p alone, while V is a function of x (or q) alone.

Now the time-derivative of the momentum p equals the Newtonian force, and so here the first Hamilton equation means that the force on the particle equals the rate at which it loses potential energy with respect to changes in x, its location. (Force equals the negative gradient of potential energy.)

The time-derivative of q means the velocity: the second Hamilton equation here means that the particle’s velocity equals the derivative of its kinetic energy with respect to its momentum. (Because the derivative with respect to p of p²/2m equals p/m = mv/m = v.)

Technique of using Hamilton's equations[link]

Hamilton's equations are used in the following way. In terms of the generalized coordinates q_i and generalized velocities q̇_i:

1. The Lagrangian is found, Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{L} = T - V

.

1. The momenta are calculated by differentiating the Lagrangian with respect to the (generalized) velocities: Failed to parse (Missing texvc executable; please see math/README to configure.): p_i(q_i, \dot q_i, t) = \frac{\partial \mathcal{L}}{\partial {\dot q_i}}

.

1. The velocities q̇_i are expressed in terms of the momenta p_i by inverting the expressions in the previous step.
2. The Hamiltonian is calculated using the usual definition of H as the Legendre transformation of L: Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H} = \sum_i {\dot q_i} \frac{\partial \mathcal{L}}{\partial {\dot q_i}}- \mathcal{L} = \sum_i {\dot q_i} p_i - \mathcal{L}

. Then the velocities are substituted for using the previous results.

1. Hamilton's equations are applied, to obtain the equations of motion of the system.

Deriving Hamilton's equations[link]

Hamilton's equations can be derived by looking at how the total differential of the Lagrangian depends on time, generalized positions Failed to parse (Missing texvc executable; please see math/README to configure.): q_i\,

and generalized velocities Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{q}_i:

^[5]

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d} \mathcal{L} = \sum_i \left ( \frac{\partial \mathcal{L}}{\partial q_i} \mathrm{d} q_i + \frac{\partial \mathcal{L}}{\partial {\dot q_i}} \mathrm{d} {\dot q_i} \right ) + \frac{\partial \mathcal{L}}{\partial t} \mathrm{d}t \,.

Now the generalized momenta were defined as Failed to parse (Missing texvc executable; please see math/README to configure.): p_i = \frac{\partial \mathcal{L}}{\partial {\dot q_i}}

and Lagrange's equations tell us that

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d}}{\mathrm{d}t} \frac{\partial \mathcal{L}}{\partial {\dot q_i}} - \frac{\partial \mathcal{L}}{\partial q_i} =0. \,

We can rearrange this to get

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial \mathcal{L}}{\partial q_i} = {\dot p}_i \,

and substitute the result into the total differential of the Lagrangian

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d} \mathcal{L} = \sum_i \left [ {\dot p}_i \mathrm{d} q_i + p_i \mathrm{d} {\dot q_i} \right] + \frac{\partial \mathcal{L}}{\partial t}\mathrm{d}t \,.

We can rewrite this as

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d} \mathcal{L} = \sum_i \left [ {\dot p}_i \mathrm{d}q_i + \mathrm{d}\left ( p_i {\dot q_i} \right ) - {\dot q_i} \mathrm{d} p_i \right ] + \frac{\partial \mathcal{L}}{\partial t}\mathrm{d}t \,

and rearrange again to get

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d} \left ( \sum_i p_i {\dot q_i} - \mathcal{L} \right ) = \sum_i \left [ -{\dot p}_i \mathrm{d} q_i + {\dot q_i} \mathrm{d}p_i \right] - \frac{\partial \mathcal{L}}{\partial t}\mathrm{d}t \,.

The term on the left-hand side is just the Hamiltonian that we have defined before, so we find that

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathrm{d} \mathcal{H} = \sum_i \left [ -{\dot p}_i \mathrm{d} q_i + {\dot q_i} \mathrm{d} p_i \right] - \frac{\partial \mathcal{L}}{\partial t}\mathrm{d}t = \sum_i \left [ \frac{\partial \mathcal{H}}{\partial q_i} \mathrm{d} q_i + \frac{\partial \mathcal{H}}{\partial p_i} \mathrm{d} p_i \right ] + \frac{\partial \mathcal{H}}{\partial t}\mathrm{d}t \,

where the second equality holds because of the definition of the total differential of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H}

in terms of its partial derivatives.  Associating terms from both sides of the equation above yields Hamilton's equations

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial \mathcal{H}}{\partial q_j} =- \dot{p}_j \,, \quad \frac{\partial \mathcal{H}}{\partial p_j} = \dot{q}_j \,, \quad \frac{\partial \mathcal{H}}{\partial t } = - {\partial \mathcal{L} \over \partial t} \,.

As a reformulation of Lagrangian mechanics[link]

Starting with Lagrangian mechanics, the equations of motion are based on generalized coordinates

Failed to parse (Missing texvc executable; please see math/README to configure.): \left\{\, q_j | j=1, \ldots,N \,\right\}

and matching generalized velocities

Failed to parse (Missing texvc executable; please see math/README to configure.): \left\{\, \dot{q}_j | j=1, \ldots ,N \,\right\}.

We write the Lagrangian as

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{L}(q_j, \dot{q}_j, t)

with the subscripted variables understood to represent all N variables of that type. Hamiltonian mechanics aims to replace the generalized velocity variables with generalized momentum variables, also known as conjugate momenta. By doing so, it is possible to handle certain systems, such as aspects of quantum mechanics, that would otherwise be even more complicated.

For each generalized velocity, there is one corresponding conjugate momentum, defined as:

Failed to parse (Missing texvc executable; please see math/README to configure.): p_j = {\partial \mathcal{L} \over \partial \dot{q}_j}.

In Cartesian coordinates, the generalized momenta are precisely the physical linear momenta. In circular polar coordinates, the generalized momentum corresponding to the angular velocity is the physical angular momentum. For an arbitrary choice of generalized coordinates, it may not be possible to obtain an intuitive interpretation of the conjugate momenta.

One thing which is not too obvious in this coordinate dependent formulation is that different generalized coordinates are really nothing more than different coordinate patches on the same symplectic manifold (see Mathematical formalism, below).

The Hamiltonian is the Legendre transform of the Lagrangian:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H}\left(q_j,p_j,t\right) = \sum_i \dot{q}_i p_i - \mathcal{L}(q_j,\dot{q}_j,t).

If the transformation equations defining the generalized coordinates are independent of t, and the Lagrangian is a sum of products of functions (in the generalized coordinates) which are homogeneous of order 0, 1 or 2, then it can be shown that H is equal to the total energy E = T + V.

Each side in the definition of Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H} produces a differential:

Failed to parse (Missing texvc executable; please see math/README to configure.): \begin{align} \mathrm{d}\mathcal{H} &= \sum_i \left[ \left({\partial \mathcal{H} \over \partial q_i}\right) \mathrm{d}q_i + \left({\partial \mathcal{H} \over \partial p_i}\right) \mathrm{d}p_i \right] + \left({\partial \mathcal{H} \over \partial t}\right) \mathrm{d}t\qquad\qquad\quad\quad \\ \\ &= \sum_i \left[ \dot{q}_i\, \mathrm{d}p_i + p_i\, \mathrm{d}\dot{q}_i - \left({\partial \mathcal{L} \over \partial q_i}\right) \mathrm{d}q_i - \left({\partial \mathcal{L} \over \partial \dot{q}_i}\right) \mathrm{d}\dot{q}_i \right] - \left({\partial \mathcal{L} \over \partial t}\right) \mathrm{d}t. \end{align}

Substituting the previous definition of the conjugate momenta into this equation and matching coefficients, we obtain the equations of motion of Hamiltonian mechanics, known as the canonical equations of Hamilton:

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial \mathcal{H}}{\partial q_j} = - \dot{p}_j, \qquad \frac{\partial \mathcal{H}}{\partial p_j} = \dot{q}_j, \qquad \frac{\partial \mathcal{H}}{\partial t } = - {\partial \mathcal{L} \over \partial t}.

Hamilton's equations are first-order differential equations, and thus easier to solve than Lagrange's equations, which are second-order. Hamilton's equations have another advantage over Lagrange's equations: if a system has a symmetry, such that a coordinate does not occur in the Hamiltonian, the corresponding momentum is conserved, and that coordinate can be ignored in the other equations of the set. Effectively, this reduces the problem from n coordinates to (n-1) coordinates. In the Lagrangian framework, of course the result that the corresponding momentum is conserved still follows immediately, but all the generalized velocities still occur in the Lagrangian - we still have to solve a system of equations in n coordinates.^[6]

The Lagrangian and Hamiltonian approaches provide the groundwork for deeper results in the theory of classical mechanics, and for formulations of quantum mechanics.

Geometry of Hamiltonian systems[link]

A Hamiltonian system may be understood as a fiber bundle E over time R, with the fibers E_t, t ∈ R being the position space. The Lagrangian is thus a function on the jet bundle J over E; taking the fiberwise Legendre transform of the Lagrangian produces a function on the dual bundle over time whose fiber at t is the cotangent space T^*E_t, which comes equipped with a natural symplectic form, and this latter function is the Hamiltonian.

Generalization to quantum mechanics through Poisson bracket[link]

Hamilton's equations above work well for classical mechanics, but not for quantum mechanics, since the differential equations discussed assume that one can specify the exact position and momentum of the particle simultaneously at any point in time. However, the equations can be further generalized to then be extended to apply to quantum mechanics as well as to classical mechanics, through the deformation of the Poisson algebra over p and q to the algebra of Moyal brackets.

Specifically, the more general form of the Hamilton's equation reads

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d}f}{\mathrm{d}t} = \{f, \mathcal{H}\} + \frac{\partial f}{\partial t}

where f is some function of p and q, and H is the Hamiltonian. To find out the rules for evaluating a Poisson bracket without resorting to differential equations, see Lie algebra; a Poisson bracket is the name for the Lie bracket in a Poisson algebra. These Poisson brackets can then be extended to Moyal brackets comporting to an inequivalent Lie algebra, as proven by H Groenewold, and thereby describe quantum mechanical diffusion in phase space (See the uncertainty principle and Weyl quantization). This more algebraic approach not only permits ultimately extending probability distributions in phase space to Wigner quasi-probability distributions, but, at the mere Poisson bracket classical setting, also provides more power in helping analyze the relevant conserved quantities in a system.

Mathematical formalism[link]

Any smooth real-valued function H on a symplectic manifold can be used to define a Hamiltonian system. The function H is known as the Hamiltonian or the energy function. The symplectic manifold is then called the phase space. The Hamiltonian induces a special vector field on the symplectic manifold, known as the Hamiltonian vector field.

The Hamiltonian vector field (a special type of symplectic vector field) induces a Hamiltonian flow on the manifold. This is a one-parameter family of transformations of the manifold (the parameter of the curves is commonly called the time); in other words an isotopy of symplectomorphisms, starting with the identity. By Liouville's theorem, each symplectomorphism preserves the volume form on the phase space. The collection of symplectomorphisms induced by the Hamiltonian flow is commonly called the Hamiltonian mechanics of the Hamiltonian system.

The symplectic structure induces a Poisson bracket. The Poisson bracket gives the space of functions on the manifold the structure of a Lie algebra.

Given a function f

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\mathrm{d}}{\mathrm{d}t} f=\frac{\partial }{\partial t} f + \{\,f,\mathcal{H}\,\}.

If we have a probability distribution, ρ, then (since the phase space velocity (Failed to parse (Missing texvc executable; please see math/README to configure.): {\dot p_i} , {\dot q _i} ) has zero divergence, and probability is conserved) its convective derivative can be shown to be zero and so

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{\partial}{\partial t} \rho = - \{\,\rho ,\mathcal{H}\,\}.

This is called Liouville's theorem. Every smooth function G over the symplectic manifold generates a one-parameter family of symplectomorphisms and if { G, H } = 0, then G is conserved and the symplectomorphisms are symmetry transformations.

A Hamiltonian may have multiple conserved quantities G_i. If the symplectic manifold has dimension 2n and there are n functionally independent conserved quantities G_i which are in involution (i.e., { G_i, G_j } = 0), then the Hamiltonian is Liouville integrable. The Liouville–Arnol'd theorem says that locally, any Liouville integrable Hamiltonian can be transformed via a symplectomorphism in a new Hamiltonian with the conserved quantities G_i as coordinates; the new coordinates are called action-angle coordinates. The transformed Hamiltonian depends only on the G_i, and hence the equations of motion have the simple form

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{G}_i = 0, \qquad \dot{\varphi}_i = F(G),

for some function F (Arnol'd et al., 1988). There is an entire field focusing on small deviations from integrable systems governed by the KAM theorem.

The integrability of Hamiltonian vector fields is an open question. In general, Hamiltonian systems are chaotic; concepts of measure, completeness, integrability and stability are poorly defined. At this time, the study of dynamical systems is primarily qualitative, and not a quantitative science.

Riemannian manifolds[link]

An important special case consists of those Hamiltonians that are quadratic forms, that is, Hamiltonians that can be written as

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H}(q,p)= \frac{1}{2} \langle p,p\rangle_q

where Failed to parse (Missing texvc executable; please see math/README to configure.): \langle\cdot,\cdot\rangle_q

is a smoothly varying inner product on the fibers Failed to parse (Missing texvc executable; please see math/README to configure.): T_q^*Q

, the cotangent space to the point q in the configuration space, sometimes called a cometric. This Hamiltonian consists entirely of the kinetic term.

If one considers a Riemannian manifold or a pseudo-Riemannian manifold, the Riemannian metric induces a linear isomorphism between the tangent and cotangent bundles. (See Musical isomorphism). Using this isomorphism, one can define a cometric. (In coordinates, the matrix defining the cometric is the inverse of the matrix defining the metric.) The solutions to the Hamilton–Jacobi equations for this Hamiltonian are then the same as the geodesics on the manifold. In particular, the Hamiltonian flow in this case is the same thing as the geodesic flow. The existence of such solutions, and the completeness of the set of solutions, are discussed in detail in the article on geodesics. See also Geodesics as Hamiltonian flows.

Sub-Riemannian manifolds[link]

When the cometric is degenerate, then it is not invertible. In this case, one does not have a Riemannian manifold, as one does not have a metric. However, the Hamiltonian still exists. In the case where the cometric is degenerate at every point q of the configuration space manifold Q, so that the rank of the cometric is less than the dimension of the manifold Q, one has a sub-Riemannian manifold.

The Hamiltonian in this case is known as a sub-Riemannian Hamiltonian. Every such Hamiltonian uniquely determines the cometric, and vice-versa. This implies that every sub-Riemannian manifold is uniquely determined by its sub-Riemannian Hamiltonian, and that the converse is true: every sub-Riemannian manifold has a unique sub-Riemannian Hamiltonian. The existence of sub-Riemannian geodesics is given by the Chow-Rashevskii theorem.

The continuous, real-valued Heisenberg group provides a simple example of a sub-Riemannian manifold. For the Heisenberg group, the Hamiltonian is given by

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H}(x,y,z,p_x,p_y,p_z)=\frac{1}{2}\left( p_x^2 + p_y^2 \right).

Failed to parse (Missing texvc executable; please see math/README to configure.): p_z

is not involved in the Hamiltonian.

Poisson algebras[link]

Hamiltonian systems can be generalized in various ways. Instead of simply looking at the algebra of smooth functions over a symplectic manifold, Hamiltonian mechanics can be formulated on general commutative unital real Poisson algebras. A state is a continuous linear functional on the Poisson algebra (equipped with some suitable topology) such that for any element A of the algebra, A² maps to a nonnegative real number.

A further generalization is given by Nambu dynamics.

Charged particle in an electromagnetic field[link]

A good illustration of Hamiltonian mechanics is given by the Hamiltonian of a charged particle in an electromagnetic field. In Cartesian coordinates (i.e. Failed to parse (Missing texvc executable; please see math/README to configure.): q_i = x_i ), the Lagrangian of a non-relativistic classical particle in an electromagnetic field is (in SI Units):

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{L} = \sum_i \tfrac{1}{2} m \dot{x}_i^2 + \sum_i e \dot{x}_i A_i - e \phi,

where e is the electric charge of the particle (not necessarily the electron charge), Failed to parse (Missing texvc executable; please see math/README to configure.): \phi

is the electric scalar potential, and the Failed to parse (Missing texvc executable; please see math/README to configure.): A_i
are the components of the magnetic vector potential (these may be modified through a gauge transformation). This is called minimal coupling.

The generalized momenta may be derived by:

Failed to parse (Missing texvc executable; please see math/README to configure.): p_j = \frac{\partial L}{ \partial \dot{x}_j} = m \dot{x}_j + e A_j.

Rearranging, we may express the velocities in terms of the momenta, as:

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{x}_j = \frac{ p_j - e A_j }{m}.

If we substitute the definition of the momenta, and the definitions of the velocities in terms of the momenta, into the definition of the Hamiltonian given above, and then simplify and rearrange, we get:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H} = \sum_i \dot{x}_i p_i - \mathcal{L} = \sum_i \frac{ (p_i - e A_i)^2 } {2 m } + e \phi.

This equation is used frequently in quantum mechanics.

Relativistic charged particle in an electromagnetic field[link]

The Lagrangian for a relativistic charged particle is given by:

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{L}[t] = - m c^2 \sqrt {1 - \frac{{\dot{\vec{x}}[t]}^2}{c^2}} - e \phi [\vec{x}[t],t] + e \dot{\vec{x}}[t] \cdot \vec{A} [\vec{x}[t],t] \,.

Thus the particle's canonical (total) momentum is

Failed to parse (Missing texvc executable; please see math/README to configure.): \vec{P}\,[t] = \frac{\partial \mathcal{L}[t]}{\partial \dot{\vec{x}}[t]} = \frac{m \dot{\vec{x}}[t]}{\sqrt {1 - \frac{{\dot{\vec{x}}[t]}^2}{c^2}}} + e \vec{A} [\vec{x}[t],t] \,,

that is, the sum of the kinetic momentum and the potential momentum.

Solving for the velocity, we get

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{\vec{x}}[t] = \frac{\vec{P}\,[t] - e \vec{A} [\vec{x}[t],t]}{\sqrt {m^2 + \frac{1}{c^2}{\left( \vec{P}\,[t] - e \vec{A} [\vec{x}[t],t] \right) }^2}} \,.

So the Hamiltonian is

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H}[t] = \dot{\vec{x}}[t] \cdot \vec{P}\,[t] - \mathcal{L}[t] = c \sqrt {m^2 c^2 + {\left( \vec{P}\,[t] - e \vec{A} [\vec{x}[t],t] \right) }^2} + e \phi [\vec{x}[t],t] \,.

From this we get the force equation (equivalent to the Euler–Lagrange equation)

Failed to parse (Missing texvc executable; please see math/README to configure.): \dot{\vec{P}} = - \frac{\partial \mathcal{H}}{\partial \vec{x}} = e (\vec{\nabla} \vec{A}) \cdot \dot{\vec{x}} - e \vec{\nabla} \phi \,

from which one can derive

Failed to parse (Missing texvc executable; please see math/README to configure.): \frac{d}{d t}\left(\frac{m \dot{\vec{x}}} {\sqrt {1 - \frac{\dot{\vec{x}}^2}{c^2}}}\right) = e \vec{E} + e \dot{\vec{x}} \times \vec{B} \,.

An equivalent expression for the Hamiltonian as function of the relativistic (kinetic) momentum, Failed to parse (Missing texvc executable; please see math/README to configure.): \vec{p}=\gamma m \dot{\vec{x}}[t] \,,

is

Failed to parse (Missing texvc executable; please see math/README to configure.): \mathcal{H}[t] = \dot{\vec{x}}[t] \cdot \vec{p}\,[t] +\frac{mc^2}{\gamma} + e \phi [\vec{x}[t],t]=\gamma mc^2+ e \phi [\vec{x}[t],t]=E+V \,.

This has the advantage that Failed to parse (Missing texvc executable; please see math/README to configure.): \vec{p}

can be measured experimentally whereas Failed to parse (Missing texvc executable; please see math/README to configure.):  \vec{P}
cannot. Notice that the Hamiltonian (total energy) can be viewed as the sum of the relativistic energy (kinetic+rest), Failed to parse (Missing texvc executable; please see math/README to configure.): E=\gamma mc^2 \,,
plus the potential energy, Failed to parse (Missing texvc executable; please see math/README to configure.): V=e \phi \,.

See also[link]

• Canonical transformation
• Classical field theory
• Classical mechanics
• Dynamical systems theory
• Hamilton–Jacobi equation
• Lagrangian mechanics
• Maxwell's equations
• Hamiltonian (quantum mechanics)
• Quantum Hamilton's equations
• Quantum field theory
• Hamiltonian optics

References[link]

Footnotes[link]

Other[link]

• Arnol'd, V. I. (1989), Mathematical Methods of Classical Mechanics, Springer-Verlag, ISBN 0-387-96890-3 
• Abraham, R.; Marsden, J.E. (1978), Foundations of Mechanics, London: Benjamin-Cummings, ISBN 0-8053-0102-X 
• Arnol'd, V. I.; Kozlov, V. V.; Neĩshtadt, A. I. (1988), Mathematical aspects of classical and celestial mechanics, 3, Springer-Verlag 
• Vinogradov, A. M.; Kupershmidt, B. A. (1981) (DjVu), The structure of Hamiltonian mechanics, London Math. Soc. Lect. Notes Ser., 60, London: Cambridge Univ. Press, http://diffiety.ac.ru/djvu/structures.djvu 

External links[link]

• Binney, James J., Classical Mechanics (lecture notes), University of Oxford, http://www-thphys.physics.ox.ac.uk/users/JamesBinney/cmech.pdf, retrieved 27 October 2010 
• Tong, David, Classical Dynamics (Cambridge lecture notes), University of Cambridge, http://www.damtp.cam.ac.uk/user/tong/dynamics.html, retrieved 27 October 2010 

• Hamilton, William Rowan, On a General Method in Dynamics, Trinity College Dublin, http://www.maths.tcd.ie/pub/HistMath/People/Hamilton/Dynamics/ 

The topic of this article may not meet Wikipedia's general notability guideline. Please help to establish notability by adding reliable, secondary sources about the topic. If notability cannot be established, the article is likely to be merged, redirected, or deleted. (December 2010)

Helen Fay Dowker (born 1965 in Manchester, Lancashire) is a British Theoretical Physicist at Imperial College London.

As a student, she was interested in wormholes and quantum cosmology. Having studied at the University of Cambridge, Dowker was awarded the Tyson Medal in 1987 and completed her Doctor of Philosophy under the supervision of Stephen Hawking^[1] in 1990.

Fay Dowker was later a Postdoctoral Researcher at the Fermi National Accelerator Laboratory, at the University of California at Santa Barbara and also the California Institute of Technology.^[2][3]

She is currently a Professor of Theoretical Physics and a member of the Theoretical Physics Group at Imperial College London and an Affiliate of the Institute for Quantum Computing. She conducts research in a number of areas of theoretical physics including quantum gravity and causal set theory.

Dowker was previously a Lecturer at Queen Mary, University of London.

References[link]

External links[link]

• Theoretical Physics Group at Imperial College London

This article about a physicist of the United Kingdom is a stub. You can help Wikipedia by expanding it. v t e

Richard Feynman
File:Richard Feynman Nobel.jpg
Born	Richard Phillips Feynman (1918-05-11)May 11, 1918 Far Rockaway, Queens, New York, U.S.
Died	February 15, 1988(1988-02-15) (aged 69) Los Angeles, California, U.S.
Residence	United States
Nationality	American
Fields	Physics
Institutions	Manhattan Project Cornell University California Institute of Technology
Alma mater	Massachusetts Institute of Technology (B.S.), Princeton University (Ph.D.)
Doctoral advisor	John Archibald Wheeler
Other academic advisors	Manuel Sandoval Vallarta
Doctoral students	F. L. Vernon, Jr.[1] Willard H. Wells[1] Al Hibbs[1] George Zweig[1] Giovanni Rossi Lomanitz[1] Thomas Curtright[1]
Other notable students	Douglas D. Osheroff Robert Barro W. Daniel Hillis
Known for	Feynman diagrams Feynman point Feynman–Kac formula Wheeler–Feynman absorber theory Bethe–Feynman formula Feynman sprinkler Feynman Long Division Puzzles Hellmann–Feynman theorem Feynman slash notation Feynman parametrization Sticky bead argument One-electron universe Quantum cellular automata
Influences	Paul Dirac
Notable awards	Albert Einstein Award (1954) E. O. Lawrence Award (1962) Nobel Prize in Physics (1965) Oersted Medal (1972) National Medal of Science (1979)
Spouse	Arline Greenbaum (m. 1941–1945) «start: (1941)–end+1: (1946)»"Marriage: Arline Greenbaum to Richard Feynman" Location: (linkback:http://en-wiki.pop.wn.com/index.php/Richard_Feynman)(died) Mary Lou Bell (m. 1952–1954) «start: (1952)–end+1: (1955)»"Marriage: Mary Lou Bell to Richard Feynman" Location: (linkback:http://en-wiki.pop.wn.com/index.php/Richard_Feynman) Gweneth Howarth (m. 1960–1988) «start: (1960)–end+1: (1989)»"Marriage: Gweneth Howarth to Richard Feynman" Location: (linkback:http://en-wiki.pop.wn.com/index.php/Richard_Feynman) (his death)
Signature
Notes He was the father of Carl Feynman and adoptive father of Michelle Feynman. He was the brother of Joan Feynman.

Richard Phillips Feynman ( /ˈfaɪnmən/; May 11, 1918 – February 15, 1988)^[2] was an American physicist known for his work in the path integral formulation of quantum mechanics, the theory of quantum electrodynamics, and the physics of the superfluidity of supercooled liquid helium, as well as in particle physics (he proposed the parton model). For his contributions to the development of quantum electrodynamics, Feynman, jointly with Julian Schwinger and Sin-Itiro Tomonaga, received the Nobel Prize in Physics in 1965. He developed a widely used pictorial representation scheme for the mathematical expressions governing the behavior of subatomic particles, which later became known as Feynman diagrams. During his lifetime, Feynman became one of the best-known scientists in the world. In a 1999 poll of 130 leading physicists worldwide by the British journal Physics World he was ranked as one of the ten greatest physicists of all time.^[3]

He assisted in the development of the atomic bomb and was a member of the panel that investigated the Space Shuttle Challenger disaster. In addition to his work in theoretical physics, Feynman has been credited with pioneering the field of quantum computing,^[4][5] and introducing the concept of nanotechnology.^[6] He held the Richard Chace Tolman professorship in theoretical physics at the California Institute of Technology.

Feynman was a keen popularizer of physics through both books and lectures, notably a 1959 talk on top-down nanotechnology called There’s Plenty of Room at the Bottom and the three volume publication of his undergraduate lectures, The Feynman Lectures on Physics. Feynman also became known through his semi-autobiographical books (Surely You’re Joking, Mr. Feynman! and What Do You Care What Other People Think?) and books written about him, such as Tuva or Bust!.

He also had a deep interest in biology, and was a friend of the geneticist and microbiologist Esther Lederberg, who developed replica plating and discovered bacteriophage lambda.^[7] They had several mutual physicist friends who, after beginning their careers in nuclear research, moved into genetics, among them Max Delbrück and Aaron Novick.

Biography[link]

Early life[link]

Richard Phillips Feynman was born on May 11, 1918,^[8] in Far Rockaway, Queens, New York.^[9] His family originated from Russia and Poland; both of his parents were Ashkenazi Jewish.^[10] They were not religious and by his youth Feynman described himself as an “avowed atheist”.^[11] Feynman (in common with the famous physicist Edward Teller) was a late talker; by his third birthday he had yet to utter a single word. The young Feynman was heavily influenced by his father, Melville, who encouraged him to ask questions to challenge orthodox thinking. From his mother, Lucille, he gained the sense of humor that he had throughout his life. As a child, he had a talent for engineering, maintained an experimental laboratory in his home, and delighted in repairing radios. His younger sister Joan also became a professional physicist.^[12][13]

Education[link]

In high school, his IQ was determined to be 125—high, but “merely respectable” according to biographer James Gleick.^[14] Feynman later scoffed at psychometric testing. In the year 1933, in which he turned 15, he taught himself trigonometry, advanced algebra, infinite series, analytic geometry, and differential and integral calculus.^[15] Before entering college, he was experimenting with and re-creating mathematical topics, such as the half-derivative, using his own notation. In high school, he was developing the mathematical intuition behind his Taylor series of mathematical operators.^[16]

His habit of direct characterization sometimes rattled more conventional thinkers; for example, one of his questions, when learning feline anatomy, was “Do you have a map of the cat?” (referring to an anatomical chart).

Feynman attended Far Rockaway High School, a school also attended by fellow laureates Burton Richter and Baruch Samuel Blumberg.^[17] A member of the Arista Honor Society, in his last year in high school, Feynman won the New York University Math Championship; the large difference between his score and those of his closest competitors shocked the judges.^[18]

He applied to Columbia University, but was not accepted.^[19][20] Instead he attended the Massachusetts Institute of Technology, where he received a bachelor’s degree in 1939, and in the same year was named a Putnam Fellow. While there, Feynman took every physics course offered, including a graduate course on theoretical physics while only in his second year.

He obtained a perfect score on the graduate school entrance exams to Princeton University in mathematics and physics—an unprecedented feat—but did rather poorly on the history and English portions.^[18] Attendees at Feynman’s first seminar included Albert Einstein, Wolfgang Pauli, and John von Neumann. He received a Ph.D. from Princeton in 1942; his thesis advisor was John Archibald Wheeler. Feynman’s thesis applied the principle of stationary action to problems of quantum mechanics, inspired by a desire to quantize the Wheeler–Feynman absorber theory of electrodynamics, laying the groundwork for the “path integral” approach and Feynman diagrams, and was titled “The Principle of Least Action in Quantum Mechanics”.

This was Richard Feynman nearing the crest of his powers. At twenty-three … there was no physicist on earth who could match his exuberant command over the native materials of theoretical science. It was not just a facility at mathematics (though it had become clear … that the mathematical machinery emerging from the Wheeler–Feynman collaboration was beyond Wheeler’s own ability). Feynman seemed to possess a frightening ease with the substance behind the equations, like Albert Einstein at the same age, like the Soviet physicist Lev Landau—but few others.

— James Gleick, Genius: The Life and Science of Richard Feynman

The Manhattan Project[link]

This unreferenced section requires citations to ensure verifiability.

Feynman (center) with Robert Oppenheimer (right) relaxing at a Los Alamos social function during the Manhattan Project

At Princeton, the physicist Robert R. Wilson encouraged Feynman to participate in the Manhattan Project—the wartime U.S. Army project at Los Alamos developing the atomic bomb. Feynman said he was persuaded to join this effort to build it before Nazi Germany developed their own bomb.

He was assigned to Hans Bethe’s theoretical division, and impressed Bethe enough to be made a group leader. He and Bethe developed the Bethe–Feynman formula for calculating the yield of a fission bomb, which built upon previous work by Robert Serber.

He immersed himself in work on the project, and was present at the Trinity bomb test. Feynman claimed to be the only person to see the explosion without the very dark glasses or welder’s lenses provided, reasoning that it was safe to look through a truck windshield, as it would screen out the harmful ultraviolet radiation.

As a junior physicist, he was not central to the project. The greater part of his work was administering the computation group of human computers in the Theoretical division (one of his students there, John G. Kemeny, later went on to co-design and co-specify the programming language BASIC). Later, with Nicholas Metropolis, he assisted in establishing the system for using IBM punched cards for computation. Feynman succeeded in solving one of the equations for the project that were posted on the blackboards. However, they did not “do the physics right” and Feynman’s solution was not used.

Feynman’s other work at Los Alamos included calculating neutron equations for the Los Alamos “Water Boiler”, a small nuclear reactor, to measure how close an assembly of fissile material was to criticality. On completing this work he was transferred to the Oak Ridge facility, where he aided engineers in devising safety procedures for material storage so that criticality accidents (for example, due to sub-critical amounts of fissile material inadvertently stored in proximity on opposite sides of a wall) could be avoided. He also did theoretical work and calculations on the proposed uranium hydride bomb, which later proved not to be feasible.

Feynman was sought out by physicist Niels Bohr for one-on-one discussions. He later discovered the reason: most physicists were too in awe of Bohr to argue with him. Feynman had no such inhibitions, vigorously pointing out anything he considered to be flawed in Bohr’s thinking. Feynman said he felt as much respect for Bohr as anyone else, but once anyone got him talking about physics, he would become so focused he forgot about social niceties.

Due to the top secret nature of the work, Los Alamos was isolated. In Feynman’s own words, “There wasn’t anything to do there”. Bored, he indulged his curiosity by learning to pick the combination locks on cabinets and desks used to secure papers. Feynman played many jokes on colleagues. In one case he found the combination to a locked filing cabinet by trying the numbers a physicist would use (it proved to be 27–18–28 after the base of natural logarithms, e = 2.71828…), and found that the three filing cabinets where a colleague kept a set of atomic bomb research notes all had the same combination. He left a series of notes as a prank, which initially spooked his colleague, Frederic de Hoffmann, into thinking a spy or saboteur had gained access to atomic bomb secrets. On several occasions, Feynman drove to Albuquerque to see his ailing wife in a car borrowed from Klaus Fuchs, who was later discovered to be a real spy for the Soviets, transporting nuclear secrets in his car to Santa Fe.

On occasion, Feynman would find an isolated section of the mesa to drum in the style of American natives; “and maybe I would dance and chant, a little”. These antics did not go unnoticed, and rumors spread about a mysterious Indian drummer called “Injun Joe”. He also became a friend of laboratory head J. Robert Oppenheimer, who unsuccessfully tried to court him away from his other commitments after the war to work at the University of California, Berkeley.

Feynman alludes to his thoughts on the justification for getting involved in the Manhattan project in The Pleasure of Finding Things Out. As mentioned earlier, he felt the possibility of Nazi Germany developing the bomb before the Allies was a compelling reason to help with its development for the US. However, he goes on to say that it was an error on his part not to reconsider the situation when Germany was defeated. In the same publication, Feynman also talks about his worries in the atomic bomb age, feeling for some considerable time that there was a high risk that the bomb would be used again soon so that it was pointless to build for the future. Later he describes this period as a “depression.”

Early academic career[link]

Following the completion of his Ph.D. in 1942, Feynman held an appointment at the University of Wisconsin–Madison as an assistant professor of physics. The appointment was spent on leave for his involvement in the Manhattan project. In 1945, he received a letter from Dean Mark Ingraham of the College of Letters and Science requesting his return to UW to teach in the coming academic year. His appointment was not extended when he did not commit to return. In a talk given several years later at UW, Feynman quipped, “It’s great to be back at the only university that ever had the good sense to fire me”.^[21]

After the war, Feynman declined an offer from the Institute for Advanced Study in Princeton, New Jersey, despite the presence there of such distinguished faculty members as Albert Einstein, Kurt Gödel, and John von Neumann. Feynman followed Hans Bethe, instead, to Cornell University, where Feynman taught theoretical physics from 1945 to 1950.^[16] During a temporary depression following the destruction of Hiroshima by the bomb produced by the Manhattan Project, he focused on complex physics problems, not for utility, but for self-satisfaction. One of these was analyzing the physics of a twirling, nutating dish as it is moving through the air. His work during this period, which used equations of rotation to express various spinning speeds, soon proved important to his Nobel Prize-winning work. Yet because he felt burned out, and had turned his attention to less immediately practical but more entertaining problems, he felt surprised by the offers of professorships from other renowned universities.^[16]

Despite yet another offer from the Institute for Advanced Study, Feynman rejected the Institute on the grounds that there were no teaching duties: Feynman felt that students were a source of inspiration and teaching a diversion during uncreative spells. Because of this, the Institute for Advanced study and Princeton University jointly offered him a package whereby he could teach at the university and also be at the Institute. That he also turned this down suggests the effects of his depression^{[not in citation given]}. (see Surely You’re Joking Mr. Feynman!). Somewhat later, feeling better, Feynman accepted an offer from the California Institute of Technology (Caltech) — and as he says in his book Surely You’re Joking Mr. Feynman! — because a desire to live in a mild climate had firmly fixed itself in his mind while installing tire chains on his car in the middle of a snowstorm in Ithaca.

Feynman has been called the “Great Explainer”.^[22] He gained a reputation for taking great care when giving explanations to his students and for making it a moral duty to make the topic accessible. His guiding principle was that if a topic could not be explained in a freshman lecture, it was not yet fully understood. Feynman gained great pleasure^[23] from coming up with such a “freshman-level” explanation, for example, of the connection between spin and statistics. What he said was that groups of particles with spin ½ “repel”, whereas groups with integer spin “clump.” This was a brilliantly simplified way of demonstrating how Fermi–Dirac statistics and Bose–Einstein statistics evolved as a consequence of studying how fermions and bosons behave under a rotation of 360°. This was also a question he pondered in his more advanced lectures and to which he demonstrated the solution in the 1986 Dirac memorial lecture.^[24] In the same lecture, he further explained that antiparticles must exist, for if particles only had positive energies, they would not be restricted to a so-called “light cone.”

He opposed rote learning or unthinking memorization and other teaching methods that emphasized form over function. He put these opinions into action whenever he could, from a conference on education in Brazil to a State Commission on school textbook selection. Clear thinking and clear presentation were fundamental prerequisites for his attention. It could be perilous even to approach him when unprepared, and he did not forget the fools or pretenders.^[25]

During one sabbatical year, he returned to Newton’s Principia Mathematica to study it anew; what he learned from Newton, he passed along to his students, such as Newton’s attempted explanation of diffraction.^{[citation needed]}

Caltech years[link]

The Feynman section at the Caltech bookstore

Feynman did significant work while at Caltech, including research in:

• Quantum electrodynamics. The theory for which Feynman won his Nobel Prize is known for its accurate predictions.^[26][27] This theory was begun in the earlier years during Feynman’s work at Princeton as a graduate student and continued while he was at Cornell. This work consisted of two distinct formulations, and it is a common error to confuse them or to merge them into one. The first is his path integral formulation, and the second is his Feynman diagrams. Both formulations contained his sum over histories method in which every possible path from one state to the next is considered, the final path being a sum over the possibilities (also referred to as sum-over-paths.)^[28] For a number of years he lectured to students at Caltech on his path integral formulation of quantum theory. The lecture notes have recently been reedited by Daniel F. Styer and published as a Dover paperback. Not only did Styer correct several hundred typographical and other minor errors, but he included many footnotes explaining, for example, several places where the author used heuristic or plausible reasoning. The second formulation of quantum electrodynamics (using Feynman diagrams) was specifically mentioned by the Nobel committee. The logical connection with the path integral formulation is interesting. Feynman did not prove that the rules for his diagrams followed mathematically from the path integral formulation. Some special cases were later proved by other people, but only in the real case, so the proofs don’t work when spin is involved. The second formulation should be thought of as starting anew, but guided by the intuitive insight provided by the first formulation. Freeman Dyson published a paper in 1949 which, among many other things, added new rules to Feynman’s which told how to actually implement Renormalization. Students everywhere learned and used the powerful new tool that Feynman had created. Eventually computer programs were written to compute Feynman diagrams, providing a tool of unprecedented power. It is possible to write such programs because the Feynman diagrams constitute a Formal language with a grammar.

• Physics of the superfluidity of supercooled liquid helium, where helium seems to display a complete lack of viscosity when flowing. Applying the Schrödinger equation to the question showed that the superfluid was displaying quantum mechanical behavior observable on a macroscopic scale. This helped with the problem of superconductivity; however, the solution eluded Feynman.^[29] It was solved with the BCS theory of superconductivity, proposed by John Bardeen, Leon Neil Cooper, and John Robert Schrieffer.

• A model of weak decay, which showed that the current coupling in the process is a combination of vector and axial currents (an example of weak decay is the decay of a neutron into an electron, a proton, and an anti-neutrino). Although E. C. George Sudarshan and Robert Marshak developed the theory nearly simultaneously, Feynman’s collaboration with Murray Gell-Mann was seen as seminal because the weak interaction was neatly described by the vector and axial currents. It thus combined the 1933 beta decay theory of Enrico Fermi with an explanation of parity violation.

Richard Feynman at Fermilab

He also developed Feynman diagrams, a bookkeeping device which helps in conceptualizing and calculating interactions between particles in spacetime, notably the interactions between electrons and their antimatter counterparts, positrons. This device allowed him, and later others, to approach time reversibility and other fundamental processes. Feynman’s mental picture for these diagrams started with the hard sphere approximation, and the interactions could be thought of as collisions at first. It was not until decades later that physicists thought of analyzing the nodes of the Feynman diagrams more closely. Feynman famously painted Feynman diagrams on the exterior of his van.^[30]

From his diagrams of a small number of particles interacting in spacetime, Feynman could then model all of physics in terms of those particles’ spins and the range of coupling of the fundamental forces.^[31] Feynman attempted an explanation of the strong interactions governing nucleons scattering called the parton model. The parton model emerged as a complement to the quark model developed by his Caltech colleague Murray Gell-Mann. The relationship between the two models was murky; Gell-Mann referred to Feynman’s partons derisively as “put-ons”. In the mid 1960s, physicists believed that quarks were just a bookkeeping device for symmetry numbers, not real particles, as the statistics of the Omega-minus particle, if it were interpreted as three identical strange quarks bound together, seemed impossible if quarks were real. The Stanford linear accelerator deep inelastic scattering experiments of the late 1960s showed, analogously to Ernest Rutherford’s experiment of scattering alpha particles on gold nuclei in 1911, that nucleons (protons and neutrons) contained point-like particles which scattered electrons. It was natural to identify these with quarks, but Feynman’s parton model attempted to interpret the experimental data in a way which did not introduce additional hypotheses. For example, the data showed that some 45% of the energy momentum was carried by electrically neutral particles in the nucleon. These electrically neutral particles are now seen to be the gluons which carry the forces between the quarks and carry also the three-valued color quantum number which solves the Omega-minus problem. Feynman did not dispute the quark model; for example, when the fifth quark was discovered in 1977, Feynman immediately pointed out to his students that the discovery implied the existence of a sixth quark, which was duly discovered in the decade after his death.

After the success of quantum electrodynamics, Feynman turned to quantum gravity. By analogy with the photon, which has spin 1, he investigated the consequences of a free massless spin 2 field, and was able to derive the Einstein field equation of general relativity, but little more.^[32] However, the computational device that Feynman discovered then for gravity, “ghosts”, which are “particles” in the interior of his diagrams which have the “wrong” connection between spin and statistics, have proved invaluable in explaining the quantum particle behavior of the Yang–Mills theories, for example QCD and the electro-weak theory.

In 1965, Feynman was appointed a foreign member of the Royal Society.^[33] At this time in the early 1960s, Feynman exhausted himself by working on multiple major projects at the same time, including a request, while at Caltech, to “spruce up” the teaching of undergraduates. After three years devoted to the task, he produced a series of lectures that eventually became the Feynman Lectures on Physics, one reason that Feynman is still regarded as one of the greatest teachers of physics. He wanted a picture of a drumhead sprinkled with powder to show the modes of vibration at the beginning of the book. Outraged by many rock and roll and drug connections that could be made from the image, the publishers changed the cover to plain red, though they included a picture of him playing drums in the foreword. Feynman later won the Oersted Medal for teaching, of which he seemed especially proud.^[34]

His students competed keenly for his attention; he was once awakened when a student solved a problem and dropped it in his mailbox; glimpsing the student sneaking across his lawn, he could not go back to sleep, and he read the student’s solution. The next morning his breakfast was interrupted by another triumphant student, but Feynman informed him that he was too late.

Partly as a way to bring publicity to progress in physics, Feynman offered $1000 prizes for two of his challenges in nanotechnology, claimed by William McLellan and Tom Newman, respectively.^[35] He was also one of the first scientists to conceive the possibility of quantum computers.

Many of his lectures and other miscellaneous talks were turned into books, including The Character of Physical Law and QED: The Strange Theory of Light and Matter. He gave lectures which his students annotated into books, such as Statistical Mechanics and Lectures on Gravity. The Feynman Lectures on Physics^[36] occupied two physicists, Robert B. Leighton and Matthew Sands as part-time co-authors for several years. Even though they were not adopted by most universities as textbooks, the books continue to be bestsellers because they provide a deep understanding of physics. As of 2005, The Feynman Lectures on Physics has sold over 1.5 million copies in English, an estimated 1 million copies in Russian, and an estimated half million copies in other languages.^{[citation needed]}

In 1974, Feynman delivered the Caltech commencement address on the topic of cargo cult science, which has the semblance of science but is only pseudoscience due to a lack of “a kind of scientific integrity, a principle of scientific thought that corresponds to a kind of utter honesty” on the part of the scientist. He instructed the graduating class that “The first principle is that you must not fool yourself—and you are the easiest person to fool. So you have to be very careful about that. After you’ve not fooled yourself, it’s easy not to fool other scientists. You just have to be honest in a conventional way after that.”^[37]

In 1984–86, he developed a variational method for the approximate calculation of path integrals which has led to a powerful method of converting divergent perturbation expansions into convergent strong-coupling expansions (variational perturbation theory) and, as a consequence, to the most accurate determination^[38] of critical exponents measured in satellite experiments.^[39]

In the late 1980s, according to “Richard Feynman and the Connection Machine”, Feynman played a crucial role in developing the first massively parallel computer, and in finding innovative uses for it in numerical computations, in building neural networks, as well as physical simulations using cellular automata (such as turbulent fluid flow), working with Stephen Wolfram at Caltech.^[40] His son Carl also played a role in the development of the original Connection Machine engineering; Feynman influencing the interconnects while his son worked on the software.

Feynman diagrams are now fundamental for string theory and M-theory, and have even been extended topologically.^{[citation needed]} The world-lines of the diagrams have developed to become tubes to allow better modeling of more complicated objects such as strings and membranes. However, shortly before his death, Feynman criticized string theory in an interview: “I don’t like that they’re not calculating anything,” he said. “I don’t like that they don’t check their ideas. I don’t like that for anything that disagrees with an experiment, they cook up an explanation—a fix-up to say, ‘Well, it still might be true.’” These words have since been much-quoted by opponents of the string-theoretic direction for particle physics.^[41]

Challenger disaster[link]

The 1986 Space Shuttle Challenger disaster

Feynman played an important role on the Presidential Rogers Commission, which investigated the Challenger disaster. During a televised hearing, Feynman demonstrated that the material used in the shuttle's O-rings became less resilient in cold weather by immersing a sample of the material in ice-cold water. The Commission ultimately determined that the disaster was caused by the primary O-ring not properly sealing due to extremely cold weather at Cape Canaveral.^[42]

Feynman devoted the latter half of his book What Do You Care What Other People Think? to his experience on the Rogers Commission, straying from his usual convention of brief, light-hearted anecdotes to deliver an extended and sober narrative. Feynman’s account reveals a disconnect between NASA’s engineers and executives that was far more striking than he expected. His interviews of NASA’s high-ranking managers revealed startling misunderstandings of elementary concepts. For instance, NASA managers claimed that there was a 1 in 100,000 chance of a catastrophic failure aboard the shuttle, but Feynman discovered that NASA's own engineers estimated the chance of a catastrophe at closer to 1 in 100. He concluded that the NASA management’s space shuttle reliability estimate was fantastically unrealistic, and was particularly angered that NASA used these figures to recruit Christa McAuliffe into the Teacher-in-Space program. He warned in his appendix to the commission’s report (which was included after he threatened not to sign the report), “For a successful technology, reality must take precedence over public relations, for nature cannot be fooled.”^[43] He also rebuked some mathematicians for their exclusivity, saying “I have great suspicion that [mathematicians] don’t know that this stuff is wrong and that they’re intimidating people.”^[44]

Ethnic identification[link]

Though raised Jewish and born to Ashkenazi parents, Feynman himself was not only atheist, but distanced himself from being labelled Jewish even on ethnic grounds. He routinely refused to be included in lists or books that classified people by race. He asked to not be included in Tina Levitan’s, The Laureates: Jewish Winners of the Nobel Prize, writing, "To select, for approbation the peculiar elements that come from some supposedly Jewish heredity is to open the door to all kinds of nonsense on racial theory," and adding "…at thirteen I was not only converted to other religious views but I also stopped believing that the Jewish people are in any way 'the chosen people'".^[45][46]

Personal life[link]

While researching for his Ph.D., Feynman married his first wife, Arline Greenbaum (often spelled Arlene). She was diagnosed with tuberculosis, but she and Feynman were careful, and he never contracted it. She succumbed to the disease in 1945. In 1946 Feynman wrote a letter to her but kept it sealed until after his death.^[47] This portion of Feynman’s life was portrayed in the 1996 film Infinity, which featured Feynman’s daughter Michelle in a cameo role.

He was married a second time in June 1952, to Mary Louise Bell of Neodesha, Kansas; this marriage was brief and unsuccessful:

He begins working calculus problems in his head as soon as he awakens. He did calculus while driving in his car, while sitting in the living room, and while lying in bed at night.

—Mary Louise Bell divorce complaint^[48]p.168

He later married Gweneth Howarth from Ripponden, Yorkshire, who shared his enthusiasm for life and spirited adventure.^[49] Besides their home in Altadena, California, they had a beach house in Baja California, purchased with the prize money from Feynman’s Nobel Prize, his one third share of $55,000. They remained married until Feynman’s death. They had a son, Carl, in 1962, and adopted a daughter, Michelle, in 1968.^[49]

Feynman had a great deal of success teaching Carl, using discussions about ants and Martians as a device for gaining perspective on problems and issues; he was surprised to learn that the same teaching devices were not useful with Michelle.^[50] Mathematics was a common interest for father and son; they both entered the computer field as consultants and were involved in advancing a new method of using multiple computers to solve complex problems—later known as parallel computing. The Jet Propulsion Laboratory retained Feynman as a computational consultant during critical missions. One co-worker characterized Feynman as akin to Don Quixote at his desk, rather than at a computer workstation, ready to do battle with the windmills.

Feynman traveled widely, notably to Brazil, where he gave courses at the CBPF (Brazilian Center for Physics Research) and near the end of his life schemed to visit the Russian land of Tuva, a dream that, because of Cold War bureaucratic problems, never became reality.^[51] The day after he died, a letter arrived for him from the Soviet government giving him authorization to travel to Tuva. Out of his enthusiastic interest in reaching Tuva came the phrase “Tuva or Bust” (also the title of a book about his efforts to get there), which was tossed about frequently amongst his circle of friends in hope that they, one day, could see it firsthand. The documentary movie Genghis Blues mentions some of his attempts to communicate with Tuva, and chronicles the successful journey there by his friends.

Responding to Hubert Humphrey’s congratulation for his Nobel Prize, Feynman admitted to a long admiration for the then vice president.^[52] In a letter to an MIT professor dated December 6, 1966, Feynman expressed interest in running for the governor of California.^[53]

Feynman took up drawing at one time and enjoyed some success under the pseudonym “Ofey”, culminating in an exhibition dedicated to his work. He learned to play a metal percussion instrument (frigideira) in a samba style in Brazil, and participated in a samba school.

In addition, he had some degree of synesthesia for equations, explaining that the letters in certain mathematical functions appeared in color for him, even though invariably printed in standard black-and-white.^[54]

According to Genius, the James Gleick–authored biography, Feynman tried LSD during his professorship at Caltech.^[18] Somewhat embarrassed by his actions, Feynman largely sidestepped the issue when dictating his anecdotes; he mentions it in passing in the “O Americano, Outra Vez” section, while the “Altered States” chapter in Surely You’re Joking, Mr. Feynman! describes only marijuana and ketamine experiences at John Lilly’s famed sensory deprivation tanks, as a way of studying consciousness.^[16] Feynman gave up alcohol when he began to show early signs of alcoholism, as he did not want to do anything that could damage his brain—the same reason given in “O Americano, Outra Vez” for his reluctance to experiment with LSD.^[16]

In Surely You’re Joking, Mr. Feynman!, he gives advice on the best way to pick up a girl in a hostess bar. At Caltech, he used a nude/topless bar as an office away from his usual office, making sketches or writing physics equations on paper placemats. When the county officials tried to close the place, all visitors except Feynman refused to testify in favor of the bar, fearing that their families or patrons would learn about their visits. Only Feynman accepted, and in court, he affirmed that the bar was a public need, stating that craftsmen, technicians, engineers, common workers “and a physics professor” frequented the establishment. While the bar lost the court case, it was allowed to remain open as a similar case was pending appeal.^[16]

Feynman has a minor acting role in the film Anti-Clock. He is credited as “The Professor.”^[55]

Death[link]

Feynman developed two rare forms of cancer, Liposarcoma and Waldenström's macroglobulinemia, dying shortly after a final attempt at surgery for the former on February 15, 1988, aged 69.^[18] His last recorded words are noted as “I’d hate to die twice. It’s so boring.”^[18][56]

Popular legacy[link]

Alan Alda, the stage, screen and television actor, studied writings about Richard Feynman’s life during the 1990s in preparation for playing the role of Feynman on stage in a yet-to-be developed play. Playwright Peter Parnell was commissioned by Alda to write a two-character play about a fictional day in the life of Feynman set two years prior to Feynman’s death from cancer, based upon Alda’s biographical research. The play, entitled QED, premiered at the Mark Taper Forum in Los Angeles, CA in 2001. The play was then presented at the Vivian Beaumont Theatre on Broadway; with both presentations starring Alan Alda as the scientist, Richard Feynman.

On May 4, 2005, the United States Postal Service issued the American Scientists commemorative set of four 37-cent self-adhesive stamps in several configurations. The scientists depicted were Richard Feynman, John von Neumann, Barbara McClintock and Josiah Willard Gibbs. Feynman’s stamp, sepia-toned, features a photograph of a 30-something Feynman and eight small Feynman diagrams.^[57] The stamps were designed by artist Victor Stabin under the direction of U.S. Postal Service art director Carl T. Herrman.^[58]

The main building for the Computing Division at Fermilab is named the “Feynman Computing Center” in his honor.^[59]

The principal character in Thomas A. McMahon’s 1970 novel, Principles of American Nuclear Chemistry: A Novel, is modeled on Feynman.^{[citation needed]}

Real Time Opera premiered its opera Feynman at the Norfolk (CT) Chamber Music Festival in June 2005.^[60]

On the 20th anniversary of Feynman’s death, composer Edward Manukyan dedicated a piece for solo clarinet to his memory.^[61] It was premiered by Doug Storey, the principal clarinetist of the Amarillo Symphony.

In 2009–11, clips of an interview with Feynman were used by composer John Boswell as part of the Symphony of Science project in the second, fifth, seventh, and eleventh installments of his videos, “We Are All Connected”, “The Poetry of Reality”, “A Wave of Reason”, and “The Quantum World”.^[62]

In 1998, a photo of Richard Feynman giving a lecture was part of the poster series commissioned by Apple Computer for their “Think Different” advertising campaign.^[63]

In 2011 Feynman was the subject of a biographical graphic novel titled simply Feynman, written by Jim Ottaviani and illustrated by Leland Myrick.^[64]

Bibliography[link]

Selected scientific works[link]

• Feynman, Richard P. (2000). Laurie M. Brown. ed. Selected Papers of Richard Feynman: With Commentary. 20th Century Physics. World Scientific. ISBN 978-981-02-4131-5. 
• Feynman, Richard P. (1942). Laurie M. Brown. ed. The Principle of Least Action in Quantum Mechanics. Ph.D. Dissertation, Princeton University. World Scientific (with title Feynman’s Thesis: a New Approach to Quantum Theory). 2005. ISBN 978-981-256-380-4. 
• Wheeler, John A.; Feynman, Richard P. (1945). "Interaction with the Absorber as the Mechanism of Radiation". Rev. Mod. Phys. 17 (2–3): 157–181. Bibcode 1945RvMP...17..157W. DOI:10.1103/RevModPhys.17.157. 
• Feynman, Richard P. (1946). A Theorem and its Application to Finite Tampers. Los Alamos Scientific Laboratory, Atomic Energy Commission. OSTI 4341197. 
• Feynman, Richard P.; Welton, T. A. (1946). Neutron Diffusion in a Space Lattice of Fissionable and Absorbing Materials. Los Alamos Scientific Laboratory, Atomic Energy Commission. OSTI 4381097. 
• Feynman, Richard P.; Metropolis, N.; Teller, E. (1947). Equations of State of Elements Based on the Generalized Fermi-Thomas Theory. Los Alamos Scientific Laboratory, Atomic Energy Commission. OSTI 4417654. 
• Feynman, Richard P. (1948a). "Space-time approach to non-relativistic quantum mechanics". Rev. Mod. Phys. 20 (2): 367–387. Bibcode 1948RvMP...20..367F. DOI:10.1103/RevModPhys.20.367. 
• Feynman, Richard P. (1948b). "Relativistic Cut-Off for Quantum Electrodynamics". Physical Review 74 (10): 1430–1438. Bibcode 1948PhRv...74.1430F. DOI:10.1103/PhysRev.74.1430. 
• Wheeler, John A.; Feynman, Richard P. (1949). "Classical Electrodynamics in Terms of Direct Interparticle Action". Rev. Mod. Phys. 21 (3): 425–433. Bibcode 1949RvMP...21..425W. DOI:10.1103/RevModPhys.21.425. 
• Feynman, Richard P. (1949). "The theory of positrons". Phys. Rev. 76 (6): 749–759. Bibcode 1949PhRv...76..749F. DOI:10.1103/PhysRev.76.749. 
• Feynman, Richard P. (1949b). "Space-Time Approach to Quantum Electrodynamic". Phys. Rev. 76 (6): 769–789. Bibcode 1949PhRv...76..769F. DOI:10.1103/PhysRev.76.769. 
• Feynman, Richard P. (1950). "Mathematical formulation of the quantum theory of electromagnetic interaction". Phys. Rev. 80 (3): 440–457. Bibcode 1950PhRv...80..440F. DOI:10.1103/PhysRev.80.440. 
• Feynman, Richard P. (1951). "An Operator Calculus Having Applications in Quantum Electrodynamics". Phys. Rev. 84: 108–128. Bibcode 1951PhRv...84..108F. DOI:10.1103/PhysRev.84.108. 
• Feynman, Richard P. (1953). "The λ-Transition in Liquid Helium". Phys. Rev. 90 (6): 1116–1117. Bibcode 1953PhRv...90.1116F. DOI:10.1103/PhysRev.90.1116.2. 
• Feynman, Richard P.; de Hoffmann, F.; Serber, R. (1955). Dispersion of the Neutron Emission in U235 Fission. Los Alamos Scientific Laboratory, Atomic Energy Commission. OSTI 4354998. 
• Feynman, Richard P. (1956). "Science and the Open Channel". Science 123 (3191): 307. February 24, 1956. Bibcode 1956Sci...123..307F. DOI:10.1126/science.123.3191.307. PMID 17774518. 
• Cohen, M.; Feynman, Richard P. (1957). "Theory of Inelastic Scattering of Cold Neutrons from Liquid Helium". Phys. Rev. 107: 13–24. Bibcode 1957PhRv..107...13C. DOI:10.1103/PhysRev.107.13. 
• Feynman, Richard P.; Vernon, F. L.; Hellwarth, R.W. (1957). "Geometric representation of the Schrödinger equation for solving maser equations". J. Appl. Phys 28: 49. Bibcode 1957JAP....28...49F. DOI:10.1063/1.1722572. 
• Feynman, Richard P. (1959). "Plenty of Room at the Bottom". Presentation to American Physical Society. http://www.its.caltech.edu/~feynman/plenty.html. 
• Edgar, R. S.; Feynman, Richard P.; Klein, S.; Lielausis, I.; Steinberg, CM (1962). "Mapping experiments with r mutants of bacteriophage T4D". Genetics 47: 179–86. February 1962. PMC 1210321. PMID 13889186. //www.pubmedcentral.nih.gov/articlerender.fcgi?tool=pmcentrez&artid=1210321. 
• Feynman, Richard P. (1966). "The Development of the Space-Time View of Quantum Electrodynamics". Science 153 (3737): 699–708. August 12, 1966. Bibcode 1966Sci...153..699F. DOI:10.1126/science.153.3737.699. PMID 17791121. 
• Feynman, Richard P. (1974a). "Structure of the proton". Science 183 (4125): 601–610. February 15, 1974. Bibcode 1974Sci...183..601F. DOI:10.1126/science.183.4125.601. PMID 17778830. 
• Feynman, Richard P. (1974). "Cargo Cult Science" (PDF). Engineering and Science 37 (7). http://calteches.library.caltech.edu/51/02/CargoCult.pdf. 
• Feynman, Richard P.; Kleinert, Hagen (1986). "Effective classical partition functions". Phys. Rev., A 34 (6): 5080–5084. December 1986. Bibcode 1986PhRvA..34.5080F. DOI:10.1103/PhysRevA.34.5080. PMID 9897894. 

Textbooks and lecture notes[link]

The Feynman Lectures on Physics is perhaps his most accessible work for anyone with an interest in physics, compiled from lectures to Caltech undergraduates in 1961–64. As news of the lectures’ lucidity grew, a number of professional physicists and graduate students began to drop in to listen. Co-authors Robert B. Leighton and Matthew Sands, colleagues of Feynman, edited and illustrated them into book form. The work has endured, and is useful to this day. They were edited and supplemented in 2005 with “Feynman’s Tips on Physics: A Problem-Solving Supplement to the Feynman Lectures on Physics” by Michael Gottlieb and Ralph Leighton (Robert Leighton’s son), with support from Kip Thorne and other physicists.

• Feynman, Richard P. (1970). The Feynman Lectures on Physics: The Definitive and Extended Edition. 3 volumes (2nd ed.). Addison Wesley (published 2005, originally published as separate volumes in 1964 and 1966). ISBN 0-8053-9045-6.  Includes Feynman’s Tips on Physics (with Michael Gottlieb and Ralph Leighton), which includes four previously unreleased lectures on problem solving, exercises by Robert Leighton and Rochus Vogt, and a historical essay by Matthew Sands.
• Feynman, Richard P. (1961). Theory of Fundamental Processes. Addison Wesley. ISBN 0-8053-2507-7. 
• Feynman, Richard P. (1962). Quantum Electrodynamics. Addison Wesley. ISBN 978-0-8053-2501-0. 
• Feynman, Richard P.; Hibbs, Albert (1965). Quantum Mechanics and Path Integrals. McGraw Hill. ISBN 0-07-020650-3. 
• Feynman, Richard P. (1967). The Character of Physical Law: The 1964 Messenger Lectures. MIT Press. ISBN 0-262-56003-8. 
• Feynman, Richard P. (1981). Statistical Mechanics: A Set of Lectures. Addison Wesley. ISBN 0-8053-2509-3. 
• Feynman, Richard P. (1985b). QED: The Strange Theory of Light and Matter. Princeton University Press. ISBN 0-691-02417-0. 
• Feynman, Richard P. (1987). Elementary Particles and the Laws of Physics: The 1986 Dirac Memorial Lectures. Cambridge University Press. ISBN 0-521-34000-4. 
• Feynman, Richard P. (1995). Brian Hatfield. ed. Lectures on Gravitation. Addison Wesley Longman. ISBN 0-201-62734-5. 
• Feynman, Richard P. (1997). Feynman’s Lost Lecture: The Motion of Planets Around the Sun (Vintage Press ed.). London: Vintage. ISBN 0-09-973621-7. 
• Feynman, Richard P. (2000). Tony Hey and Robin W. Allen. ed. Feynman Lectures on Computation. Perseus Books Group. ISBN 0-7382-0296-7. 

Popular works[link]

• Surely You’re Joking, Mr. Feynman!: Adventures of a Curious Character, with contributions by Ralph Leighton, W. W. Norton & Co, 1985, ISBN 0-393-01921-7.
• What Do You Care What Other People Think?: Further Adventures of a Curious Character, with contributions by Ralph Leighton, W. W. Norton & Co, 1988, ISBN 0-393-02659-0.
• No Ordinary Genius: The Illustrated Richard Feynman, ed. Christopher Sykes, W. W. Norton & Co, 1996, ISBN 0-393-31393-X.
• Six Easy Pieces: Essentials of Physics Explained by Its Most Brilliant Teacher, Perseus Books, 1994, ISBN 0-201-40955-0.
• Six Not So Easy Pieces: Einstein’s Relativity, Symmetry and Space-Time, Addison Wesley, 1997, ISBN 0-201-15026-3.
• The Meaning of It All: Thoughts of a Citizen Scientist, Perseus Publishing, 1999, ISBN 0-7382-0166-9.
• The Pleasure of Finding Things Out: The Best Short Works of Richard P. Feynman, edited by Jeffrey Robbins, Perseus Books, 1999, ISBN 0-7382-0108-1.
• Classic Feynman: All the Adventures of a Curious Character, edited by Ralph Leighton, W. W. Norton & Co, 2005, ISBN 0-393-06132-9. Chronologically reordered omnibus volume of Surely You’re Joking, Mr. Feynman! and What Do You Care What Other People Think?, with a bundled CD containing one of Feynman’s signature lectures.
• Quantum Man, Atlas books, 2011, Lawrence M. Krauss, ISBN 978039306471

Audio and video recordings[link]

• Safecracker Suite (a collection of drum pieces interspersed with Feynman telling anecdotes)
• Los Alamos From Below (audio, talk given by Feynman at Santa Barbara on February 6, 1975)
• Six Easy Pieces (original lectures upon which the book is based)
• Six Not So Easy Pieces (original lectures upon which the book is based)
• The Feynman Lectures on Physics: The Complete Audio Collection
• Samples of Feynman’s drumming, chanting and speech are included in the songs “Tuva Groove (Bolur Daa-Bol, Bolbas Daa-Bol)” and “Kargyraa Rap (Dürgen Chugaa)” on the album Back Tuva Future, The Adventure Continues by Kongar-ool Ondar. The hidden track on this album also includes excerpts from lectures without musical background.
• The Messenger Lectures, given at Cornell in 1964, in which he explains basic topics in physics. Available on Project Tuva for free (See also the book The Character of Physical Law)
• Take the world from another point of view [videorecording] / with Richard Feynman; Films for the Hu (1972)
• The Douglas Robb Memorial Lectures Four public lectures of which the four chapters of the book QED: The Strange Theory of Light and Matter are transcripts. (1979)
• The Pleasure of Finding Things Out (1981) (not to be confused with the later published book of same title)
• Richard Feynman: Fun to Imagine Collection, BBC Archive of 6 short films of Feynman talking in a style that is accessible to all about the physics behind common to all experiences. (1983)
• Elementary Particles and the Laws of Physics (1986)
• The Last Journey of a Genius, a BBC TV production in association with WGBH Boston (1989)
• Tiny Machines: The Feynman Talk on Nanotechnology (video, 1984)
• Computers From the Inside Out (video)
• Quantum Mechanical View of Reality: Workshop at Esalen (video, 1983)
• Idiosyncratic Thinking Workshop (video, 1985)
• Bits and Pieces — From Richard’s Life and Times (video, 1988)
• Strangeness Minus Three (video, BBC Horizon 1964)
• No Ordinary Genius (video, Cristopher Sykes Documentary)
• Richard Feynman — The Best Mind Since Einstein (video, Documentary)
• The Motion of Planets Around the Sun (audio, sometimes titled “Feynman’s Lost Lecture”)
• Nature of Matter (audio)

See also[link]

Notes[link]