- published: 26 Mar 2014
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The Lagrangian, L, of a dynamical system is a function that summarizes the dynamics of the system. It is named after Joseph Louis Lagrange. The concept of a Lagrangian was originally introduced in a reformulation of classical mechanics by Irish mathematician William Rowan Hamilton known as Lagrangian mechanics.
In classical mechanics, the Lagrangian is defined as the kinetic energy, T, of the system minus its potential energy, V. In symbols,
If the Lagrangian of a system is known, then the equations of motion of the system may be obtained by a direct substitution of the expression for the Lagrangian into the Euler–Lagrange equation.
The trajectory of a thrown ball is characterized by the sum of the Lagrangian values at each time being a minimum.
The Lagrangian L can be calculated at several instants of time t, and a graph of L against t can be drawn. The area under the curve is the action. Any different path between the initial and final positions leads to a larger action than that chosen by nature. Nature chooses the smallest action - this is the Principle of Least Action.
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