Orbit (dynamics)

In mathematics, in the study of dynamical systems, an orbit is a collection of points related by the evolution function of the dynamical system. The orbit is a subset of the phase space and the set of all orbits is a partition of the phase space, that is, different orbits do not intersect in the phase space. Understanding the properties of orbits by using topological methods is one of the objectives of the modern theory of dynamical systems.

For discrete-time dynamical systems, the orbits are sequences; for real dynamical systems, the orbits are curves; and for holomorphic dynamical systems, the orbits are Riemann surfaces.

Definition

Given a dynamical system (T, M, Φ) with T a group, M a set and Φ the evolution function

we define

then the set

is called orbit through x. An orbit which consists of a single point is called constant orbit. A non-constant orbit is called closed or periodic if there exists a t in T so that

for every point x on the orbit.

Real dynamical system

Given a real dynamical system (R, M, Φ), I(x) is an open interval in the real numbers, that is . For any x in M

Orbit

In physics, an orbit is the gravitationally curved path of an object about a point in space, for example the orbit of a planet about a star. Orbits of planets are typically elliptical, and the central mass being orbited is at a focal point of the ellipse.

Current understanding of the mechanics of orbital motion is based on Albert Einstein's general theory of relativity, which accounts for gravity as due to curvature of space-time, with orbits following geodesics. For ease of calculation, relativity is commonly approximated by the force-based theory of universal gravitation based on Kepler's laws of planetary motion.

History

Historically, the apparent motions of the planets were described geometrically (without regard to gravity) in terms of epicycles, which are the sums of numerous circular motions. Theories of this kind predicted paths of the planets moderately well, until Johannes Kepler was able to show that the motions of planets were elliptical motions.

The celestial spheres model is a geocentric model of the solar system originally used to explain the apparent motion of the planets in the sky. It posited the existence of perfect moving spheres or rings to which the stars and planets were attached. After the planets' motions were more accurately measured, theoretical mechanisms such as deferent and epicycles were added. Although it was capable of accurately predicting the planets' positions in the sky, more and more epicycles were required as the measurements became more accurate, hence the model became more and more unwieldy.

ArcelorMittal Orbit

The ArcelorMittal Orbit is a 114.5 metre (376 feet) tall sculpture and observation tower in the Queen Elizabeth Olympic Park in Stratford, London. It is Britain's largest piece of public art, and is intended to be a permanent lasting legacy of London's hosting of the 2012 Summer Olympic and Paralympic Games, assisting in the post-Olympics regeneration of the Stratford area. Sited between the Olympic Stadium and the Aquatics Centre, it allows visitors to view the whole Olympic Park from two observation platforms.

Orbit was designed by Turner-Prize winning artist Sir Anish Kapoor and Cecil Balmond of engineering Group Arup. Announced on 31 March 2010, it was expected to be completed by December 2011. The project came about after Mayor of London Boris Johnson and Olympics Minister Tessa Jowell decided in 2008 that the Olympic Park needed "something extra". Designers were asked for ideas for an "Olympic tower" at least 100 metres (330 ft) high, and Orbit was the unanimous choice from proposals considered by a nine-person advisory panel.

Orbit (horse)

Orbit (foaled 1885) was a Thoroughbred racehorse. He was trained at Kingsclere by John Porter for the 1st Duke of Westminster. As a three-year-old he won the Eclipse Stakes.

Breeding

Orbit was the son of Epsom Derby and Champion Stakes winner Bend Or. His dam was Fair Alice, a daughter of July Stakes winner Cambuscan.

Racing career

Orbit won three races as a two-year-old; the Criterion Nursery Handicap at Newmarket, the Kempton Park Champion Nursery Handicap and the Daveridge Stakes. Orbit started his three-year-old career by winning the Craven Stakes at Newmarket by ¾ length from Cotillon. His next race came in the 2000 Guineas at Newmarket. Friar's Balsam started as the 1/3 favourite for the race, with Ayrshire at 100/12 and Orbit at 100/8. Orbit ran on well in the closing stages to finish in third place. Ayrshire won the race by two lengths from Johnny Morgan, who was a head in front of Orbit. After winning the 2000 Guineas win Ayrshire started as the 5/6 favourite for the Epsom Derby and Orbit was second favourite at 11/2. Orbit could only finish in fifth place, over seven lengths behind winner Ayrshire. He then finished second in the Triennial Stakes at Ascot. Orbit started as the 9/4 favourite for the Eclipse Stakes and in the final 100 yards of the race Orbit gradually edged away from stablemate Ossory and beat him by a length.