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group theory


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The study of the symmetries that define the properties of a system. Invariance under symmetry operations enables much about a system to be deduced without knowing explicitly the solutions to the equations of motion. Newton's law of gravitation, for instance, exhibits spherical symmetry. The force of gravity due to the attraction of a planet to a star is the same for all positions that are equidistant from the centre of mass of the star. However, the possible trajectories of the planet include non-symmetric elliptical orbits. These elliptical orbits are solutions to the Newtonian equations. However, one discovers on solving them that the planet does not move at a constant speed around the ellipse: it speeds up when it approaches perihelion and slows down approaching aphelion, which is consistent with what one might expect from a spherically symmetric force law. This behaviour, first formulated as one of Kepler's laws of planetary motion, is now accepted as a result of the conservation of angular momentum.

This association of a dynamical symmetry with a conservation law was first suggested by A. E. Noether in 1918 (see Noether's theorem). For example, the laws of physics are invariant under translations in time; they are the same today as they were yesterday. Noether's theorem relates this invariance to the conservation of energy. If a system is invariant under translations in space, linear momentum is conserved.

The symmetry operations on any physical system must possess the following properties:1.Closure. If Ri and Rj are in the set of all symmetry operations, then the combination, RiRj – meaning: first perform Ri, then perform Rj – is also a member of the set; that is, there exists an Rk, which is a member of the set such that Rk = RiRj.2.Identity. There is an element I, which is also a member of the set of symmetry operations, such that I Ri = Ri I = Ri, for all elements Ri in the set.3.Inverse. For every element Ri there is an inverse, Ri−1, such that RiRi−1 = Ri−1Ri = I.4.Associativity. Ri (RjRk) = (RiRj) Rk.These are the defining properties of a group in group theory. Group elements need not commute; i.e., Ri RjRj Ri, in general; if all the elements do commute then the group is said to be Abelian. Though translations in space and time are Abelian, it is easily verified that rotations about axes in 3D space are not. Groups can be finite (as the group of rotations of an equilateral triangle) or infinite (for example, the set of all integers, with addition used to combine the members). Groups can also be classed as continuous or discrete. An example of a continuous group is the group of all continuous translations of a point on a spherical surface. The symmetries of the star-planet system are the elements of this spherical group. Discrete groups have elements that may be labelled by an index that takes on only integer values. All finite groups and some infinite groups, such as the group of integers described above, are discrete.

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Subjects: Physics.


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