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: