In classical and quantum mechanics, invariance of space under translation results in momentum being an invariant and the
conservation of momentum, whereas invariance of the origin of time, i.e. translation in time, results in energy being an invariant and the
conservation of energy. In general, by
Noether's theorem, any invariance of a physical system under a
continuous symmetry leads to a fundamental
conservation law. In
crystals, the
electron density is periodic and invariant with respect to discrete translations by unit cell vectors. In very few materials, this symmetry can be broken due to enhanced
electron correlations. Another examples of physical invariants are the
speed of light, and
charge and
mass of a particle observed from two
reference frames moving with respect to one another (invariance under a spacetime
Lorentz transformation), and invariance of
time and
acceleration under a
Galilean transformation between two such frames moving at low velocities. Quantities can be invariant under some common transformations but not under others. For example, the velocity of a particle is invariant when switching coordinate representations from rectangular to curvilinear coordinates, but is not invariant when transforming between frames of reference that are moving with respect to each other. Other quantities, like the speed of light, are always invariant. Physical laws are said to be invariant under transformations when their predictions remain unchanged. This generally means that the form of the law (e.g. the type of differential equations used to describe the law) is unchanged in transformations so that no additional or different solutions are obtained.
Covariance and contravariance generalize the mathematical properties of
invariance in
tensor mathematics, and are frequently used in
electromagnetism,
special relativity, and
general relativity. ==Informal usage==