Analytical Mechanics - Symmetry, Conservation, and Noether's Theorem

Symmetry, Conservation, and Noether's Theorem

Symmetry transformations in classical space and time

Each transformation can be described by an operator (i.e. function acting on the position r or momentum p variables to change them). The following are the cases when the operator does not change r or p, i.e. symmetries.

Transformation	Operator	Position	Momentum
Translational symmetry
Time translations
Rotational invariance
Galilean transformations
Parity
T-symmetry

where R(n̂, θ) is the rotation matrix about an axis defined by the unit vector n̂ and angle θ.

Noether's theorem

Noether's theorem states that a continuous symmetry transformation of the action corresponds to a conservation law, i.e. the action (and hence the Lagrangian) doesn't change under a transformation parameterized by a parameter s:

the Lagrangian describes the same motion independent of s, which can be length, angle of rotation, or time. The corresponding momenta to q will be conserved.