[NOTES/CM-10012] Continuous Symmetry Transformation

Noether's theorem gives a constant of motion \(G\) for every continuous symmetry transformation. One can express this constant of motion in terms phase space variables.

In the phase space formulation, the constant of motion \(G\) given by Noether's theorem, expressed in terms of coordinates and momenta generates the infinitesimal symmetry transformation.

Since \(G\) is a constant of motion, it commutes with the Hamiltonian, the change in Hamiltonian \(\{H,G\}_\text{PB}\) is zero and hence the Hamiltonian is invariant under the transformations.

For a system of particles having Lagrangian invariant under translations, the constant of motion turns out to be total momentum \(\sum_\alpha p_\alpha\) of the system.

If the Lagrangian of a system of particles is invariant under rotations, the constant of motion turn out to be the components of total angular momentum \(\vec{L}=\sum_\alpha \vec{r}_\alpha\times \vec{p}_\alpha\). We will explicitly demonstrate some of the above statements.