Past exam of the mathematics course of the University of Cambridge / 2024 / ii / Paper 2 / 8E (Classical Dynamics)

The generalized momentum conjugate to $q$ is

A coordinate $q$ is an ignorable coordinate when $L$ has no explicit dependence on it, so $\partial L/\partial q=0$. Its Euler--Lagrange equation iswhich proves that $p_q$ is conserved.

Let $q=(q_1,q_2,q_3)$ and $n=(1,1,1)$. Sincethe Lagrangian depends on $q$ only through $|q{|}^2$ and $(q\cdot n{)}^2$, and on the velocity through $|\dot{q}{|}^2$. Rotations about the $n$-axis preserve all three quantities, proving the invariance.

The axially symmetric isotropic-mass Lagrangian therefore conserves the axial angular momentumup to an irrelevant normalization of $n$. Since $L$ has no explicit time dependence, conservation of energy from time-translation invariance gives the second conserved quantity