What are the integrals of motion of a system with the following Lagrangian?
$L=a\dot{\phi_1}^2+b\dot{\phi_2}^2+c\cos(\phi_1-\phi_2)$?
where $a,b,c$ are constants, $\phi_1,\phi_2$ are angles and $\dot{\phi_i}$ represents differentiation wrt time.
I believe the Hamiltonian is conserved, but are there any more?
Perhaps there is an isotropy of space here, since $\phi_1,\phi_2$ only exist as a difference $\phi_1-\phi_2$? So angular momentum?
Are the above 2 right? Are there any more?
Thanks.
ADDED: "integrals of motion" are sometimes referred to elsewhere as "constants of motions" or "conserved quantities".