Are these integrals of motion? 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".

This is easy. The potential is translation invariant so you get the sum of the momenta as first integral. More interesting is to add another variable and a term like  d $\cos(\phi_2-\phi_3)$. It is related with  a root system of type $A_2$ and can be generalized to $A_n$ or any simple Lie algebra.
