Are these integrals of motion?

Question

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".

Answer 1

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.

Answer 2

Just write down the motion equations and you will get $$ a\ddot\phi_1=-c\sin(\phi_1-\phi_2) $$ $$ a\ddot\phi_2=c\sin(\phi_1-\phi_2). $$ Now, sum these two equations and you will get $$ \dot\phi_1+\dot\phi_2=constant. $$ Indeed, it is not difficult to realize that a change of coordinates to $\Phi_1=\phi_1+\phi_2$ and $\Phi_2=\phi_1-\phi_2$ can make all things somewhat clearer.