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".

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.
