# 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.
• Thank you, Pantelis. I have not encountered "root system" and "Lie algebra" before... how do they relate to the constants of motion? Also, by translation, do you mean taking $\phi\to \phi+\phi_0$? – Angle Jul 9 '12 at 18:42
• @Angle Yes that's correct. If you replace $\phi_i$ with $\phi_i+t$ the potential is invariant. You can think of it as an action of the additive group of reals on the phase space. This symmetry gives a contant of motion by Noether's Theorem. The expressions $\phi_1- \phi_2$, $\phi_2-\phi_3$ are the simple roots of a Lie algebra of type $A_2$. It helps you get a Lax pair and then the additional constants of motion. But for this simple case see solution by Jon. – PAD Jul 9 '12 at 22:04
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.