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What are the integrals of motion of a system with the following Lagrangian?


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?


ADDED: "integrals of motion" are sometimes referred to elsewhere as "constants of motions" or "conserved quantities".

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up vote 1 down vote accepted

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.

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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.

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