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Let's make the problem simple and consider the situation there's a system of $N$ particles moving in three-dimensional space is described by the Lagrangian:

$$ L = \frac 12 \sum ^N_{i=1}m_i\dot {\boldsymbol x}_i^2 - \frac 12 \sum _{i=1}^N \sum _{j=1}^N U_{ij}(\boldsymbol A \cdot (\boldsymbol x_i-\boldsymbol x_j)) $$

in which each pair $\left(i,j\right)$ of particles is allowed to have a different interaction potential $U_{ij}\left(a\right) = U_{ji}\left(a\right)$, and $A$ is a constant vector which is the same for all pairs. Essentially I'm saying that the particles only have pairwise interaction that is dependent on their separation along the direction of the A vector.

There's some obvious constants of motion, for example:

  • The component of the linear momentum orthogonal to the $\boldsymbol{A}$ vector of each particle, giving $2N$ constants of motion
  • Total linear momentum along the A vector
  • The total energy.

However, I am told that I am able to find $3N+2$ constants of motion from Noether's theorem (or otherwise). Here I've only found $2N+2$ constants of motion. What are the other constants of motion?

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Rotate to coordinates where $A$ is a multiple of the first unit vector. This separates variables, and you can easily find the constants of motion.

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Angular momentum about $\vec A$ is conserved for each particle, giving an extra number N constants of the motion. This comes from the rotational invariance of the Lagrangian about $\vec A$.

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