Determining if constants of motion are independent

Say, in Hamiltonian mechanics, we know two constants of motion, $A$ and $B$. It could be proven that the quantity $[A,B]$ is also a constant of motion, where $[A,B]$ denotes the Poisson brackets of $A$ and $B$.

As an example, consider the Hamiltonian: $$H = \frac 1 2 p_R^2 + \frac{p_\phi^2}{2R^2} + \frac A R.$$

H itself obviously is a constant of motion. Further, we can show that the quantity $p_\phi$ and the quantity $C$, given by: $$C = p_Rp_\phi \sin \phi + \frac{p_\phi^2}{R} \cos \phi + A \cos \phi.$$

are both constants of motion. However, how do we know if $[p_{φ},C]$ is another independent constant of motion? Or more generally, how do we know, amongst the 4 constants of motions (H, $p_\phi$, $C$, $[p_{φ},C]$), how many of them are independent?

2 Answers

Picking up on the check mentioned in J.G.'s answer:

1. For a $2n$ dimensional phase space, there are at most $2n$ independent constants of motion.

2. Similarly, there are at most $2n-1$ independent integrals of motion, since they are by definition not allowed to depend explicitly on time $t$.

3. More generally, given $N$ integrals of motion $I_1, \ldots, I_N$ of the the phase space variables $z^1, \ldots, z^{2n}$, the number of independent integrals of motion in the point $z$ is given by the rank of the rectangular matrix $$\left(\frac{\partial I_k(z)}{\partial z^{\ell}}\right)_{1\leq k\leq N, 1\leq \ell\leq 2n} .$$ See also e.g. this Phys.SE post. OP's case has $n=2$.

4. OP's example: The angular variable $\phi$ is a cyclic variable so $p_{\phi}$ is an integral of motion. The Hamiltonian $$H~:=~\frac{p_r^2}{2}+\frac{p_{\phi}^2}{2r^2}+\frac{A}{r}~=~\frac{p_r^2}{2}+\frac{W^2}{2}-\frac{A}{2p_{\phi}^2}, \qquad W~:=~\frac{p_{\phi}}{r}+\frac{A}{p_{\phi}},$$ is an integral of motion different from $p_{\phi}$ (because of the $p_r$ & $W$ dependence). One may check that the 1-parameter family $$B(\alpha)~:=~p_r \sin(\phi+\alpha) + W \cos(\phi+\alpha)$$ is an integral of motion different from $p_{\phi}$ and $H$ (because of the $\phi$ dependence). However, any $B(\alpha)$ can be written in terms of just $B(0)$ and $B(\pi/2)$ via the addition formulas for sine & cosine. And $B(\pi/2)$ is not independent, since $$B(0)^2+B(\pi/2)^2~=~2H+\frac{A}{p_{\phi}^2} .$$ So the 1-parameter family $B(\alpha)$ contains only one new integral of motion. Altogether the system is maximally superintegrable with 3 independent integrals of motion. OP's integral of motion is $C:=p_{\phi}B(0)$.

• What about the specific case mentioned in the question, where I have generated 4 constants of motions (3 of them I know and the 4th one I've generated from Poisson's Brackets)? For sure there's maximum 3 independent constants of motions (2n-1, as you've mentioned), but how do I know exactly how many of them are independent? – user148792 May 27 '18 at 2:14
• I updated the answer. – Qmechanic May 27 '18 at 3:44
• Actually can I do it by inspection instead? Am I correct that the 4 constants of motions I've listed above actually contains 3 independent constants of motion? – user148792 May 27 '18 at 12:36

The standard technique is to count the symmetries and, if we seem to have found too many conserved charges, check for linear dependence (or in some cases, other relationships) between them. See Appendix D of my thesis for an example. And sometimes two conserved currents give the same charge, due to their difference integrating to zero; see Appendix E of the same for an example.

• What about the case, say, I am given 4 complicated expressions of constants of motion and neither can I check linearly dependence nor give a physical interpretation (e.g. check if they give same charge)? Are there a general (perhaps bullet proof way) of determining if those 4 constants of motions are independent? – user148792 May 27 '18 at 2:05
• PS: The 4 constants of motions I am referring to are those mentioned in the question. – user148792 May 27 '18 at 2:11
• @delickcrow123 Since $\phi$ is cyclic, consider the special case $p_\phi=0$. – J.G. May 27 '18 at 6:14
• Actually can I do it by inspection instead? Am I correct that the 4 constants of motions I've listed above actually contains 3 independent constants of motion? – user148792 May 27 '18 at 12:36