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From Landau & Lifshitz, Classical Mechanics, the number of integrals of independent integrals of motion for a system of $s$ degrees of freedom is $2s-1$.

I am considering a spherical pendulum in absence of gravity. For this problem $s=2$, so I expect 3 independent integrals of motion. The first two are the angular momentum in the $\theta$ and $\phi$ direction. The energy is also a constant of motion, but in the absence of motion it is a function of the angular momentum since $$L=T=\frac{m\ell^2}{2}(\dot{\theta}^2+\dot{\phi}^2\sin^2(\theta)).$$

I have 3 integrals of motion but 2 are independent. Did I miss one, or have I misunderstood the statement from Landau? The linear momentum should not be conserved, so should I include integrals of motion of ignorable coordinates?

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Hints:

  1. Write the 3D angular momentum vector $\vec{L}$ in spherical coordinates.

  2. Note that $\vec{L}$ depends on the angular coordinates $\theta$ and $\varphi$ (and their time derivatives), but not the radial coordinate $r$ (and its time derivative).

  3. Claim that $\vec{L}$ are three independent integrals of motion for a particle confined on the two-sphere $S^2$.

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  • $\begingroup$ How can I claim that L is three integrals of motion when I only have 2 generalized coordinates? Shouldn't each integral of motion only correspond to a degree of freedom? Maybe my idea that the integrals of motion correspond to the degrees of freedom is incorrect. $\endgroup$
    – Anode
    Sep 9, 2014 at 18:58
  • $\begingroup$ There are two DOF on $S^2$. A maximally superintegrable system with two DOF has three integrals of motion, cf. e.g. my Phys.SE answer here. $\endgroup$
    – Qmechanic
    Sep 10, 2014 at 15:58

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