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Consider an arbitrary dimension $n>3$. What are the independent first integrals for a particle?

The Hamiltonian is

$$ H = \frac{p^2}{2m} +V (|r|) . $$

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    $\begingroup$ Angular momentum $\endgroup$ – qfzklm Sep 1 '15 at 14:51
  • $\begingroup$ But what is it in higher dimensions? $\endgroup$ – kaiser Sep 1 '15 at 14:52
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    $\begingroup$ The angular momentum can be obtained by $L_{ij} = r_i p_j - r_j p_i$. Remember that the angular momentum is not a real vector but a Pseudo-vector. You can read this page physics.stackexchange.com/questions/9864/… $\endgroup$ – qfzklm Sep 1 '15 at 14:58
  • $\begingroup$ Yes, it is straightforward to check that they are first integrals. But, there is some problem. There are $n(n-1)/2$ such quantities, while the degrees of freedom of the system is just $n$! $\endgroup$ – kaiser Sep 1 '15 at 15:28
  • $\begingroup$ Sorry, I just noticed that they are not commutative. $\endgroup$ – kaiser Sep 1 '15 at 16:09

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