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

The Hamiltonian is

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

Qmechanic
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kaiser
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    Angular momentum – qfzklm Sep 01 '15 at 14:51
  • But what is it in higher dimensions? – kaiser Sep 01 '15 at 14:52
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    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 http://physics.stackexchange.com/questions/9864/how-to-define-angular-momentum-in-other-than-three-dimensions – qfzklm Sep 01 '15 at 14:58
  • 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 $n$! – kaiser Sep 01 '15 at 15:28
  • Sorry, I just noticed that they are not commutative. – kaiser Sep 01 '15 at 16:09

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