# First integrals for a particle in a central-force field

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

The Hamiltonian is

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

• Angular momentum – qfzklm Sep 1 '15 at 14:51
• But what is it in higher dimensions? – kaiser Sep 1 '15 at 14:52
• 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/… – qfzklm Sep 1 '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 1 '15 at 15:28
• Sorry, I just noticed that they are not commutative. – kaiser Sep 1 '15 at 16:09