# Reducing degrees of freedom of a Hamiltonian by restricting first integrals

Suppose I have a Hamiltonian $H$ with a first integral $C$. Then by fixing $C=0$ is it always possible to reduce the degrees of freedom by 1?

This is typically done in the reduction of the motion in the central force field. However I cannot find any literature on extension to the general case.

Thanks in advance.

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## 1 Answer

The first question has a positive answer: if there is a first integral, the space where the movement occurs can be reduced by 1. The most prominent example is when there is energy conservation, on which the system is restricted to a 2N-1 space.

The general case (2N dimensions) is treated in the same manner (see e.g. goldstein's Classical Mechanics). However, it is not true that if you have M first integrals, you can reduce the system to a 2N-M space. The Poisson brackets between then must be zero in order to you be able to reduce the dimension. In this case, the constant of motions are said involution.

In case you find N constant of motions involution, the system is said integrable, and the movement is restricted to a N-dimensional torus.

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The poisson bracket needs to be non-zero right? Suppose that the poisson bracket is zero of the first integrals $C_1,C_2$. Then we can find co-ordinates $x_1=C_1$ and $x_2=C_2$ by the Caratheodory-Jacobi-Lie theorem. –  Novo Jul 4 '12 at 9:09
–  Novo Jul 4 '12 at 9:14