Mathematics Stack Exchange is a question and answer site for people studying math at any level and professionals in related fields. Join them; it only takes a minute:

Sign up
Here's how it works:
  1. Anybody can ask a question
  2. Anybody can answer
  3. The best answers are voted up and rise to the top

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.

share|cite|improve this question

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.

share|cite|improve this answer
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
Also see… – Novo Jul 4 '12 at 9:14

Your Answer


By posting your answer, you agree to the privacy policy and terms of service.

Not the answer you're looking for? Browse other questions tagged or ask your own question.