If I have an autonomous series of differential equations dxidt = Ai(x1,...,xn) with the condition that n∑i=1∂Ai∂xi = 0 in all regions of phase space, can this be written as a Hamiltonian system in terms of some generalized position and momentum coordinates?
Answer
Comments to the question (v1):
Let there be given an n-dimensional manifold M with a smooth vector field X∈Γ(TM).
If (x1,…,xn) is some local coordinates on M, then the vector field takes the form X = Xi(x)∂∂xi, and one may study the autonomous first-order ODE dxi(t)dt = Xi(x(t)). Note that the ODE (B) transforms covariantly under change of coordinates.
If X does not vanish in a point p∈M, then one may choose a local coordinate neighborhood U⊆M of p, with local coordinates (y1,…,yn), so that X = ∂∂y1. This procedure is sometimes called stratification or straightening out of a vector field. It is a special case of Frobenius theorem.
The ODE (B) then becomes dyidt = δi1 in the local coordinate neighborhood U⊆M.
If one chooses the Poisson bracket in the obvious way, i.e. {yi,y2}PB = δi1,etc, then one may bring the ODE (4) on Hamiltonian form dyidt = {yi,y2}PB in the local coordinate neighborhood U⊆M.
If the dimension n is even, then the Poisson bracket (E) can be chosen to be non-degenerate.
The question of the existence of a global Hamiltonian formulation is much more subtle, even for n=2. See also e.g. this and this related Phys.SE posts.
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