Sunday, 10 December 2017

classical mechanics - When can an autonomous system be written using a Hamiltonian?


If I have an autonomous series of differential equations dxidt = Ai(x1,...,xn)

with the condition that ni=1Aixi = 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):





  1. Let there be given an n-dimensional manifold M with a smooth vector field XΓ(TM).




  2. 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.




  3. If X does not vanish in a point pM, then one may choose a local coordinate neighborhood UM 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.





  4. The ODE (B) then becomes dyidt = δi1

    in the local coordinate neighborhood UM.




  5. 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 UM.




  6. If the dimension n is even, then the Poisson bracket (E) can be chosen to be non-degenerate.





  7. 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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