5
$\begingroup$

If I have an autonomous series of differential equations $$\tag{1} \frac{dx_i}{dt} ~=~ A_i(x_1,...,x_n)$$ with the condition that $$\tag{2} \sum_{i=1}^n\frac{\partial A_i}{\partial x_i}~=~0$$ in all regions of phase space, can this be written as a Hamiltonian system in terms of some generalized position and momentum coordinates?

$\endgroup$
  • $\begingroup$ Comment to the question (v1): Note that while the ODE (1) is covariant under coordinate transformations, the divergence-free condition (2) is not, unless we introduce (and specify a choice of) a volume-form. $\endgroup$ – Qmechanic Apr 9 '15 at 22:44
5
$\begingroup$

Comments to the question (v1):

  1. Let there be given an $n$-dimensional manifold $M$ with a smooth vector field $X\in \Gamma(TM)$.

  2. If $(x^1, \ldots, x^n)$ is some local coordinates on $M$, then the vector field takes the form $$\tag{A} X~=~X^i(x)\frac{\partial}{\partial x^i},$$ and one may study the autonomous first-order ODE $$\tag{B} \frac{dx^i(t)}{dt}~=~ X^i(x(t)).$$ Note that the ODE (B) transforms covariantly under change of coordinates.

  3. If $X$ does not vanish in a point $p\in M$, then one may choose a local coordinate neighborhood $U\subseteq M$ of $p$, with local coordinates $(y^1, \ldots, y^n)$, so that $$\tag{C} X~=~\frac{\partial}{\partial y^1}.$$ 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 $$\tag{D} \frac{dy^i}{dt}~=~ \delta^i_1$$ in the local coordinate neighborhood $U\subseteq M$.

  5. If one chooses the Poisson bracket in the obvious way, i.e. $$\tag{E}\{y^i,y^2\}_{PB}~=~\delta^i_1,\qquad \text{etc},$$ then one may bring the ODE (4) on Hamiltonian form $$ \tag{F} \frac{dy^i}{dt}~=~ \{ y^i, y^2\}_{PB}$$ in the local coordinate neighborhood $U\subseteq M$.

  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.

$\endgroup$
  • $\begingroup$ What kind of conditions would be required for the existence of a global Hamiltonian formulation? I take it that analyticity of the A_i(x)'s would not be sufficient? $\endgroup$ – djbinder Apr 10 '15 at 2:17
  • $\begingroup$ Analyticity of the vector field is not enough, cf. e.g. this counterexample. $\endgroup$ – Qmechanic Apr 10 '15 at 9:49
  • $\begingroup$ Correction to the answer (v2): The word ODE (4) below eq. (E) should be ODE (D). $\endgroup$ – Qmechanic Aug 13 '15 at 10:19
  • $\begingroup$ For the related question whether there exists a conserved energy function (rather than a full Hamiltonian formulation), see this Phys.SE post. $\endgroup$ – Qmechanic Aug 13 '15 at 13:27

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

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