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I met the following expression in some article "... where $\Phi_t$ is the time-1 flow map of the Hamiltonian vector field produced by the Hamiltonian function $H$ = ..." I haven't met any explicit definition of such thing yet. Would you please give a clear definition of " time-$t$ flow map of a Hamiltonian vector field " ? For example, what is the time-$t$ flow map of the vector field produced by $$H: \mathbb{R}^d \to \mathbb{R}^d: q:=(q_1, ...,q_d)\mapsto H(q)= q_d.$$ Thank you in advanced

A. PI
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1 Answers1

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I believe you are reading symplectic geometry. Let$(M,\omega)$ a symplectic manifold $H:M\rightarrow R$ an Hamiltonian (a differentiable function), write $i_X\omega=-dH$. You obtain a (Hamiltonian) vector field on $M$ whose flow is the Hamiltonian flow.

Example: $M=R^{2n}, \omega= \sum dx_i\wedge dx_{n+1}$.

Tsemo Aristide
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  • Not at all. I am working on some construction relevant to some ODE of the form $\dot{z} = J \nabla H(z)$ and at some step, I need to find an explicit definition of the time-$t$ flow map of the kinetic energy of $H$. – A. PI Jul 19 '16 at 23:51
  • If you look carefully, you'll see that $J\nabla H$ is the Hamiltonian field for the example above, I guess $J$ is the complex structure. – Tsemo Aristide Jul 19 '16 at 23:55
  • So what is the definition ? – A. PI Jul 20 '16 at 00:23
  • @A.MONET: There's some confusion here. In the example in your question, you're saying that $H$ is the vector field, and then you give a formula which is not a vector field but a function $H\colon \mathbb{R}^d \to \mathbb{R}$. But when one talks about Hamiltonian vector fields, $H$ is a function, and the associated vector field is $J \nabla H$, where $J$ is a so-called Poisson matrix. In order to say what this vector field is is in your example, we'd need to know what $J$ is in your case. Then when you have the vector field, you integrate it in the usual way to obtain the flow. – Hans Lundmark Jul 20 '16 at 06:56
  • That was I red in this Article page 5 after equation (41). The matrix $J$ is some matrix that satisfies $J^{-1}= -J$ – A. PI Jul 20 '16 at 10:00
  • And I want to say: what is the time-$t$ flow map of the vector field produced by the function $H$ ... – A. PI Jul 20 '16 at 10:06
  • As you know, a flow $\mathcal{F}$ on a topological space $X$ is a mapping $\mathcal{F}\colon X \times \mathbb{R} \mapsto X$. When you have a flow you can define a time-$T$ mapping as $F^{T} (x) = \mathcal{F}(x, T)$. Any vector field on a compact manifold (no matter Hamiltonian or not) defines the flow on this manifold. – Evgeny Jul 23 '16 at 09:18