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Rasalhague

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*The Road to Reality*, § 20.2, Roger Penrose talks about a "vector field on the phase space [itex]T^*(C)[/itex]", where [itex]C[/itex] is a configuration space. He calls this vector field "the Hamiltonian flow", draws it as little arrows in Fig. 20.5 (that's his typical way of drawing tangent vectors, in contrast to the little squares or tablet shapes he uses to depict cotangent vectors), and expresses it in two different notations:

[tex]\left \{ H, \enspace \right \} = \frac{\partial \mathcal{H}}{\partial p_r} \frac{\partial }{\partial q^r} - \frac{\partial \mathcal{H}}{\partial q^r}\frac{\partial }{\partial p_r}[/tex]

Does this mean that the Hamiltonian flow is a section of the tangent bundle of the cotangent bundle of the configuration space, i.e. the tangent bundle of the phase space, i.e. [itex]T(T^*(C))[/itex]?

On Wikipedia "Hamiltonian flow" redirects to "Hamiltonian vector field", as if they might be synonymous, but the article mentions in passing "the flow of a Hamiltonian vector field" (without defining it), as if the author of that part considered them not synonymous.

The article Hamiltonian systems at Scholarpedia has the equation:

[tex]i_X \omega \equiv \omega(X, \; \cdot \;) = dH[/tex]

which I suppose could be turned around to give

[tex]\omega^{-1}(dH, \; \cdot \; ) = X.[/tex]

The word "flow" crops up a few times in the latter article, but isn't explained at an introductory level. If not identical to the Hamiltonian vector field, but closely enough related for Penrose to treat them as the same thing, I wonder if the Hamiltonian flow is [itex]dH[/itex], to those authors who make a distinction, a section of the

*cotangent bundle*of the cotangent bundle of the configuration space.

The Wikipedia definition of flow in general, at first sight, seems like a slightly different way of formalizing the idea that Wald, Isham and this page call a curve, which I've more often seen called a parameterization (of a curve, the curve being thought of as what Wald, Isham etc. would call the image of a curve). But I could well be mistaken.