Hamiltonian Flow: Meaning & Definition

In summary: I'm not sure I could define either, but we're probably both visualizing correctly, so, all the same, good enough for me.In summary, Roger Penrose discusses the concept of a vector field on a phase space called the Hamiltonian flow, which he expresses in two different notations. This vector field is a section of the tangent bundle of the cotangent bundle of the configuration space. While the terms "Hamiltonian flow" and "Hamiltonian vector field" are often used interchangeably, they are not precisely synonymous. The flow of a vector field is a map that describes the movement of a particle in the velocity field, and the Hamiltonian flow is the flow associated with a Hamiltonian function on a symplectic manifold
  • #1
Rasalhague
1,387
2
In 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.
 
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  • #2
Rasalhague said:
In 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]?

Yes, precisely!

Rasalhague said:
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.

They are basically synonymous, but not precisely synonymous. Given a vector field X on a manifold, its flow is a map f: M x R-->M, such that f(p,t) is the point in M where a particle starting at p would be after voyaging for a time t in the velocity field X. Reciprocally, given a map f: M x R-->M with the properties of a flow, differentiating it, you can find a unique vector field X having f has its flow. (Called the "infinitesimal generator" of f in physics) These are not difficult ideas but I must admit that this is not transparent from browsing wikiepdia. As usual, Lee's Introduction to Smooth Manifold has the clearest exposition of these notions. (And it even has a few pages about Hamiltonian dynamics I do believe.)

Rasalhague said:
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]

Given a hamiltonian function H on a symplectic manifold (M,w), the hamiltonian vector field associated to H, also called the symplectic gradient of H, is the vector field X obtained by "dualizing dH" via w has you wrote. The hamiltonian flow is then simply the flow of X. But apparently, Penrose chose to depart from usual terminology a little and call X itself the hamiltonian flow of H.

Observe that the ordinary gradient of a function H is the vector field obtained by "dualizing dH" as above, not via w, but via the scalar product, or more generally a riemannian metric.
 
  • #3
Thanks, Quasar. That helps a lot. The concept of flow sounds very much like stream lines, field lines, geodesics...
 

Related to Hamiltonian Flow: Meaning & Definition

What is Hamiltonian Flow?

Hamiltonian Flow is a mathematical concept that describes the motion of a system over time. It is named after the physicist and mathematician William Rowan Hamilton and is often used in the study of dynamical systems.

What is the definition of Hamiltonian Flow?

The definition of Hamiltonian Flow is a set of equations that describe the evolution of a dynamical system over time. These equations are derived from Hamilton's equations of motion and describe the change in position and momentum of a system as it moves through time.

What is the difference between Hamiltonian Flow and Lagrangian Flow?

Hamiltonian Flow and Lagrangian Flow are two different approaches to describing the motion of a system. While Hamiltonian Flow uses Hamilton's equations of motion to describe the evolution of a system, Lagrangian Flow uses Lagrange's equations. The main difference between the two is that Hamiltonian Flow is based on the concept of energy, while Lagrangian Flow is based on the concept of generalized coordinates.

How is Hamiltonian Flow used in science?

Hamiltonian Flow is used in a variety of scientific fields, including physics, mathematics, and engineering. It is particularly useful in the study of dynamical systems, which are systems that change over time. Hamiltonian Flow can be used to analyze the behavior and stability of these systems, as well as to make predictions about their future states.

What are some real-world applications of Hamiltonian Flow?

Hamiltonian Flow has many practical applications in the real world. It is used in fields such as celestial mechanics, fluid dynamics, and quantum mechanics to model the behavior of complex systems. It is also used in the design and control of mechanical and electrical systems, as well as in the development of new technologies such as GPS and satellite navigation systems.

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