Reference for a type of "multi-hamiltonian" system

+ 1 like - 0 dislike
279 views

Let $H_1,H_2\in\mathcal{C}^1(\mathbb{R}^3;\mathbb{R})$ be two scalar fields. Consider a trajectory $\vec{x}(t)\in\mathbb{R}^3$ such that, for all observable $f\in\mathcal{C}^1(\mathbb{R}^3;\mathbb{R})$,

$$\frac{\mathrm{d}}{\mathrm{d}t}f(x)=\det\big(\nabla f,\nabla H_1, \nabla H_2\big)=\frac{\partial(f,H_1,H_2)}{\partial\vec{x}}.$$

This dynamical system recalls a Hamiltonian system with hamiltonian $H$ on the phase space $\lbrace(x,p)\in\mathbb{R}^2\rbrace$ such that for all observable $f\in\mathcal{C}^1(\mathbb{R}^2;\mathbb{R})$:

$$\frac{\mathrm{d}}{\mathrm{d}t}f(x)=\det\big(\nabla f,\nabla H\big)=\frac{\partial(f,H)}{\partial(x,p)}=\frac{\partial f}{\partial x}\frac{\partial H}{\partial p}-\frac{\partial f}{\partial p}\frac{\partial H}{\partial x}=\big\lbrace f,H\big\rbrace,$$

the Poisson bracket. Hence I would like to say that my dynamical system is a kind of "multi-hamiltonian" system. Is there any reference in which this kind of generalisation is studied?

Edit: it can be generalised to a system with $d-1$ scalar fields $(H_i)$ on $\mathbb{R}^d$ satisfying:

$$\frac{\mathrm{d}}{\mathrm{d}t}f(x)=\det\big(\nabla f,\nabla H_1,... \nabla H_{d-1}\big)=\frac{\partial(f,H_1,...,H_{d-1})}{\partial\vec{x}}.$$

edited Nov 16, 2020

For $d-1>1$ this would need more momenta variables for the determinant to be meaningful. How can these variables be defined? I have not seen a theory for which this is necessary.

+ 1 like - 0 dislike

Intriguingly enough, a quick search for "multi Hamiltonian physics" does not give any meaningful result; the mechanics described by the OP is Nambu mechanics, and is linked to nonassociative algebras appearing in e.g. M-theory.

In contrast to the construction by the OP, Nambu generalizes the Poisson bracket (rather than using a higher-dimensional matrix determinant) and writes the equations of motion
$$\frac{df}{dt}=\{f,H_1,H_2,\ldots,H_n\}$$

See
https://arxiv.org/pdf/hep-th/0212267
https://arxiv.org/abs/1903.05673

answered Dec 17, 2020 by (85 points)
+ 0 like - 0 dislike

I guess in a complicated (for example, multiparticle compound) system, when you manage to separate some variables (often called collective (or normal or "elementary") modes), their equations are "independent" and governed with different "Hamiltonians" (or differential equations), although belonging to and describing this compound system.

answered Dec 21, 2020 by (92 points)

 Please use answers only to (at least partly) answer questions. To comment, discuss, or ask for clarification, leave a comment instead. To mask links under text, please type your text, highlight it, and click the "link" button. You can then enter your link URL. Please consult the FAQ for as to how to format your post. This is the answer box; if you want to write a comment instead, please use the 'add comment' button. Live preview (may slow down editor)   Preview Your name to display (optional): Email me at this address if my answer is selected or commented on: Privacy: Your email address will only be used for sending these notifications. Anti-spam verification: If you are a human please identify the position of the character covered by the symbol $\varnothing$ in the following word:p$\hbar$ysicsOverfl$\varnothing$wThen drag the red bullet below over the corresponding character of our banner. When you drop it there, the bullet changes to green (on slow internet connections after a few seconds). To avoid this verification in future, please log in or register.