# Reference for a type of "multi-hamiltonian" system

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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:

$$\frac{\mathrm{d}}{\mathrm{d}t}f(x)=\det\big(\nabla f,\nabla H_1,... \nabla H_{d-1}\big)=\frac{\partial(f,H_1,...,H_{d-1})}{\partial\vec{x}}.$$ asked Nov 16, 2020
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.

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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 (45 points)
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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 (112 points)

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