# 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
edited Nov 16

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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