Reference for a type of "multi-hamiltonian" system

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:

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

• How does your proposal behave under changes of coordinates? I ask to see if we can figure out if there is any further geometric meaning to it. Nov 16, 2020 at 1:12
• A set of new coordinates $\vec{y}\in\mathbb{R}^3$ satisfying $\partial\vec{y}/\partial \vec{x}=1$ preserves the equation of evolution (due to the fact that $\partial(f,H_1,H_2)/\partial\vec{x}=\partial(f,H_1,H_2)/\partial{\vec{y}}\times\partial\vec{y}/\partial\vec{x}$). This is analogous to a conanical change of coordinates for a usual Hamiltonian system. Nov 16, 2020 at 1:32
• In particular, the orders of the $x$ dimensions seems tied to the orders of the $H$'s, so it might be better to think of it as something like $(x, p_{1}, p_{2})$ rather than $\vec x$ Nov 16, 2020 at 2:07
• My first question would be, " can you reduce this down to a normal Hamiltonian system by eliminating the third variable"? Nov 16, 2020 at 2:09
• If we assume that $\partial(x_1,x_2,H_2)/\partial\vec{x}=1$, then $\partial(f,H_1,H_2)/\partial\vec{x}=\partial(f,H_1,H_2)/\partial(x_1,x_2,H_2)=\partial(f,H_1)/\partial(x_1,x_2)$ (after development toward the third column), and the system is then hamilonian in this specific case. Nov 16, 2020 at 3:02

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 $$\frac{df}{dt}=\{f,H_1,H_2,…,H_n\}$$