# Questions tagged [poisson-geometry]

For questions about Poisson manifolds, Poisson brackets and their geometric properties.

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### Motivation for Lie bracket on $\Omega^1(P)$ for a Poisson manifold $P$

A Poisson structure on a manifold $P$ is usually taken as a binary operation $$\pi=\{-,-\}_P:C^\infty(P)\times C^\infty(P)\rightarrow C^\infty(P)$$ satisfying certain conditions. In many books/notes, ...
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### Is there a natural way to build a volume form on an oriented Poisson manifold?

Editet question Let $(M, \pi)$ be an oriented Poisson manifold. Is there a natural way to build a volume form from the Poisson bivector $\pi$? Original question It is always possible to build a volume ...
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### Poisson bracket and co-adjoint orbits for $sl(2)$

So I am trying to do this problem from Peter Olver's book Application of Lie groups to differential equations and I am wondering if somebody could check my work because I am not really sure about it ...
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### Sphere is a symplectic submanifold of Lie algebra $S0(3)^*$

I'm working through this example in Peter Olver's textbook Application of Lie Groups to Differential equations and I am having some trouble and was wondering if somebody could point out where I went ...
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### Rotations in $R^2$ are Poisson maps

So I am trying to learn about Hamiltonian systems from Peter Olver's book, applications of Lie groups to differential equations. Right now I am on the section talking about Poisson maps and there is ...
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### Proof of the Marsden-Ratiu Poisson reduction theorem.

I am trying to understand the proof of the Marsden-Ratiu theorem on Poisson reduction by distributions (J. E. Marsden and T. S. Ratiu. “Reduction of Poisson manifolds”. In: Letters in Mathematical ...
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### How to check that the Poisson bracket of traces of matrix powers is zero?

The canonical Lie-Poisson bracket on functions on the space $\mathfrak g$ of n×n square matrices is given by: $$\{f_1,f_2\}(a)=\langle a, [df_1(a),df_2(a)] \rangle,$$ where $a \in {\mathfrak g}^*$, [,]...
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### Showing that the moment map $\langle \mathbf{J}(z),\xi\rangle=\mathbf{i}_{\xi_P}(\Theta)(z)$ is equivariant

I am studying through Introduction to mechanics and symmetry by Marsden and Ratiu, specifically the chapter on Momentum maps, and wanted some confirmation as to whether my argument for the following ...
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### Proof functions in involution for R matrix Lie-Poisson bracket

I'm reading the well-known article of Reyman and Semenov-Tian-Shansky called "Group-Theoretical Methods in the Theory of Finite-Dimensional Integrable Systems". Suppose $\mathfrak{g}$ is a ...
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### Poisson structure identity

I'm attempting a question under Hamiltonian Dynamics. We are given that $\omega^{ab}$ is an antisymmetric matrix such that it's components depend on coordinates $x^a$ and such that the Poisson bracket ...
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### Generators in the sense of the theory of canonical transformations

I was studying this paper 1 and I’m stuck at eq.13. I have really no idea why they wrote that equation. I’m especially confused about $K=pq$ and $X=q^2$. Maybe I have to study something about that, ...
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### Poisson Bracket on non-smooth manifold

It is known that any Poisson bracket on a smooth variety $X$ corresponds to a bivector field. What is the simplest example of a non-smooth variety with a Poisson bracket that does not correspond to a ...
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### Describing the Poisson Structure on $\mathbb{R}^3$ to show $\mathbb{R}^3$ is a Poisson manifold.

We say that $M$ is a Poisson manifold if $M$ is a smooth manifold with a Lie bracket $\{\cdot,\cdot\}$ on $C^{\infty}(M)$ where $\{f,g\}=p(df\wedge dg)$, $p\in\Gamma(\Lambda^2TM)$. I try to understand ...
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### Symplectic Leaves of $\mathfrak{su}(3)^*$
I am trying to determine the symplectic leaves of the singular distribution of $\mathfrak{su}(3)^*$ induced by the linear poisson bracket. Pick an element $x=(x^1,x^2,...,x^8)\in \mathfrak{su}(3)^*$. ...