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Questions tagged [poisson-geometry]

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

In differential geometry, a Poisson manifold is a smooth manifold equipped with a Poisson bracket (Also called Poisson structure). Poisson manifolds are a generalisation of the symplectic manifold, which in turn are a generalisation of the phase space from Hamiltonian mechanics.

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An analogue of the Poisson bracket in contact geometry?

McDuff and Salamon define an analogue of the Poisson bracket in contact geometry on page 135 in the third version of Introduction to Symplectic Topology. The definition is the following. Let $(M,\alpha)$ be a contact manifold with a globally…
Chris Kuo
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Relation between Lie bracket and Poisson bracket

For any vector field $X$ on a smooth manifold $Q$, define $f_X : T^* Q \to \mathbb{R}, \omega \mapsto \omega(X_x)$ for $\omega \in T_x^* Q$. We also have that $\{ \cdot ,\cdot\}$ is an arbitrary Poisson bracket and $[\cdot,\cdot]$ is the Lie…
SFSH
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Intuition about Poisson bracket

I've been reading about Hamiltonian mechanics which in its mathematical description uses Poisson manifolds From my limited understanding, on a Poisson manifold $M$ we can look at the Poisson bracket as a gadget that gives a smooth vector field…
ante.ceperic
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An Hamiltonian diffeomorphism is also a Poisson diffeomorphism

Let $(M,\{-,-\})$ be a Poisson manifold. An Hamiltonian isotopy is a smooth family of diffeomorphisms $\{\varphi^t:M\to M\}_{t\in [0,1]}$ such that $\varphi^0=\text{id}_M$ there exists a smooth family of functions $\{h_t:M\to \mathbb{R}\}_{t\in…
Kandinskij
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Finite dimensional Lie subalgebras of the polynomial Poisson algebra

Let $R_n=\mathbb{C}[q_1, \ldots, q_n, p_1, \ldots p_n]$ be a polynomial algebra over complex numbers with even number of variables. Then $R$ admits Poisson bracket defined on linear functions as follows $$ \{p_i, q_j\} =\delta_{ij},\\ \{q_i,…
Alex
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Hausdorff dimension of Hamiltonian orbit closure and symplectic leaves

Let $\dot{x} = \Pi \cdot \nabla H$ be a smooth Hamiltonian-Poisson system on $\mathbb{R}^n$. $H: \mathbb{R}^n \to \mathbb{R}$ is the Hamiltonian and $\Pi = (\Pi^{ij})$ is a skew-symmetric matrix of functions $\mathbb{R}^n \to \mathbb{R}$ satisfying…
Ricardo Buring
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Definition of Poisson Bracket

Context: Let $f,g :T^*M\rightarrow \mathbb{R}$, the Poisson Bracket was defined classically as $$\{f,g\}=\sum\limits_{i=1}^n\frac{\partial f}{\partial q^i}\frac{\partial g}{\partial p_i}-\frac{\partial f}{\partial p_i}\frac{\partial g}{\partial…
Powder
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Not understanding the foliation structure of a Poisson manifold

Consider a Poisson manifold $(M, P)$ (no assumption about the rank or regularity of $P$). For each possible rank $2r$ of $P$, let $(M_r, P_r)$ be the corresponding symplectic leaf of dimension $2r$, with symplectic form $\Omega_r$ and associated…
Alex M.
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What does it mean for a Symplectic Form to be invariant under Group Action?

This should be a very basic question for people familiar with differential manifolds. I'm more or less new to the field so let me apologize in advance for ill-defined questions if arising. I split the question into 3 more specific ones. Let…
user192049
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Distribution of distance from origin to any point of Poisson point process

Assume a homogeneous Poisson point process in a plane (2D) with density $\lambda$. Let $n$, the number of points, be random according to the homogeneous Poisson point process. Let $\{r_1, r_2, \ldots, r_n\}$ be the set of radial distances of the…
eHH
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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 $a \in {\mathfrak g}^*$, [,] is the commutator and $\langle, \rangle$ is the…
IntegrableSystemsEnthusiast
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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 problem is correct. I have added quite a bit of…
J.V.Gaiter
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Is there a known Hamiltonian for the Lorenz-63 system?

It has been shown that Chua's system has a Hamiltonian-Poisson realization (Arieşanu 2013). That is, there exists a Hamiltonian $H=f(x)$ over $x\in \mathbb{R}^3$ and an skew-symmetric matrix $\Pi\in \mathbb{R}^{3,3}$ such that $\dot{x} = \Pi \cdot…
BAYMAX
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Is there a theory of "quadratic" Hamiltonian evolutions on Poisson manifolds?

I am dealing with a PDE which can be written in the form $$\frac{d}{dt} f(t) = \{a, f(t)\} + \{\{b, f(t)\}, f(t)\}$$ A Hamiltonian equation on a Poisson manifold has the following form: $$\frac{d}{dt} f(t) = \{a, f(t)\}$$ My equation seems to be a…
Robert Wegner
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Poisson manifolds that do not come from quotient of symplectic manifold by Lie groups?

Given a lie group $G$ that acts over a symplectic manifold $(M,\omega)$ then its quotient $M/G$ (under mild hipothesis) is a Poisson manifold. This is an very useful way to construct Poisson manifolds from symplectic manifolds alreadly known. I am…
FUUNK1000
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