Questions tagged [poisson-geometry]

Question 1

I'm a graduate student in Mathematics, currently learning Dirac structures in Differential Geometry. However, I cannot make sense of how the Dorfman bracket condition comes all of a sudden. It ...

Question 2

Let $G$ be a Lie group and $\mathfrak g$ it's Lie algebra. I'm trying to check that the Poisson structure on the quotient manifold $\mathfrak g^* \cong T^*G/G$ is given by $$\{f,g\}_{\mathfrak g^*}(\...

Question 3

I am currently wondering whether it is possible to express the variation of the action in the least action principle as a Poisson bracket. To be more precise, let $q$ be a system of coordinates, $p$ ...

Question 4

The following question is stated as exercise 10.2-2, p. 338 in the book "Introduction to Mechanics and Symmetry - 2nd Edition" by Marsden, Ratiu. Unless otherwise stated, all references ...

Question 5

In the book "Introduction to Mechanics and Symmetry - 2nd edition", by Marsden, Ratiu, in exercise 10.1-1 on p. 331 one is asked to define a Poisson bracket on the product $P_1 \times P_2$ ...

Question 6

In the book "Introduction to Mechanics and Symmetry - 2nd Edition" by Marsden, Ratiu the Lie-Poisson Bracket of an ideal fluid is defined on p. 20, resp. p. 329. On p. 335 the helicity ...

Question 7

This is a follow up question to (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 $n$, ...

Question 8

On the one hand, on a symplectic manifold $(M,\omega)$, one has the notion of Liouville (or complete) integrability, where one has a complete set of Poisson-commuting functions $\{h_i\}_{i\leq n}$: $$\...

Question 9

Let $M$ be a symplectic manifold, $\Psi=(\psi^1,\ldots,\psi^k):M\to \mathbb{R}^{k}$ a smooth map, and $c$ a regular value. Consider a submanifold $N=\Psi^{-1}(c)\hookrightarrow M$. Show that $N$ is ...

Question 10

(Silly question I'm afraid...) Consider symplectic manifolds $(M_i,\omega_i)$, with Poisson bracket $\{\cdot ,\cdot \}_{i}$, $i=1,2$, and let $\phi:M_1\to M_2$ be a smooth map. Prove that if $\phi$ ...

Question 11

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, ...

Question 12

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}=T_e G$. The Kirillov-Kostantstructure Poisson structure on $\mathfrak{g}^*$ is defined as $\{ f,g\} (p)=p([f_{*p},g_{*p}])$ where $f,g\in C^{\...

Question 13

Suppose a bivector field $\pi^{ij}$ such that $\pi^{ij}=-\pi^{ji}$, $\pi^{ij}\partial_{i}f\partial_{k}g=\{f, g\}$ defines a Poisson bracket $\{,\}$ on a smooth manifold (Einstein's summation is ...

Question 14

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} ...

Question 15

Let $\omega \in \Omega^2(M)$ be a non-degenerate 2-form. For a function $f \in C^\infty(M)$, define the vector field $X_f ∈ \mathscr X(M)$ via the relation: $$i_{X_f} \omega = df$$, where the first ...

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Questions tagged [poisson-geometry]

What could have been a good motivation / intuition behind the Dorfman Bracket condition?

Poisson structure on dual of lie algebra

Is it possible to express the least action principle as a Poisson bracket?

Casimir functional for Vlasov-Poisson bracket

Poisson bracket for product of poisson manifolds

Verification that Helicity is a Casimir function of the Lie-Poisson Bracket for an ideal fluid

Distribution of reference angle from x-axis to any point of Poisson point process

Casimir functions and Liouville integrability

Level-set submanifold is symplectic iff Poisson bracket matrix is nonsingular

Diffeomorphism is Poisson map iff is symplectomorphism

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

Some confusion in Kirillov-Kostant structure

Jacobi identity for Poisson bracket in local coordinates

Is there a theory of "quadratic" Hamiltonian evolutions on Poisson manifolds?

Prove $dω = 0 \iff$ {·, ·} is a Poisson bracket knowing that $\{f, g\} := X_f (g)$ and $i_{X_f} \omega = df$

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