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Does there exists a symplectic formulation of statistical physics? I know that thermodynamics can be written in a symplectic language and of course classical mechanics is intrinsically formulated sym …

The classical phase space of the spin degrees of freedom is represented by the two-sphere with a symplectic form given by $s$ times the standard volume form. This is clearly no cotangent bundle to som …

This is always the case (using "naturality" as Paul Bryan suggested in the comments). Let $f: M \to N$ be a smooth map between Riemannian manifolds $(M,g)$ and $(N, h)$. Let $g^\flat: TM \to T^*M$ de …

Here is an interpretation using symmetry reduction, but without explicitly using the Lenz-Runge vector (it's essentially an extended version of the example given in Cushman & Bates "Global aspects of …

Smooth vector fields are in a one-to-one relationship with flows $\Phi: D \subseteq M\times \mathbb{R} \rightarrow M$, $$X_m = {\frac{d}{d t}}_{t=0} \Phi(m, t),$$ and by the symplectic form also with …

Let $M$ be compact and connected. For every closed $1$-form $\alpha$ on $M$ consider the Roger $2$-cocyle $\Psi_\alpha$ on the Lie algebra $\operatorname{ham}(M, \omega)$ defined by $$ \Psi_\alpha(X_f …

The following textbooks contain a proof of the symplectic slice theorem: Juan-Pablo Ortega and Tudor S. Ratiu: Momentum Maps and Hamiltonian Reduction Gerd Rudolph and Matthias Schmidt: Differential …

Let $G$ be a compact Lie group with complexification $G^c$, and consider a principal $G^c$-bundle $P^c \to M$ together with a reduction $P \subseteq P^c$ to $G$. Assume that $M$ is a Riemann surface. …

I'm looking for a reference for a special type of Lagrangian embedding in a prequantizable symplectic manifold. The setting is a symplectic manifold $(M, \omega)$, whose symplectic form is the curvat …

What are examples of non-equivariant momentum maps? Off the top of my hat, I know about the following examples: the action of translations of a symplectic vector space (yielding the Heisenberg group …

A gauge transformation can be viewed as a section of $P \times_G G$, where $G$ acts via conjugation. Hence the Lie algebra $\mathfrak{gau}$ of infinitesimal gauge transformation is the space of sectio …

Let $ (M, \omega) $ be a symplectic manifold. The de Rham class of $\omega$ induces a homomorphism $[\omega]: H_2(M) \to \mathbb{R}$, whose image $\Gamma_{\omega} \subseteq \mathbb{R}$ is called the g …

There is indeed a deep relation between Lagrangian submanifolds, the Fourier transformation and microlocal analysis. This is extensively discussed in Bates and Weinstein: Lectures on the Geometry of Q …

I'm not aware of a momentum map interpretation of the Einsteins's equation, but you can bring Einstein's equations in a Hamiltonian form with momentum map constraints (this is due to Fischer & Marsden …

In the general case, the reduced space $\mu^{-1}(c) / G_c$ is what is called a stratified symplectic space. This means, that for every orbit type $(H)$ the orbit type subset $\mu^{-1}(c)_{(H)} / G_c$ …