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Results tagged with sg.symplectic-geometry
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user 394
Hamiltonian systems, symplectic flows, classical integrable systems
31
votes
Accepted
How to see the Phase Space of a Physical System as the Cotangent Bundle
Let's start by answering the first question.
Let $M$ be any manifold. Consider a physical system consisting of a point-particle moving on $M$. What are the configurations of this physical system? …
- 33.3k
27
votes
Accepted
The Jacobi Identity for the Poisson Bracket
The Jacobi identity for the Poisson bracket does indeed follow from the fact that $d\Omega =0$.
I claim that (twice) the Jacobi identity for functions $f,g,h$ is precisely
$$d\Omega(X_f,X_g,X_h) = 0. …
- 33.3k
17
votes
Accepted
Why can we define the moment map in this way (i.e. why is this form exact)?
Both answers are "No."
There are well-known obstructions to the existence of an equivariant momentum mapping arising from the action by symplectomorphisms of a group $G$ on a symplectic manifold. Th …
- 33.3k
7
votes
Accepted
Question about the dimension of a Contact (Symplectic) manifold
And by popular request, here's my comment as an answer :)
Your guess is correct. Contact structures are structures associated to a one-form $\alpha$ with maximal rank. There are two cases:
for odd …
- 33.3k
6
votes
Accepted
Trivial canonical bundle of a Ricci-flat, simplyconnected Kähler manifold
The answer to the first question is that the Ricci tensor defines a (1,1) form (called the Ricci form) and this is the curvature of the connection on $K_M$ induced by the Levi-Civita connection on $T …
- 33.3k
5
votes
Reference for Almost-Kahler geometry
The paper that started this all is the one by Gray and Hervella where they classified the different types of almost Hermitian structures. It's a classic and still very much well worth reading:
The s …
- 33.3k
4
votes
Accepted
coisotropic submanifolds
Locally, any codimension-$k$ submanifold can be described as the zero locus of $k$ smooth functions $f_1,\dots,f_k$. (This is true globally if and only if the submanifold has trivial normal bundle.) …
- 33.3k
3
votes
Is the 'massive' Calogero-Moser system still integrable?
Just to add to Gjergji Zaimi's answer, Harry Braden has sent me the expressions for the conserved charges responsible for the integrability of the $N=3$ model:
The total momentum $P = p_1 + p_2 + p_ …
- 33.3k
2
votes
Accepted
coisotropic submanifolds on poisson manifolds
The answer is 'Yes', at least with some additional transversality conditions.
Corollary (1.2.6) in Weinstein's Coisotropic calculus and Poisson groupoids states that
Let $M$ be a submanifold of t …
- 33.3k
1
vote
Question about the definition of hamiltonian group action.
Your belief is only partially correct. The existence of a momentum map requires two conditions:
First, as you point out, if $X^*$ is the fundamental vector field corresponding to $X \in \mathfrak{g …
- 33.3k