What is symplectic geometry? [closed]
Solution 1:
Quick "fake" answer: In classical mechanics one usually describes a particle measuring its position $q_1, \dots, q_n$ and momentum $p_1, \dots, p_n$. To describe how these change one needs to introduce a "Hamiltonian", i.e. a function measuring the energy of the system.
For a particle of mass $m$ moving in the ordinary space $\mathbb R^n$ it is: $$H(q, p) = \frac{p_1^2 + \dots + p_n^2}{2m} + V(q)$$ where $V\colon \mathbb R^n\to\mathbb R$ is the "potential energy" of the particle. Then one solves a system of ODEs: $$\begin{cases} \dot p_i = -\frac{\partial H}{\partial q_i} \\ \dot q_i = \frac{\partial H}{\partial p_i} \end{cases}$$
For example if you plug $n=1$ and $V(q) = kq^2/2$, you will get an ordinary harmonic oscillator $q(t)=A\cos(\omega t+\phi)$, $\omega^2=k/m$. (Similarly you get an expression for the momentum $p$).
Now let's generalize. One starts with a configuration space that is a manifold $M$, used to measure the position of the particle. Local coordinates are our $q_1, \dots, q_n$. Then one introduces the phase space $P=T^*M$ on which the local coordinates are $q_1, \dots, q_n, p_1, \dots, p_n$. The motion of the particle can be described by a path on $P$, which measures not only the position but the momentum as well. We do this by introducing a function $H\colon P\to \mathbb R$ and we try to find a vector field on $P$ such that: $$i_X\omega=-dH,$$
where $\omega = dp_1 \wedge dq_1 + \dots + dp_n\wedge dq_n$ in local coordinates. (It is not obvious that it is globally defined). This (not incidentally) looks similar to the expression $\omega(\textbf x, \textbf y)$ you have written down in the question.
The point is that the whole dynamics is in fact encoded in the symplectic 2-form $\omega$. (If you have a Hamiltonian describing a particle, just find a vector field and solve an ODE to get the path).
Generalizing even further let's think about a symplectic manifold $(P, \omega)$ where $\omega$ is a distinguished 2-form with 'nice' properties (it's assumed to be closed and nondegenerate). In particular this gives some topological restrictions on $P$ – for example $P$ needs to be even-dimensional and orientable, with $\omega\wedge \dots\wedge \omega$ acting as a volume form.
Obviously one can organize such manifolds into a category and ask the usual questions – can we characterize them up to an isomorphism? (Called 'symplectomorphism'; strongly related to 'canonical transformations' of physics). Can we introduce any invariants? (Apparently there are no local ones as every symplectic manifold locally looks like $\mathbb R^{2n}$ with the symplectic form from your question).
As we can do classical mechanics on such manifolds, can we 'quantize' them and do quantum mechanics?
We have a nice additional structure – how does it interfere with a Riemannian metric or complex structure (what leads to Kähler geometry and Calabi-Yau manifolds of string theory).
... and similar questions seem to be so ubiquitous that I'd risk to say: every modern differential geometer needs to learn symplectic geometry.
Full answer: This is too broad subject to describe it fully here. But definitely it's worth to study. I recommend:
- Cohn's post,
- Webster's post,
- Cannas da Silva's notes,
- Meinrenken's notes,
- Butterfield's On Symplectic Reduction in Classical Mechanics,
- Arnold's Mathematical Methods of Classical Mechanics,
- Abraham and Marsden's Foundations of Mechanics,
- McDuff and Salamon's Introduction to Symplectic Topology.