EDIT: Much thanks for answers. As was pointed out, the question as it stands is a little too broad. Nevertheless, I don't want to delete it, because I think that such introduction-style questions can be answered without writing a book, rather something more like an introduction to a book and fits here. Moreover, commenters have linked to great resources, and this question might help someone else. I made a follow up strictly narrower question instead.

First some background, so that you know where I came from. But the question in the title stands as it is, if you want to answer without appealing to what is below, please do.

I am currently learning about Lie groups. One of the first things that I've seen are the classical groups, and the classical group that I want to talk about today is the symplectic group $\mathrm{Sp}(n,\mathbb{F})$.

The definition of $\mathrm{Sp}(n,\mathbb{F})$ I am familiar with is as follows:

Let $\omega$ be an skew-symmetric bilinear form on $\mathbb{F}^{2n}$, which is unique up to change of basis. It is given by the formula $$\omega(\mathbf{x},\mathbf{y}) = \sum_{i=1}^n{x_iy_{i+n}-y_ix_{i+n}}$$

Why is this symplectic form important?

We can then write out the definition

$$\mathrm{Sp}(n,\mathbb{F}) = \left\{ A: \mathbb{F}^{2n} \to \mathbb{F}^{2n} \mid \omega(A\mathbf{x},A\mathbf{y}) = \omega(\mathbf{x},\mathbf{y}) \text{ for all } \mathbf{x,y} \in \mathbb{F}^{2n}\right\}$$

I can see the analogue of $O(n,\mathbb{F})$. We also have some bilinear form that needs to be preserved, namely the inner product $\langle \cdot,\cdot\rangle$. But more importantly, elements of $O(n,\mathbb{F})$ are really easy to visualize, because I intuitively know what a rigid transformation is. So the important question for me is

How to visualize symplectic transformations?

And I tried to research this question, and I stumbled upon the topic of symplectic linear spaces and symplectic manifolds. A symplectic vector space is defined analogous to Euclidean vector space, but the inner product is again substituted by symplectic form.

What is a symplectic vector space, intuitively?

I saw that the intuition behind these things should be that $\mathbb{R}^{2n}$ should be treated as a space of positions and velocities, a phase space. And I don't understand it. But I feel that physical intuition would be really helpful.

What is the connection of classical mechanics with symplectic geometry?

I don't know classical mechanics, sadly, so a quick mathematical rundown would be appreciated.

All the questions that I've asked above could be summarized to one question:

What is symplectic geometry?

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    $\begingroup$ While I think this is an excellent question in the abstract, it really falls under the 'too broad' category. From the help center, on what types of questions to avoid asking: "Your questions should be reasonably scoped. If you can imagine an entire book that answers your question, you’re asking too much." There are multiple books that try to answer this question, so... $\endgroup$ – Steven Stadnicki Apr 7 at 16:23
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    $\begingroup$ A very nice book to study is Arnol'd "Mathematical methods of classical mechanics" where he goes from the formalisms of Newton's mechanic to Lagrange formulations to Hamilton mechanics. The last one is an (somewhat) synonym for "symplectic geometry", most of the research topics of the latter are motivated by the former. $\endgroup$ – Lutz Lehmann Apr 7 at 16:45
  • $\begingroup$ I'd suggest Ana Cannas da Silva's notes: people.math.ethz.ch/~acannas/Papers/lsg.pdf $\endgroup$ – Paweł Czyż Apr 7 at 17:03
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    $\begingroup$ The best treatment I've seen on this is McDuff/Salamon. $\endgroup$ – Mnifldz Apr 7 at 17:15
  • $\begingroup$ @StevenStadnicki I don't imagine an entire book that answers my question, rather an entire introduction to a book. I think these introduction-style questions should have introduction-style answers, therefore drastically bounding their scope. Take the algebraic geometry one for example (with, to be honest, I was somewhat inspired by). $\endgroup$ – mzg147 Apr 7 at 22:33

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:

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