Intuitive

Our everyday world is ruled by Euclidean geometry (and by its extension, Riemannian geometry); we can measure distances in it, and velocities. Far away from our daily experience, and much more subtle, is the mechanical phase space world, in which all the phenomena related to simultaneous consideration of position and variation of position; a deep understanding of this world requires the recourse to a somewhat counter-intuitive geometry, the symplectic geometry of Hamiltonian mechanics. Symplectic geometry is highly counter-intuitive; the notion of length does not make sense there, while the notion of area does. This "areal" nature of symplectic geometry, which was not realized until very recently, has led to unexpected mathematical developments, starting in the mid 1980's with Gromovís discovery of a "non-squeezing" phenomenon which is reminiscent of the quantum uncertainty principle—but in a totally classical setting! Symplectic Capacities and the Geometry of Uncertainty by Maurice de Gosson et. al.

• A Fight to Fix Geometry’s Foundations by Kevin Hartnett

Concrete

A simple way of putting it is that a two-form a way of measuring area in multivariable calculus. I believe the significance for physics boils down to the following: it turns out that a two-form is precisely what is required to translate an energy functional on phase space (a Hamiltonian) into a flow (a vector field). [See Wikipedia for how the translation goes, or read Arnold's book Mathematical Methods of Classical Mechanics, or a similar reference.] The flow describes time evolution of the system; the equations which define it are Hamilton's equations. One property these flows have is that they preserve the symplectic form; this is just a formal consequence of the recipe for going from Hamiltonian to flow using the form. So, having contemplated momentum, here we find ourselves able to describe how systems evolve using the phase space T*M, where not only is there an extremely natural extra structure (the canonical symplectic form), but also that structure happens to b preserved by the physical evolution of the system. That's pretty nice! Even better, this is a good way of expressing conservation laws. When physical evolution preserves something, that's a conservation law. So in some sense, "conservation of symplectic form" is the second most basic conservation law. (The most basic is conservation of energy, which is essentially the definition of the Hamiltonian flow.) You can use conservation of symplectic form to prove the existence of other conserved quantities when your system is invariant under symmetries (this is Noether's theorem, which can also be proved in other ways, I think, but they probably boil down to the same argument ultimately). http://qr.ae/TUTIn9

• Why symplectic geometry is the natural setting for classical mechanics by Henry Cohn
• Chapter "Phase Spaces of Mechanical Systems are Symplectic Manifolds" in the book Symmetry in Mechanics by Stephanie Frank Singer
• See also Chapter 1 in Principles Of Newtonian And Quantum Mechanics by Gosson

Abstract

"The symplectic geometry arises from the understanding of the fact that the transformations of the phase flows of the dynamical systems of classical mechanics and of variational calculus ~and hence also of optimal control theory! belong to a narrower class of diffeomorphisms of the phase space, than the incompressible ones. Namely, they preserve the so-called symplectic structure of the phase space—a closed nondegenerate differential two-form. This form can be integrated along two-dimensional surfaces in the phase space. The integral, which is called the Poincare´ integral invariant, is preserved by the phase flows of Hamilton dynamical systems. The diffeomorphisms, preserving the symplectic structure—they are called symplectomorphisms—form a group and have peculiar geometrical and topological properties. For instance, they preserve the natural volume element of the phase space ~the exterior power of the symplectic structure 2-form! and hence cannot have attractors"

Symplectic geometry and topology by V. I. Arnold

What's the relation to the symplectic groups?

• See http://math.ucr.edu/home/baez/symplectic.html

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Why is it interesting?

As each skylark must display its comb, so every branch of mathematics must finally display symplectization. In mathematics there exist operations on different levels: function acting on numbers, operators acting on functions, functors acting on operators, and so on. Symplectization belongs to the small set of highest level operations, acting not on details (functions, operators, functions=, but on all the mathematics at once.

Catastrophe Theory, by V. Arnold

The word symplectic was coined by Hermann Weyl in his famous treatise The Classical groups […]

Weyl devoted very little space to the symplectic group, it was then a rather baffling oddity which presumably existed for some purpose, though it was not clear what. Now we know: the purpose is dynamics.

In ordinary euclidean geometry the central concept is distance. To capture the notion of distance algebraically we use the inner (or scalar) product $ x.y$ of two vectors $x$ and $y$. […] All the basic concepts of euclidean geometry can be obtained from the inner product. […] The inner product is a bilinear form - the terms look like $x_i y_j$. Replacing it with other bilinear forms creates new kinds of geometry. Symplectic geometry corresponds to the form $x_1 y_2 -x_2 y_1$, which is the area of the parallelogram formed by the vectors $x$ and $y$. […] The symplectic form provides the plane with a new kind of geometry, in which very vector has length zero and is at right angles to itself. […] Can such bizarre geometries be of practical relevance? Indeed they can: they are the geometries of classical mechanics.

In Hamilton's formalism, mechanics systems are described by the position coordinates $q_1,\ldots,q_n$, momentum coordinates $p_1,\ldots,p_n$m and a function $H$ of these coordinates (nowadays called the hamiltonian) which can be thought of as the total energy. Newtons equations of motion take the elegant form $dq/dt=\partial H/\partial p,$ $dp/dt= -\partial H/\partial q$. When solving Hamilton's equations it is often useful to change coordinates. but if the position coordinates are transformed in some way, then the corresponding momenta must be transformed consistently. Pursuing this idea, it turns out that such transformations have to be the symplectic analogies of rigid euclidean motions. The natural coordinate changes in dynamics are symplectic. This is a consequence of the asymmetry in Hamitlon's equations, whereby $dq/dt$ is plus $\partial H/\partial p$, but $dp/dt$ is minus $\partial H/\partial q$, that minus sign again.

https://www.nature.com/nature/journal/v329/n6134/pdf/329017a0.pdf

I’ve tried to show you that the symplectic structure on the phase spaces of classical mechanics, and the lesser-known but utterly analogous one on the phase spaces of thermodynamics, is a natural outgrowth of utterly trivial reflections on the process of minimizing or maximizing a function S on a manifold Q.

The first derivative test tells us to look for points with

$$d S = 0$$

while the commutativity of partial derivatives says that

$$d^2 S = 0$$

everywhere—and this gives Hamilton’s equations and the Maxwell relations.

https://johncarlosbaez.wordpress.com/2012/01/23/classical-mechanics-versus-thermodynamics-part-2/

Hamilton's equations push us toward the viewpoint where $p$ and $q$ have equal status as coordinates on the phase space $X$. Soon, we'll drop the requirement that $X\subseteq T^\ast Q$ where $Q$ is a configuration space. $X$ will just be a manifold equipped with enough structure to write down Hamilton's equations starting from any $H \colon X\rightarrow\mathbb{R}$.

The coordinate-free description of this structure is the major 20th century contribution to mechanics: a symplectic structure.

This is important. You might have some particles moving on a manifold like $S^3$, which is not symplectic. So the Hamiltonian mechanics point of view says that the abstract manifold that you are really interested in is something different: it must be a symplectic manifold. That's the phase space $X$. Lectures on Classical Mechanics by J. Baez

The mathematical structure underlying both classical and quantum dynamical behaviour arises from symplectic geometry. It turns out that, in the quantum case, the symplectic geometry is non-commutative, while in the classical case, it is commutative.https://arxiv.org/pdf/1602.06071.pdf

Further Reading:

• THE SYMPLECTIZATION OF SCIENCE by Mark J. Gotay et. al.