What is the motivation and intuition behind the symplectic form?

Question

That's the definition we got for the symplectic form:

Let $$\omega : \: \mathbb{C}^n \times \mathbb{C}^n \rightarrow \mathbb{C}$$ be a bilinear, anti-symmetric and non-degenerate ($\forall_{y \in \mathbb{C}^n} \: \omega(x,y)=0 \: \Rightarrow \: x=0$) map.

Let $e_1, ..., e_n$ be the base of $\mathbb{C}^n$ which means, that for $x, y \in \mathbb{C}^n$ we got $x = \sum x_i e_i$, $y = \sum y_i e_i$ Then, our symplectic form is equal to $$\omega(x,y) = x^T \Omega y$$

My question here is: What does this symplectic form intuitively tell us? How can I understand it in an intuitive way (and not abstract like here)?

Answer 1

Here are some informal sources of intuition and motivation:

• Why symplectic geometry is the natural setting for classical mechanics by Henry Cohn.

• What is a symplectic manifold, really? by Ben Webster.

For some technical detail about the latter, see also my questions Assignment of energy functions to flows is "equivariant"? and Pushforward of Hamiltonian vector field by (reverse) Hamiltonian flow is Hamiltonian.

Answer 2

Here is a recommendation: Have a look in Vladimir Arnold´s book Mathematical Methods of Classical Mechanics. There the geometric meaning behind the formulae is explained.