Why is it important that Hamilton's equations have the four symplectic properties and what do they mean?

Question

The symplectic properties are:

1. time invariance
2. conservation of energy
3. the element of phase space volume is invariant to coordinate transformations
4. the volume the phase space element is invariant with respect to time

I'm most inerested in what 3 and 4 mean and why they are important.

Answer 1

Coordinate invariance guarantees that the phase space $M$ can be endowed with a symplectic 2-form $\omega$ which locally is given by $\omega = dq^i \wedge dp_i$. This form is closed ($d\omega = 0$) and nondegenerate, i.e. $\omega^n$ is a volume form, where $2n$ is the dimension of $M$. The other conditions say that for any Hamiltonian function $H$, the induced flow on $M$ preserves both $H$ and $\omega$. It turns out that proving this is totally trivial using the language of symplectic geometry.

So why bother? By reformulating mechanics in terms of symplectic manifolds, we now have all the modern tools of differential geometry and topology at our disposal.

Example. Using Morse theory, for any (sufficiently nice) Hamiltonian, we can obtain bounds on the number of equilibrium solutions in terms of the Betti numbers of $M$.

Example. Gromov nonsqueezing. I won't state it, but it is a classical analogue of the uncertainty principle. In particular, it shows that certain reasonable phenomena are impossible for Hamiltonian systems.

Example. Reduced phase spaces (symplectic reduction). This is very clean from the geometric point of view, but very messy in local coordinates.

Example. Integrable systems, spectral curves, etc. Again, hard to imagine formulating a lot of this without geometry at our disposal.

Example. Deformation quantization, geometric quantization (and others). This gives some insight into what quantization actually is, or at least should be. Dirac-style canonical quantization might work for lots of simple systems, but there are subtleties involved and for more complicated systems it is not at all obvious what the right quantization is. The geometric point of view also relates quantization to representation theory (see e.g. the work of Kostant).

Answer 2

I think the properties 3 and 4 are important because in these way the probability distribution in phase space is conserved and the information is also conserved. Some systems dont have this property, these are chaotic. In these systems the volume in phase space could increase until filling the complete phase space in a fractal way.