Symplectic geometry of phase space

[CrazyNeutrino]

What is a symplectic manifold or symplectic geometry? (In intuitive terms please)
I have a vague understanding that it involves some metric that assigns an area to a position and conjugate momentum that happens to be preserved. What is 'special' about Hamilton's formulation that makes it more useful than Lagrange's? Why is it that phase space has this special geometry but state space does not? What makes the generalized momentum a more useful coordinate than the generalized velocity?

 

[Orodruin]

A symplectic manifold is a manifold equipped with a symplectic form. The symplectic form is a non-degenerate 2-form, which makes it an anti-symmetric rather than symmetric tensor. It is not a metric.

Hamilton’s formalism reduces the equations of motion to the flow of a Hamiltonian vector field, which is a set of first order differential equations and the symmetries of the time evolution phase space are often more manifest and boil down to the Poisson bracket.

State space and phase space are different manifolds (the latter being the cotangent bundle of the former).

Using the generalised momenta as coordinates, the symplectic form takes a much easier form than if you use generalised velocities. Much like using Cartesian coordinates instead of curvilinear ones makes the metric take an easier form in Euclidean space.

 

[CrazyNeutrino]

Thank you, this is the most concise explanation I've seen yet!

 

[vanhees71]

A very good (but of course also very mathematical) explaning these things in detail is

V. I. Arnold, Mathematical Methods of Classical Mechanics, Springer Verlag (1989)