Inculcation Symplectic geometry

parent: inculcation

motivating Hamiltonian vector fields §

Given a smooth $H:{\mathbb{R}}^2\to \mathbb{R}$, how can we produce a vector field that “preserves” this function? Well Hamilton’s equations on ${\mathbb{R}}^2$

$$\left[\begin{array}{c}\dot{x}\\ \dot{y}\end{array}\right]=\left[\begin{array}{cc}0& -1\\ 1& 0\end{array}\right]\left[\begin{array}{c}\frac{\partial H}{\partial x}\\ \frac{\partial H}{\partial y}\end{array}\right]$$

produces a vector field perpendicular to the gradient of $H$ which does the job! But in doing so, this “Hamiltonian vector field” of $H$ produces nice geometrical properties. For example it has zero divergence, implying its flow preserves area (in ${\mathbb{R}}^2$ )!

Symplectic geometry helps study the geometry of such vector fields in a general setting on “Symplectic manifolds” which are smooth manifolds with the minimum structure needed to define Hamiltonian vector fields.