The n-Category Café

Quantization and Cohomology (Week 4)

Posted by John Baez

Here are the notes for the October 24th class on Quantization and Cohomology:

• Week 4 (Oct. 24) - Hamiltonian dynamics and symplectic geometry. Hamiltonian vector fields. Getting Hamiltonian vector fields from a symplectic structure. The canonical 1-form on a cotangent bundle, and how this gives a symplectic structure.
  • Homework: show the symplectic structure $\omega = dp_i \wedge dq^i$ on the cotangent bundle gives $\omega(v_H, -) = dH$, where the Hamiltonian vector field $v_H$ is given by $$v_H = \frac{\partial H}{\partial p_i}\frac{\partial}{\partial q^i} - \frac{\partial H}{\partial q_i}\frac{\partial}{\partial p^i}$$