Action-angle Coordinates

A semi-casual introduction to Action-angle coordinates in Hamiltonian dynamics.

Transcript

1. ACTION-ANGLE COORDINATES Adrian Price-Whelan
4. n second order 2n first order
5. n=3, 6D phase-space
6. Integrable systems There exist n independent integrals of motion In 2n-dimensional phase-space:
7. Integrable systems There exist n independent integrals of motion Motion is confined to an n-dimensional manifold In 2n-dimensional phase-space:
8. Integrable systems Bounded motion periodicity
9. Integrable systems What if…
10. Integrable systems
11. Action-angle coords.
12. Action-angle coords.
13. Action-angle coords.
20. cool. SO WHAT?
21. Tidal stream formation Hessian
22. Tidal stream formation
23. Position Velocity
24. Angles Actions
25. so how do we find actions?
26. This is not a solved problem!
27. Kaasalainen & Binney McGill & Binney Sanders & Binney Bovy