Geometric Mechanics

I’m thrilled to say that I’m going to be able to take a readings course on mechanics next semester with a math professor. I’m going to follow along in Jose’s and Saletan’s [J,S] text Classical Dyanmics and supplement it with Arnold’s text and various web sources like John Baez’s1 notes. The goal is to learn the geometry behind mechanics. Physics, where I am, is largely taught as a bunch of separate seemingly unrelated topics. Quantum has those kets and bras and Hilbert space, while mechanics has Lagrangians and Statistical Mechanics has partition functions and Hamiltonians (though not as operators). However I know that these areas (along with optics, EM, thermo, etc) are all connected by deep geometric results.

While on winter break I’m working through some material to get a head start. Right now I’m up to the point where [J,S] have derived the coordinate free Lagrangian:
L_{\Delta} \theta_L-dL=0

where \theta_L=\frac{\partial L}{\partial \dot{q^{\alpha}}}dq^{\alpha}

Well that looks neat, but what does it mean? \theta_L is a one-form field and we’re looking at how it evolves along \Delta which is the vector field
\Delta = \dot{q}^{\alpha}\frac{\partial}{\partial q^{\alpha}}+\ddot{q}^{\alpha}\frac{\partial}{\partial \dot{q}^{\alpha}}
This field has vectors parallel to the possible trajectories of the system. The time evolution of a dynamical variable f(q,\dot{q}) is the Lie derivative, L_{\Delta}f. So what does it mean to ask how the one form \theta_L evolves along these trajectories? Even more so what does the Lagrangian even mean? What does it represent about the system? The notes by Baez (above) seem to offer some possible insight and maybe I’ll have more to add later.

I’ll grant that this post doesn’t answer any questions as much as it asks them. This will often be the case and I hope to return and post updates to these posts as time goes on. Essentially this will be a online record of my attempt to learn some beautiful physics and mathematics.

1: John Baez is a hoot to read. He refers to stuff like hyperreals in his mechanics book!


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