Classical Mechanics versus Thermodynamics (Part 2)

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Classical Mechanics versus Thermodynamics (Part 2)

January 24, 2012 by doxdrum

I showed you last time that in many branches of physics—including classical mechanics and thermodynamics—we can see our task as minimizing or maximizing some function. Today I want to show how we get from that task to symplectic geometry. So, suppose we have a smooth function $S: Q \to \mathbb{R}$ where $Q$ is some manifold. A minimum or maximum of $S$ can only occur at a point where $d S = 0$ Here the differential $d S$ which is a 1-form on $Q.$ If we pick local …