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

$latex S: Q \to \mathbb{R} $

where $latex Q$ is some manifold. A minimum or maximum of $latex S$ can only occur at a point where

$latex d S = 0$

Here the differential $latex d S$ which is a 1-form on $latex Q.$ If we pick local coordinates $latex q^i$ in some open set of $latex Q,$ then we have

$latex \displaystyle {d S = \frac{\partial S}{\partial q^i} dq^i } $

and these derivatives $latex \displaystyle{ \frac{\partial S}{\partial q^i} } $ are very interesting. Let’s see why:

**Example 1.** In classical mechanics, consider a particle on a manifold $latex X.$ Suppose the particle…

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