Susskind Classical Mechanics Lecture 3

Source

write this last

Consider:

Really means the action is stationary

What are the laws determining equilibrium?

Forces will be zero, and if the forces are derived from potential (as all real forces are) then:

$\frac{d V}{d X} = 0$

These points will be at the minima and maxima of $V (x)$

In multiple dimensions

$\frac{\partial V (X)}{\partial X_{i}} = 0$

Summary of law of equilibrium $δ v = 0$ for any small change in position for however many dimensions

Functions that minimise some quantity

Example 1, shortest line between 2 points on a plane

Points $x_{1} y_{1}, x_{2} y_{2}$

If line between described as $y = f (x)$

Looking at small change $d S$ in distance along curve for small change in $x, y$

${d S}^{2} = {d x}^{2} + {d y}^{2}$ (Pythagoras)

$d S = \sqrt{1 + (\frac{d y}{d x})^{2}} d x$

So total distance is

$S = \int \sqrt{1 + (\frac{d y}{d x})^{2}} d x$

To find the shortest line, that is the minimum so

$δ \int \sqrt{1 + (\frac{d y}{d x})^{2}} d x = 0$