# Principle of Least Action

1. Jan 27, 2007

### HalfManHalfAmazing

1. The problem statement, all variables and given/known data
The Lagrangian of a mass in a uniform gravitational field can be written as follows: $$L = \frac{1}{2}m\dot{y}^2 + mgy$$

Consider all differentiale functions y(t) such that y(t1) = y1 and y(t2) = y2 where y1 and y2 are fixed values. Show that the action is a minimum for the function defining the true motion.

2. Relevant equations

I believe this simply an 'example' of prooving the principle of least action.

3. The attempt at a solution

I am wondering where to start. I derived the equation of true motion of the particle. Should I know plus in the given Lagrangian to the action integral and then then add a pertubation to the equation of true motion and then plug THAT into the action and show that the action of the perturbed equation of motion must be greater then the action of the original?

Thanks guys!

2. Jan 28, 2007

### AlephZero

Your idea is one way of doing it.

Another way would be to find the function that minimizes the action using calculus of variations, and see that it is the same as the true motion which you derived indepedently.

3. Jan 28, 2007

### HalfManHalfAmazing

We have not been taught calculus of variations yet, though I may look into that. Thanks!