# Determine path of point mass using the Hamilton's principle

I am very new in this field but I try to solve a problem by using the Hamilton's principle and afterwards I want to compare the solution by solving the same problem using conservation laws. What I want to do is to determine the path of the mass. Here is what I have:

I have got a body of mass $m$. This body moves uniform from point $A$ to point $B$ by crossing point $C$. The $x$-Axis is a wall. So I don't need $g$ in my calculations.

What I want to do is to determine the path of this body by using Hamilton's principle.

I think, that I only need the kinetic energy which I can put into

$$S = \int_{t_0}^{t_1} L \ dt = \int_{t_0}^{t_1} T-V \ dt = \int_{t_0}^{t_1} \frac{m}{2}v^2 \ dt = \int_{t_0}^{t_1} \frac{m}{2}\left(\frac{dx}{dt}\right)^2 \ dt.$$

This is what I get with $V = 0$. But I still need the path of the mass. How do I get this path? And how can I determine the path by using conservation laws? And how do cohere my angles $a$ and $b$?

In the case of no potential, the speed of your particle is constant. Over any interval $S = \tfrac{1}{2}mv^2 T$ where $T$ is a length of time.
$$T_{AC} + T_{CB} > T_{AB}$$
This will definitely be true since $d = vt$ so we can multiple both sides by $v$:
$$\overline{AC} + \overline{CB} > \overline{AB}$$