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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:

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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

\begin{equation} 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. \end{equation}

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 $b$?

Can someone please help me to understand this problem?

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1 Answer 1

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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.

Hamilton's Least Action principle should state:

$$T_{AC} + T_{CB} > T_{AB}$$

This will definitely be true since $d = vt$ so we can multiple both sides by $v$:

$$ \overline{AC} + \overline{CB} > \overline{AB}$$

This is known as the triangle inequality in math, that the sum of two sides of a triangle is bigger than the third.

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