# Lagrange/Hamilton system

1. Mar 3, 2008

### jacobrhcp

[SOLVED] Lagrange/Hamilton system

1. The problem statement, all variables and given/known data

There is a circle without mass, radius r. On the edge of the circle there is a mouse, being forced to move around the circle.

The angle the mouse makes with respect to the centre of the circle is called $$\theta(t)$$

At the same time, the circle is hold at it's place at a point Q on the edge and is rotated around this point Q with constant angular velocity $$\omega$$. Forget about friction.

write down the equation of motion of the mouse.

3. The attempt at a solution

the number of degrees of freedom for this system is 2, $$\omega$$ and $$\theta$$

The Lagrangian L=K-V
$$V=0, K=\frac{m v^2}{2}$$

where $$v=v_c+v_\theta$$
and $$v_c$$ is the velocity of the centre of the circle, and $$v_\theta$$ is the velocity of the mouse around the circle.

$$v_c=\omega l$$
$$v_\theta=\theta' l$$

so $$\delta J = \delta {\int_{t_1}}^{t_2} \frac{m (\omega l + \theta' l)^2}{2} dt = \delta {\int_{t_1}}^{t_2} \frac{m (\omega^2 l^2 + 2 \omega \theta' l^2 + \theta'^2 l^2)}{2} dt = 0$$

Now I have two problems.

1) is this a good way to write down the Lagrangian?
2) how do I get the $$"\delta"$$ inside the integral in a good way?

I suppose after that it's integration by parts and setting the integrand equal to zero.

EDIT: I solved the problem myselve today =)

thanks for anyone who was willing to look at it.

Last edited: Mar 4, 2008