# Spring Pendulum - Lagrangian Mechanics

## Homework Equations

Euler-Lagrange Equation
$\frac{\partial{L}}{\partial{q}} - \frac{d}{dt}\frac{\partial{L}}{\partial{\dot{q}}} = 0$
$L = T - V$

## The Attempt at a Solution

a. The potential energy V is the potential energy from the spring and the gravitational potential energy. The kinetic energy is the energy in the radial direction and in the theta direction.
$$L = \frac{1}{2}m(\dot{r}^2+\dot{\theta}^2(r_0+r)^2) + mg(r_0+r)cos\theta - \frac{1}{2}kr^2$$

b. Use one Euler-Lagrange equation for r and one for theta and I get: $$m\dot{\theta}^2 + mgcos\theta - kr = \frac{d}{dt}(m\dot{r})$$ $$-mg(r_0+r)sin\theta = \frac{d}{dt}(m\dot{\theta}(r_0+r)^2)$$
Which are the F=ma equation for the radial direction and the Torque = (d/dt) Angular momentum equation.

c. Confused here, if angular position and velocity are to be fixed, I assume they mean $\theta$ and $\dot{\theta}$, do I consider them both to be zero, or just a constant number? And if the position is fixed doesn't that mean that the velocity should be zero? Is this a differential equation I have to solve?

d. and e. Need c to continue.

Any help is appreciated, thanks !

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vela
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