Initial Conditions Applied to a Lagrangian

[vs74043]

Homework Statement

The scenario is a pendulum of length l and mass m₂ attached to a mass of m₁ which is allowed to slide along the horizontal with no friction. The support mass moves along in the X direction and the pendulum swings through the x-y plane with an angle θ with the vertical. After finding the Lagrangian and the Euler-Lagrange equations for each of the two variables X and θ, apply the conditions
$$\theta(0)=\theta_{0}<<1\text{ but greater than 0}\\
\dot{\theta}(0)=0\\
X(0)=0\\
\dot{X}(0)=0$$
to get the motion for θ and X.

Homework Equations

I have derived the Lagrangian of this to be
$$L=\frac{1}{2}(m_{1}+m_{2})\dot{X}^{2}+m_{2}l\cos\theta\dot{\theta}\dot{X}+\frac{1}{2}m_{2}l^{2}\dot{\theta}^{2}+m_{2}gl\cos\theta$$

The Attempt at a Solution

 

[BvU]

Hello vs,

Post needs some adjusting to the PF guidelines. What relevant equations do you have to continue this exercise ? What you mention now is an intermediate result (which I think is right, but I didn't check it - will do so if you show the steps ).