Pendulum-with-free-frictionless-support

This program simulates the movement of a pendulum with a free frictionless support on the x-coordinate. See the following image:

To calculate the path of the masses i defined an ODE with the equations that define the Lagrangian. We have two masses the support $M_1$, and the pendulum $M_2$. Then we define the position and velocitis of each particle with $x$ and $\theta$:

$$ \begin{array}{cc} x_1 = x & y_1 = 0 \\ x_2 = x + lsin(\theta) & y_2 = -lcos(\theta) \end{array} $$

Therefore we have the velocitie for the particle one is $(\dot{x},0)$, and for the particle two is $(\dot{x} + l\dot{\theta}cos(\theta), l\dot{\theta}sin(\theta))$. Calculating the kinetic energy and the potencial energy, we obtain the next lagrangian:

$$ \begin{aligned} L &= T - U = (T_1 + T_2) - (U_1 + U_2) \\ & = \frac{1}{2}(M_1+M_2)\dot{x}^2 + M_2l\dot{\theta}\dot{x}cos(\theta) + \frac{1}{2}M_2l^2\dot{\theta}^2 + mglcos(\theta) \end{aligned} $$

Then computing the Euler-Lagrange equations: $$\frac{d}{dt}\frac{\partial L}{\partial\dot{x}} - \frac{\partial L}{\partial x} = 0$$ and $$\frac{d}{dt}\frac{\partial L}{\partial\dot{\theta}} - \frac{\partial L}{\partial \theta} = 0$$ we obtain the next two equations:

$$ \begin{aligned} (E_1):& \frac{1}{2}(M_1 + M_2)\ddot{x} + M_2l\ddot{\theta}cos(\theta) - M_2l\dot{\theta}^2sin(\theta) = 0 \\ (E_2): & l\ddot{\theta} + \ddot{x}cos(\theta) + gsin(\theta) = 0 \end{aligned} $$

After that we only have to rename the variables $v:= \dot{x}$ and $w:=\dot{\theta}$ define our EDO and apply for example a Runge Kutta so we can approximate the trayectori. Here we have an example with initial conditions $x = -7.5; \theta = 120º; v = 1; w = 0$: