# Introducing Randomness into Lagrangian Mechanics

Let's say at $t_o$ we have a ball rolling along a (rigid) tight rope. Is there anyway that we can solve for the trajectory of the ball knowing that at some $t'$ there will be a random constraint force in the N/E/S/W plane that will make it fall off the rope?

In essence is it it possible to analytically solve for the family of trajectories that could arise from this random perturbation in the framework of Lagrangian mechanics? In this case it's easy to visualize that the trajectory would take one of two forms, the first a momentary perturbation and the second a continuous froce that's initially randomly tuned:

1st:

$x = x_0 + v_0t + H(t'){\int^{t' + \delta t}_{t'} F_{a_x}dt'\over m}t$

$y = y_0 + H(t'){\int^{t' + \delta t}_{t'} F_{a_y}dt'\over m}t$

$z = z_0-H(t'){1\over2} gt^2$

2nd:

$x = x_0 + v_0t + H(t'){F_{a_x}\over m}t^2$

$y = y_0 + H(t'){F_{a_y}\over m}t^2$

$z = z_0-H(t'){1\over2} gt^2$

Where $H(t')$ is the step function at the time of the perturbation $F_a$ and the integral is my attempt at trying to describe the impulse imparted by $F_a$.

But is this able to be generalized in anyway to arbitrary solve problems in the framework of Lagrangian mechanics?

## 1 Answer

In your problem you have a constant force $\mathbf F$ that switches on at a time t'. You can then write the potential $U$ as $$U=H(t-t')\mathbf F\cdot\mathbf r$$ so that the Lagrangian follows directly as $L = T - U$, with $T$ given by the kinetic term of a point mass, if it is ok to neglect rotations, or that of a rolling ball otherwise. This Lagrangian reproduces the equation of motion you are looking for.

As for the solution of these equations, I don't see other way than to split it into different domains, according to $H(t-t')$.

• ah, I screwed that up. I meant to have an impulse style perturbation and didn't pay enough attention to how I should properly do my unit correction. I'll add the case I meant to describe over the current one. – Skyler Jan 31 '15 at 23:29