# Homework Help: Numerical: Shortest Time of Travel of a Particle experiencing a Time-Varying Force

1. Sep 24, 2012

### ecastro

1. The problem statement, all variables and given/known data
The problem is to calculate the equation of the path that a particle will travel in the least time if this particle is receiving a time-varying force. The force is more likely a Gaussian White Noise, F(t).

2. Relevant equations
Trying to relate it with the brachistochrone problem,
$t = \int \frac{ds}{v}$
Where $ds$ is the space coordinate of the system and $v$ is the velocity of the particle, which can be calculated by the given force. Letting,

$v = \int F(t) dt$

Then direct substitution to the brachistochrone equation.

3. The attempt at a solution
If all of my assumptions on solving the problem are correct, then,
$t = \int \frac{ds}{\int F(t) dt}$
And since the velocity of the particle is time-dependent, then it goes out of the integral, which is then,
$t = \frac{1}{\int F(t) dt}\int ds$

As seen in the last equation, I arrived at an integral which have an obvious result, the equation must be a line to have the least time of travel. The problem is, I do not know if it is valid to use the given brachistochrone equation when the force is time varying and I also need to do it numerically, so I do not know where the factor $\frac{1}{\int F(t) dt}$ comes into when done numerically.