Calculating Least Time Path of a Particle Under Varying Force

[ecastro]

Homework Statement

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).

Homework 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.

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.

 

[mark726]

Thank you for sharing your attempt at solving this problem. You have correctly identified the brachistochrone problem as a potential approach to solving this problem. However, there are a few key points that I would like to address in regards to your solution.

Firstly, it is important to note that the brachistochrone problem assumes a constant gravitational force acting on the particle, not a time-varying force. Therefore, it may not be applicable in this scenario.

Secondly, even if we assume that the brachistochrone problem can be applied to this situation, your solution does not take into account the time-varying nature of the force. As you mentioned, the velocity of the particle is dependent on time, so simply taking it out of the integral is not a valid approach.

To properly solve this problem, you may need to use numerical methods and simulations to analyze the motion of the particle under the influence of a time-varying force. This may involve breaking the motion into small time intervals and calculating the velocity and position at each interval, taking into account the changing force at each point.

I hope this helps guide you towards a more accurate solution. Good luck with your research!