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

  1. Sep 24, 2012 #1
    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,
    [itex]t = \int \frac{ds}{v}[/itex]
    Where [itex]ds[/itex] is the space coordinate of the system and [itex]v[/itex] is the velocity of the particle, which can be calculated by the given force. Letting,

    [itex]v = \int F(t) dt[/itex]

    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,
    [itex]t = \int \frac{ds}{\int F(t) dt}[/itex]
    And since the velocity of the particle is time-dependent, then it goes out of the integral, which is then,
    [itex]t = \frac{1}{\int F(t) dt}\int ds[/itex]

    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 [itex]\frac{1}{\int F(t) dt}[/itex] comes into when done numerically.
     
  2. jcsd
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