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Calculus of Variations: Minimizing Fuel Consumption w/ v(t)

  1. May 20, 2015 #1
    1. The problem statement, all variables and given/known data
    (I'm learning all of this in German, so I apologize if something is translated incorrectly.) So last week we started calculus of variations, and I'm rather confused about how to approach the following problem:

    The fuel consumption of a vehicle per unit of time is expressed as follows:

    In which the vehicle travels a distance D in a given time T. v(t) is the speed of the vehicle (a, b are constants). The beginning condition is v(0)=0.

    For what v(t) is the fuel consumption minimal and compare this consumption with another v(t) contanting a constant acceleration.

    A tip is then given: Find the functional J[v] which reflects the fuel consumption and the functional for the condition. Take note that v(T) isn't given, but that the stationary J[v] implies a boundary condition of a(T) = 0.
    2. Relevant equations
    Based on the problems we received last week, I assume y(t)+εη(t) is necessary for the minimization of this problem.

    3. The attempt at a solution
    So I'm not sure how to determine the functional J(v) that reflects the fuel consumption, but I have assumed the velocity function is any sort of y(t)+εη(t) which fulfills the condition that df/dt = 0. (Which means it is extremal)

    5. Übungsblatt Theo - Seite 1.jpg

    Otherwise I'm pretty lost here guys. Thanks so much for the help!
    Last edited: May 20, 2015
  2. jcsd
  3. May 20, 2015 #2


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    Since ##\frac{df}{dt}## is the rate of fuel consumption, then the total fuel consumption is the integral of that with respect to ##t##, isn't it? That's what you want to minimize.
  4. May 20, 2015 #3

    Ray Vickson

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    If I understand correctly, your problem is
    [tex] \begin{array}{rl}\min & \int_0^T (a v + b \dot{v}^2 ) \, dt,\\
    \text{subject to}& \int_0^T v \, dt = D
    \end{array} [/tex]
    This is a constrained calculus-of-variations problem. The constraint can be handled using a Lagrange multiplier method; see, eg.,
    http://www.mpri.lsu.edu/textbook/chapter8-b.htm#integral ---the section titled 'constrained variational problems--integral constraint'.

    Strictly speaking, the second integral above is 'displacement', not 'distance', but if ##v(t) \geq 0## throughout ##[0,T]## these two concepts are the same.
  5. May 20, 2015 #4
    Wow, Ray! That was a massive help. I can safely say my first experience on the Physics Forums was fantastic thanks to you!
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