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

[Hunter Bliss]

Homework Statement

(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.

Homework Equations

Based on the problems we received last week, I assume y(t)+εη(t) is necessary for the minimization of this problem.

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)

Otherwise I'm pretty lost here guys. Thanks so much for the help!

 

[Dick]

Homework Statement

(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.

Homework Equations

Based on the problems we received last week, I assume y(t)+εη(t) is necessary for the minimization of this problem.

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)

View attachment 83768

Otherwise I'm pretty lost here guys. Thanks so much for the help!

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.

 

[Ray Vickson]

Homework Statement

(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.

Homework Equations

Based on the problems we received last week, I assume y(t)+εη(t) is necessary for the minimization of this problem.

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)

View attachment 83768

Otherwise I'm pretty lost here guys. Thanks so much for the help!

If I understand correctly, your problem is
$$\begin{array}{rl}\min & \int_0^T (a v + b \dot{v}^2 ) \, dt,\\ \text{subject to}& \int_0^T v \, dt = D \end{array}$$
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.

 

[Hunter Bliss]

Wow, Ray! That was a massive help. I can safely say my first experience on the Physics Forums was fantastic thanks to you!