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

In summary, the conversation discusses a problem in calculus of variations involving the minimization of fuel consumption for a vehicle traveling a certain distance in a given time. The main task is to find the functional J[v] that reflects the fuel consumption and satisfies the condition of a stationary J[v] implying a boundary condition of a(T) = 0. The problem can be solved using a constrained calculus-of-variations approach with the use of a Lagrange multiplier.
  • #1
Hunter Bliss
4
1

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:

BlIvZXf.png

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)

5. Übungsblatt Theo - Seite 1.jpg


Otherwise I'm pretty lost here guys. Thanks so much for the help!
 
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  • #2
Hunter Bliss said:

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:

BlIvZXf.png

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.
 
  • #3
Hunter Bliss said:

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:

BlIvZXf.png

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
[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.
 
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Likes Hunter Bliss
  • #4
Wow, Ray! That was a massive help. I can safely say my first experience on the Physics Forums was fantastic thanks to you!
 

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

1. What is the principle behind the Calculus of Variations?

The principle behind the Calculus of Variations is to find the path or curve that minimizes a specified functional, which is a functional of a function or a set of functions. In other words, it is a method for finding the function that makes a certain quantity as small or as large as possible.

2. How is Calculus of Variations applied to minimizing fuel consumption?

In the context of minimizing fuel consumption, the Calculus of Variations is used to find the optimal trajectory for a vehicle by minimizing the total energy used during the journey. This involves finding the function that minimizes the integral of the vehicle's velocity with respect to time, subject to certain constraints such as distance, time, and acceleration.

3. What is the role of v(t) in Calculus of Variations for minimizing fuel consumption?

The function v(t) represents the velocity of the vehicle at any given time t. It is the variable that is being optimized in order to minimize fuel consumption. By finding the function that minimizes the integral of v(t), we can determine the optimal velocity profile for the vehicle to minimize energy usage.

4. What are some practical applications of minimizing fuel consumption using Calculus of Variations?

Aside from optimizing vehicle trajectories, minimizing fuel consumption using Calculus of Variations has practical applications in various fields such as aerospace engineering, robotics, and economics. It can also be used to optimize the path of a rocket or satellite in space, or to minimize energy usage in industrial processes.

5. What are some challenges associated with using Calculus of Variations to minimize fuel consumption?

One of the main challenges is the complexity of the optimization problem, which often involves solving a system of differential equations. This can be computationally intensive and require advanced mathematical techniques. Additionally, real-world factors such as changing environmental conditions and vehicle dynamics can make it difficult to accurately predict and optimize fuel consumption using Calculus of Variations.

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