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Introduction to the Calculus of Variations and Control with Modern Applications

Ibrahim Blair

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Published in Introduction To The Calculus Of Variations And Control With Modern Applications (Chapman Hall/CRC Applied Mathematics Nonlinear Science 28)

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The calculus of variations is a mathematical technique that can be used to find the extreme values of a functional. A functional is a function that takes a function as its argument. The calculus of variations can be used to solve a wide variety of problems, including problems in physics, engineering, and economics.

In this article, we will introduce the calculus of variations and discuss some of its applications. We will begin by defining the concept of a functional and then we will discuss the Euler-Lagrange equation, which is the main equation used in the calculus of variations. We will then discuss some examples of how the calculus of variations can be used to solve real-world problems.

Introduction to the Calculus of Variations and Control with Modern Applications (Chapman Hall/CRC Applied Mathematics Nonlinear Science 28)

Introduction to the Calculus of Variations and Control with Modern Applications (Chapman & Hall/CRC Applied Mathematics & Nonlinear Science Book 28)

by John A. Burns

5 out of 5

Language	:	English
File size	:	8320 KB
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Print length	:	562 pages

A functional is a function that takes a function as its argument. In other words, a functional is a function of a function. Functionals are often used to represent the total cost, energy, or other quantity associated with a particular function.

For example, the functional

$$J[y] = \int_a^b f(x, y, y') dx$$

represents the total energy of a vibrating string. In this functional, $y$ is the displacement of the string, $f$ is the energy density, and $a$ and $b$ are the endpoints of the string.

The Euler-Lagrange equation is the main equation used in the calculus of variations. The Euler-Lagrange equation is a differential equation that must be satisfied by any function that minimizes or maximizes a functional.

The Euler-Lagrange equation is given by

$$\frac{\partial L}{\partial y}- \frac{d}{dx}\left(\frac{\partial L}{\partial y'}\right) = 0,$$

where $L$ is the Lagrangian. The Lagrangian is a function that is defined by

$$L(x, y, y') = f(x, y, y') - y'g(x, y),$$

where $g$ is a function that is called the potential function.

The calculus of variations has a wide variety of applications in physics, engineering, and economics. Some of the most common applications include:

Minimizing the energy of a vibrating string. The calculus of variations can be used to find the shape of a vibrating string that minimizes its total energy. This problem is important in the design of musical instruments.
Maximizing the efficiency of a heat engine. The calculus of variations can be used to find the shape of a heat engine that maximizes its efficiency. This problem is important in the design of power plants.
Optimizing the trajectory of a spacecraft. The calculus of variations can be used to find the trajectory of a spacecraft that minimizes the amount of fuel required to reach its destination. This problem is important in the design of space missions.

The calculus of variations is a powerful mathematical technique that can be used to solve a wide variety of problems. In this article, we have introduced the calculus of variations and discussed some of its applications. We have also discussed the Euler-Lagrange equation, which is the main equation used in the calculus of variations.

We hope that this article has given you a better understanding of the calculus of variations. If you have any questions, please feel free to contact us.

Introduction to the Calculus of Variations and Control with Modern Applications (Chapman & Hall/CRC Applied Mathematics & Nonlinear Science Book 28)

by John A. Burns

5 out of 5