Calculus of Variations

18MAT31 - ENGINEERING MATHEMATICS 3 | Module-5 VTU Notes

VTU | 18MAT31 -MODULE-5

Calculus of Variations is a branch of mathematics that deals with finding the optimal function or curve that minimizes or maximizes a certain functional. A functional is a function of a function, and the goal is to find the function that extremizes the functional while satisfying certain boundary conditions. This field has applications in physics, engineering, economics, and other areas where optimization is important.

Key Points:

1. Functional: In calculus of variations, instead of dealing with functions of one or more variables, you work with functionals, which are mappings from a space of functions to the real numbers.

2. Variational Problem: The central problem in calculus of variations is to find the function that extremizes (minimizes or maximizes) a given functional. This involves finding the function that makes the functional stationary (has a derivative of zero).

3. Euler-Lagrange Equation: The Euler-Lagrange equation is a differential equation that describes the stationary points of a functional. It provides necessary conditions for a function to be an extremum of the functional.

4. Boundary Conditions: Variational problems often involve boundary conditions, which are constraints on the values of the function at the endpoints of the interval.

5. Applications: Calculus of variations has applications in various fields, including classical mechanics (principle of least action), optimal control theory, geometry (geodesics), economics (utility maximization), and more.

Example: Brachistochrone Problem

The Brachistochrone problem is a classic example in calculus of variations. It asks for the curve between two points A and B along which a bead, subject only to gravity, will slide in the shortest time. Let's consider the problem of finding the shape of the curve y(x) that minimizes the time taken by the bead to travel from A to B.

The functional to minimize is the time of travel, given by:

T=∫xA​xB​​2gy​1+y′2​​dx

where y' is the derivative of y with respect to x, and g is the acceleration due to gravity.

Using the Euler-Lagrange equation, we find the differential equation that characterizes the optimal curve:

dxd​(1+y′2​y′​)−21​2gy​1​=0

Solving this differential equation subject to appropriate boundary conditions (y(x_A) = y_A and y(x_B) = y_B), we obtain the shape of the curve that minimizes the travel time.

In summary, calculus of variations is a mathematical framework used to find functions that extremize functionals. It has practical applications in optimizing various processes, and the Brachistochrone problem is a classic example that demonstrates how it can be used to solve real-world problems.