Calculus of Variations

1. Introduction
2. Fundamental Concepts
3. The Euler-Lagrange Equation
4. Solution Methods
5. Applications
6. Advanced Topics
7. Quiz

1. Introduction to Calculus of Variations

The Calculus of Variations is a field of mathematical analysis that deals with maximizing or minimizing functionals, which are mappings from a set of functions to the real numbers. It emerged in the 17th century with the famous brachistochrone problem posed by Johann Bernoulli in 1696.

Historical Example: The Brachistochrone Problem

Given two points A and B in a vertical plane, find the curve along which a particle would travel from A to B in the least time, assuming the particle starts at rest and is acted upon only by gravity.

The solution to this problem is a cycloid, which surprisingly is not a straight line (which would give the shortest distance). This illustrates how the calculus of variations can lead to counterintuitive results.

Unlike ordinary calculus which deals with finding extrema of functions, the calculus of variations seeks extrema of functionals, which are essentially "functions of functions."

2. Fundamental Concepts

2.1 Functionals

A functional \(J[y]\) maps a function \(y(x)\) to a real number. It is typically expressed as an integral:

\[J[y] = \int_{x_1}^{x_2} F(x, y, y') \, dx\]

where \(F(x, y, y')\) is a function of the independent variable \(x\), the dependent variable \(y\), and its derivative \(y'\).

2.2 Variations and Extrema

To find the function \(y(x)\) that extremizes (maximizes or minimizes) the functional \(J[y]\), we consider small variations of the function:

\[y(x) \rightarrow y(x) + \epsilon \eta(x)\]

where \(\epsilon\) is a small parameter and \(\eta(x)\) is an arbitrary function that vanishes at the endpoints (i.e., \(\eta(x_1) = \eta(x_2) = 0\)).

For \(J[y]\) to be at an extremum, the first variation must vanish:

\[\left. \frac{d}{d\epsilon} J[y + \epsilon \eta] \right|_{\epsilon=0} = 0\]

This leads to the Euler-Lagrange equation, which is the fundamental equation of the calculus of variations.

3. The Euler-Lagrange Equation

The Euler-Lagrange equation is the necessary condition for a function to extremize a functional. For a functional of the form \(J[y] = \int_{x_1}^{x_2} F(x, y, y') \, dx\), the Euler-Lagrange equation is:

\[\frac{\partial F}{\partial y} - \frac{d}{dx}\left(\frac{\partial F}{\partial y'}\right) = 0\]

3.1 Derivation

The derivation of the Euler-Lagrange equation involves the following steps:

Consider the variation \(J[y + \epsilon \eta] - J[y]\).
Expand this in powers of \(\epsilon\) and extract the term proportional to \(\epsilon\) (first variation).
Set this first variation to zero for all possible \(\eta(x)\).
Apply integration by parts to transform terms involving \(\eta'(x)\).
Use the fundamental lemma of calculus of variations to obtain the Euler-Lagrange equation.

Example: Shortest Path Between Two Points

To find the curve of shortest length connecting two points in a plane, we minimize the arc length functional:

\[J[y] = \int_{x_1}^{x_2} \sqrt{1 + (y')^2} \, dx\]

Here, \(F(x, y, y') = \sqrt{1 + (y')^2}\). Applying the Euler-Lagrange equation:

\[\frac{\partial F}{\partial y} - \frac{d}{dx}\left(\frac{\partial F}{\partial y'}\right) = 0\] \[0 - \frac{d}{dx}\left(\frac{y'}{\sqrt{1 + (y')^2}}\right) = 0\]

This implies that \(\frac{y'}{\sqrt{1 + (y')^2}} = C\) (constant), which leads to \(y' = \text{constant}\), meaning the solution is a straight line, as expected.

4. Solution Methods

4.1 Direct Integration

In some simple cases, the Euler-Lagrange equation can be directly integrated to find the extremal function.

Example: When F Does Not Depend on y

If \(F = F(x, y')\) does not depend explicitly on \(y\), then \(\frac{\partial F}{\partial y} = 0\), and the Euler-Lagrange equation becomes:

\[\frac{d}{dx}\left(\frac{\partial F}{\partial y'}\right) = 0\]

This implies that \(\frac{\partial F}{\partial y'} = C\) (constant), which is a first integral of the Euler-Lagrange equation.

4.2 Beltrami Identity

If \(F = F(y, y')\) does not depend explicitly on \(x\), we can use the Beltrami identity:

\[F - y' \frac{\partial F}{\partial y'} = C\]

This provides a first integral of the Euler-Lagrange equation and often simplifies the solution process.

4.3 Variational Methods

When direct integration is not possible, various numerical and approximate methods can be used:

Ritz Method: Approximate the solution using a linear combination of basis functions with adjustable parameters.
Finite Element Method: Divide the domain into small elements and approximate the solution within each element.
Gradient Descent: Iteratively improve an initial guess by moving in the direction of steepest descent of the functional.

Example: Ritz Method for Minimal Surface of Revolution

To find the curve that, when rotated around the x-axis, produces the surface of minimum area (catenary), we can approximate the solution using the Ritz method with a trial function:

\[y(x) \approx a_0 + a_1 x + a_2 x^2 + \ldots + a_n x^n\]

We then substitute this into the functional and minimize with respect to the coefficients \(a_0, a_1, \ldots, a_n\).

5. Applications

5.1 Physics

The calculus of variations has numerous applications in physics:

Example: Principle of Least Action

In classical mechanics, the path taken by a physical system between two states is the one that minimizes the action functional:

\[S[q] = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt\]

where \(L\) is the Lagrangian of the system, typically \(L = T - V\) (kinetic energy minus potential energy). Applying the Euler-Lagrange equation leads to Newton's equations of motion.

5.2 Geometry

Minimal surfaces, geodesics, and isoperimetric problems are geometric applications of the calculus of variations.

Example: Geodesics on a Surface

The shortest path between two points on a curved surface is a geodesic. For a surface with metric \(ds^2 = E \, dx^2 + 2F \, dx \, dy + G \, dy^2\), the length functional is:

\[L[y] = \int_{x_1}^{x_2} \sqrt{E + 2F y' + G (y')^2} \, dx\]

The Euler-Lagrange equation leads to the geodesic equation.

5.3 Engineering and Optimization

The calculus of variations is used in control theory, optimal design, and various engineering problems.

Example: Optimal Control

In optimal control theory, we seek to find the control input \(u(t)\) that minimizes a cost functional:

\[J[u] = \int_{t_0}^{t_f} f(x, u, t) \, dt\]

subject to the system dynamics \(\dot{x} = g(x, u, t)\). The solution involves Pontryagin's maximum principle, which is a generalization of the Euler-Lagrange equation.

6. Advanced Topics

6.1 Multiple Variables

The calculus of variations extends to functions of multiple variables, leading to partial differential equations such as the Laplace equation and the Poisson equation.

Example: Dirichlet's Principle

Dirichlet's principle states that the function minimizing the energy functional:

\[E[u] = \int_\Omega \frac{1}{2}|\nabla u|^2 \, dx\]

subject to boundary conditions \(u = g\) on \(\partial\Omega\), satisfies Laplace's equation \(\nabla^2 u = 0\) in \(\Omega\).

6.2 Constraints

Many variational problems involve constraints, which can be handled using Lagrange multipliers.

Example: Isoperimetric Problem

Find the curve of fixed length \(L\) that encloses the maximum area. The area functional is:

\[A[y] = \int_{x_1}^{x_2} y \, dx\]

subject to the constraint:

\[L[y] = \int_{x_1}^{x_2} \sqrt{1 + (y')^2} \, dx = L_0\]

Using a Lagrange multiplier \(\lambda\), we minimize \(J[y] = -A[y] + \lambda (L[y] - L_0)\). The solution is a circle.

6.3 Second Variation and Sufficient Conditions

The second variation provides sufficient conditions for a function to be a minimum or maximum of the functional, analogous to the second derivative test in ordinary calculus.