Calculus of variation

Learning Objectives:

Understand the foundational concepts of the calculus of variations.
Learn techniques to find functions that extremize a given functional.
Apply the Euler-Lagrange equation to physical problems such as the brachistochrone, geodesics, and the principle of least action.

Key Concepts / Definitions:

Functional: A quantity that depends on a function and possibly its derivatives, typically expressed as an integral.
Variation ($\delta y$): A small arbitrary change in the function $y(x)$ used to probe how the functional changes.
Euler-Lagrange Equation: A differential equation derived from the condition that a functional is stationary (has an extremum).

Theory and Explanation: The calculus of variations deals with finding functions that make a given functional attain a stationary value (usually a minimum or maximum). A typical problem is to find a function $y(x)$ that extremizes the integral:

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

The basic technique involves:

Introducing a small variation: $y(x) \rightarrow y(x) + \epsilon \eta(x)$, where $\eta(x)$ is an arbitrary smooth function with $\eta(x_1) = \eta(x_2) = 0$.
Calculating the first-order change in $I[y]$ with respect to $\epsilon$.
Setting $\delta I = 0$ for all $\eta(x)$ leads to the Euler-Lagrange equation:

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

This equation provides the necessary condition for the function $y(x)$ to make $I[y]$ stationary.

Applications:

Principle of Least Action in mechanics.
Geodesics on surfaces.
Brachistochrone problem in dynamics.
Optics: Fermat’s principle of least time.

Mathematical Formulation: Consider a functional:

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

To find $y(x)$ such that $I[y]$ is extremized, apply the Euler-Lagrange equation:

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

Special Cases:

If $f$ does not explicitly depend on $y$:
$\frac{d}{dx} \left( \frac{\partial f}{\partial y'} \right) = 0$
If $f$ does not explicitly depend on $x$ (Beltrami identity):
$f - y' \frac{\partial f}{\partial y'} = \text{constant}$

Solved Examples:

Example 1:
Problem: Find the curve $y(x)$ between two points that minimizes the integral
$I[y] = \int_{x_1}^{x_2} y'^2 \, dx$
Solution:
Here, $f = y’^2$, so
$\frac{\partial f}{\partial y} = 0, \quad \frac{\partial f}{\partial y'} = 2y'$
Then:
$\frac{d}{dx}(2y') = 0 \Rightarrow y'' = 0$
Solving:
$y(x) = Ax + B$
which is a straight line — the shortest path between two points.
Example 2:
Problem: Use the calculus of variations to find the curve of quickest descent (brachistochrone problem).
Solution:
The time of descent is given by:
$T[y] = \int_{x_1}^{x_2} \sqrt{\frac{1 + y'^2}{2gy}} \, dx$
Applying the Euler-Lagrange equation leads to a complex differential equation whose solution is a cycloid — the curve traced by a point on the rim of a rolling circle.

Important Points / Summary:

The Euler-Lagrange equation gives the condition for a function to extremize a functional.
Constraints can be handled using the method of Lagrange multipliers.
Applications span classical mechanics, optics, and geometry.

Practice Questions:

Short Answer:
1. What is a functional? Give an example.
2. State the Euler-Lagrange equation and explain its significance.
Numerical:
1. Find the function $y(x)$ that minimizes $\int_0^1 (y’)^2 \, dx$ with boundary conditions $y(0)=0$, $y(1)=1$.
2. Solve the Euler-Lagrange equation for $f = y^2 + (y’)^2$.
MCQs:
1. The Euler-Lagrange equation is derived from the condition:
  a) $\delta I = \text{maximum}$
  b) $\delta I = \text{minimum}$
  c) $\delta I = 0$
  d) $\delta I = \infty$
2. Which of the following is not an application of calculus of variations?
  a) Geodesics
  b) Snell’s law
  c) Newton’s second law
  d) Principle of least action

Applications of the Calculus of Variations

The calculus of variations plays a crucial role in many physical and geometric problems where a functional (usually representing energy, time, or length) must be minimized or maximized. Below are four fundamental applications:

1. Principle of Least Action in Mechanics

Statement

In classical mechanics, the motion of a particle is such that it minimizes (or makes stationary) the action functional:

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

Here, $ L = T - V $ is the Lagrangian, where $ T $ is the kinetic energy and $ V $ is the potential energy. The function $ q(t) $ describes the generalized coordinates of the system.

Euler–Lagrange Equation

The extremum of the action occurs when:

\[\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0\]

This is known as the Euler–Lagrange equation, and it leads to Newton’s laws when applied in the appropriate context.

Significance

Provides a powerful reformulation of classical mechanics.
Fundamental in quantum mechanics, field theory, and general relativity.
Basis of the Lagrangian and Hamiltonian formalisms.

2. Geodesics on Surfaces

Statement

A geodesic is the shortest path between two points on a curved surface. For example, great circles are geodesics on a sphere.

Problem

Given a surface described (parametrically or by a constraint), find the curve $ \gamma(t) = (x(t), y(t), z(t)) $ that minimizes the arc length:

\[S[\gamma] = \int_a^b \sqrt{\dot{x}^2 + \dot{y}^2 + \dot{z}^2} \, dt\]

Or, in a curved coordinate system with metric $ g_{ij} $:

\[S[\gamma] = \int_a^b \sqrt{g_{ij} \frac{dx^i}{dt} \frac{dx^j}{dt}} \, dt\]

Example: Sphere

On a sphere of radius $ R $, the metric is:

\[ds^2 = R^2(d\theta^2 + \sin^2 \theta\, d\phi^2)\]

Minimizing the arc length leads to the equation of great circles.

Applications

General relativity: particles follow geodesics in spacetime.
Navigation: shortest paths on Earth.
Computer graphics and robotics.

3. Brachistochrone Problem in Dynamics

Statement

Find the curve between two points (not vertically aligned) along which a particle slides under gravity in the least time, assuming no friction.

Functional to Minimize

\[T[y] = \int_{x_1}^{x_2} \sqrt{\frac{1 + (y')^2}{2gy}} \, dx\]

Using the calculus of variations and the Beltrami identity, one derives the curve of fastest descent.

Solution

The solution is a cycloid, given parametrically as:

\[x = a(\theta - \sin \theta), \quad y = a(1 - \cos \theta)\]

Significance

Originated the field of variational calculus (posed by Johann Bernoulli in 1696).
Early example where minimizing time (not distance or energy) leads to a surprising result.
Important in physics and engineering for time-optimization problems.

4. Optics: Fermat’s Principle of Least Time

Statement

Fermat’s principle states that light takes the path which minimizes the time taken to travel between two points.

Mathematical Formulation

If the speed of light varies with position, say $ v(x) = \frac{1}{n(x)} $, where $ n(x) $ is the refractive index, then the time taken is:

\[T[y] = \int_{x_1}^{x_2} \frac{\sqrt{1 + (y')^2}}{v(x, y)} \, dx\]

This is a variational problem where the path of light $ y(x) $ extremizes the travel time.

Example: Snell’s Law

Applying the calculus of variations to two media with different refractive indices leads to:

\[\frac{\sin \theta_1}{v_1} = \frac{\sin \theta_2}{v_2}\]

This is Snell’s law of refraction.

Significance

Foundation of geometrical optics.
Connects variational principles to physical phenomena.
Analogous to least action in mechanics and leads to ray-tracing methods.