📖 What is Calculus of Variations?
Functionals and Variations
Functional is a mapping that takes a function as input and produces a real number as output:
\[
J[y] = \int_{x_1}^{x_2} F(x, y(x), y'(x)) dx
\]
Variational problem: Find a function \(y(x)\) that extremizes the functional \(J[y]\)
Variation \(\delta y\): Infinitesimal change of the function \(y(x)\)
\[
y(x) \to y(x) + \epsilon \eta(x), \quad \eta(x_1) = \eta(x_2) = 0
\]
The condition that the first variation of the functional vanishes gives the extremal condition:
\[
\delta J = \frac{d}{d\epsilon}J[y + \epsilon\eta]\bigg|_{\epsilon=0} = 0
\]
Euler-Lagrange Equation
A function \(y(x)\) that extremizes the functional \(J[y] = \int F(x, y, y') dx\) satisfies the following differential equation:
\[
\frac{\partial F}{\partial y} - \frac{d}{dx}\left(\frac{\partial F}{\partial y'}\right) = 0
\]
This is called the Euler-Lagrange equation.
Physical Significance
- Principle of least action: The motion of a physical system occurs along a path that extremizes the action integral
- Energy minimization: Equilibrium states minimize the energy functional
- Elastic deformation: Deformation of elastic bodies minimizes strain energy
- Shape optimization: Optimize performance functionals in structural design
Summary
- Calculus of variations is a method for finding functions that extremize functionals, with the Euler-Lagrange equation as its foundation
- The brachistochrone curve (curve of fastest descent) is a cycloid and represents a classical application of calculus of variations
- Geodesics are shortest paths on surfaces; on a sphere, they are great circles
- The principle of least action is a fundamental principle of physics and forms the basis of Lagrangian mechanics
- In the isoperimetric problem, the figure that maximizes area for a given perimeter is a circle
- The Galerkin method is a powerful technique for solving partial differential equations in weak form
- The finite element method is based on variational principles and is widely applied to elastic body deformation and shape optimization
- In materials science, energy minimization principles and shape optimization are of practical importance