Variational Calculus

Variational Calculus and Functional Optimisation

Introduction to Functional Optimisation

Functional optimization deals with finding functions (rather than finite-dimensional vectors) that optimize a functional.

A functional is a mapping from a space of functions to real numbers: $L:{\mathbb{R}}^{\infty }\to \mathbb{R}$ (technically, $H^1\to \mathbb{R}$).

Concept of Functional and Variational Calculus

• In classical (discrete) optimisation, we minimize a function of $D$ variables
• In variational calculus, we consider a continuum of variables $y(x)$, where $x$ is continuous – it can be seen as a limit $D\to \infty$

Variation of a Functional

For a function $y(x)$ and a “direction” $\delta y(x)$ (which is another function), the variation of objective function is:

$\delta L(y;\delta y):=L(y+\epsilon \delta y)-L(y)$

The “derivative” of this variation should be zero for all admissible $\delta y$:

${\lim }_{\epsilon \to 0}\frac{\delta L(y;\delta y)}{\epsilon }=0\forall \text{ admissible }\delta y(x)$

Euler-Lagrange Equation

For a functional of the form:

$L(y)={\int }_a^{b}f(y,y^{\prime },x)dx$

The necessary condition for an extremum is given by the Euler-Lagrange equation:

$\frac{\partial f}{\partial y}-\frac{d}{dx}\left(\frac{\partial f}{\partial y^{\prime }}\right)=0$

Example: The Brachistochrone Problem

Problem Statement: Given two points A and B, find the path joining these points such that a bead sliding down under gravity travels from A to B in the shortest time.

Objective Function: $L(y)={\int }_A^B\sqrt{\frac{1+(dy/dx{)}^2}{2gy}}dx$

This is a classic problem that can be solved using the Euler-Lagrange equation. The solution is a cycloid.

Example: Minimum Distance Path

Objective Function: $L(y)={\int }_A^B\sqrt{1+y^{\prime 2}}dx$

Here:

• $f(y,y^{\prime },x)=\sqrt{1+y^{\prime 2}}$
• $\partial f/\partial y=0$
• $\partial f/\partial y^{\prime }=\frac{y^{\prime }}{\sqrt{1+y^{\prime 2}}}$

Applying the Euler-Lagrange equation yields $y^{\prime }=\beta$ (constant), so the solution is a straight line.

Key Points about Functional Optimisation

1. Instead of algebraic equations, we get differential equations as the first-order necessary conditions
2. These differential equations, along with boundary conditions, need to be solved to find the extrema
3. Finite element method (FEM) is based on variational calculus (minimizing total system energy)
4. Lagrange multiplier and KKT conditions extend to the continuum limit of variational calculus
5. Contact and plasticity in FEM (constraints) also lead to KKT conditions