SYSTEMATIC MATHEMATICS

Calculus of Variations

Ordinary calculus chooses the best number; this course asks how to choose the best whole curve, surface profile, field, or controlled path. Begin with a concrete question: among nearby curves with fixed endpoints, which change makes length or energy go up or down? A variation is introduced as one small, named change to an entire function before any Euler–Lagrange formula appears. From that picture the course derives boundary conditions, constraints, symmetry, second-order minimum tests, existence methods, weak formulations, optimal control, and numerical approximation. An equation that is merely necessary is never called a minimum without the additional evidence.

Before this course: Completed Real Analysis, Measure & Lebesgue Integration, Multivariable Calculus, Linear Algebra, Differential Equations & Dynamical Systems, Partial Differential Equations, Vector Calculus, and Numerical Analysis. Learners should be comfortable with rigorous limits, integration, norms and inner products, boundary-value problems, quadratic forms, weak PDE formulations, and basic finite elements. Weak derivatives, weak convergence, subsequence extraction, and weak lower semicontinuity are rebuilt inside this course rather than assumed.

COURSE FACTSLevel, chapters, units, prerequisite, and outcome
Chapter 1

Functionals and admissible function spaces

Objective: Why must the norm or topology on X be declared?

A functional assigns a number to an entire curve or field; its domain must encode smoothness and boundary data before “nearby” functions or extrema have meaning.

Read the admissible family before differentiating: A functional J:X→R maps a function u in an admissible set X to a scalar; a typical integral functional is J[u]=∫_a^bL(x,u,u′)dx with fixed, free, or constrained traces. The result to establish is: If X is convex and L(x,y,p) is convex in (y,p), then J is convex on X; if strict convexity acts on every nonzero admissible difference, a minimizer is unique. Keep the topology, endpoint traces, constraints, regularity, and allowed signs of each variation attached to every step.

The worked question is: For J[u]=∫₀¹(u′)²dx, compare u=x and u=x² under endpoints u(0)=0,u(1)=1. Begin with “Both functions satisfy the endpoint conditions.” and finish with “For u=x², J=∫₀¹4x²dx=4/3.” Retain every integration-by-parts boundary term, then decide whether the route proves only stationarity or an actual minimum.

If X is convex and L(x,y,p) is convex in (y,p), then J is convex on X; if strict convexity acts on every nonzero admissible difference, a minimizer is unique.

For u,v∈X and 0≤t≤1, convexity of X makes tu+(1−t)v admissible.

Apply pointwise convexity of L to the values and derivatives of this combination.

Integrate the pointwise inequality; strict inequality for nonidentical functions gives uniqueness.

For J[u]=∫₀¹(u′)²dx, compare u=x and u=x² under endpoints u(0)=0,u(1)=1.

  1. Both functions satisfy the endpoint conditions.
  2. For u=x, J=∫₀¹1dx=1.
  3. For u=x², J=∫₀¹4x²dx=4/3.

Result: Within these two admissible candidates, u=x has smaller energy 1 than 4/3; this comparison alone does not yet prove global minimality.