The calculus of variations is a powerful mathematical tool for finding optimal functions. It's used to solve problems like finding the shortest path between two points or the shape of a hanging chain.

At its core, the calculus of variations uses the Euler-Lagrange equation to find stationary points of functionals. This leads to solutions for various optimization problems in physics, engineering, and economics.

Calculus of Variations

Euler-Lagrange equations derivation

• Provide necessary conditions for a function to be a stationary point of a functional (mapping from vector space to real numbers)
• Consider functional $J[y] = \int_{x_1}^{x_2} F(x, y(x), y'(x)) dx$ and assume $y(x)$ is a stationary point
• Introduce small variation $\epsilon \eta(x)$ around $y(x)$ and expand $J[y + \epsilon \eta]$ using Taylor series
• Set first-order term to zero, integrate by parts, and apply fundamental lemma of calculus of variations
• Resulting Euler-Lagrange equation: $\frac{\partial F}{\partial y} - \frac{d}{dx} \frac{\partial F}{\partial y'} = 0$
• Express balance between change in $F$ with respect to $y$ and change in $F$ with respect to $y'$
• Solutions make functional $J[y]$ stationary (minimum, maximum, or saddle points)

Variational problem solutions

• Identify functional $J[y] = \int_{x_1}^{x_2} F(x, y(x), y'(x)) dx$ to be minimized or maximized
• Write Euler-Lagrange equation: $\frac{\partial F}{\partial y} - \frac{d}{dx} \frac{\partial F}{\partial y'} = 0$
• Solve differential equation for $y(x)$ subject to boundary conditions
• Verify solution satisfies sufficient conditions for optimality (if applicable)
• Examples:
  • Geodesics: Find shortest curve connecting two points on a surface
  • Brachistochrone: Find curve of fastest descent between two points under gravity
  • Isoperimetric problems: Maximize or minimize quantity subject to perimeter or area constraint

Advanced Topics in Variational Calculus

Legendre transform in variations

• Convert between Lagrangian (generalized coordinates and velocities) and Hamiltonian (generalized coordinates and momenta) formulations
• Legendre transform definition: Given $f(x)$, transform $g(p) = \sup_x (px - f(x))$
• Relates Lagrangian $L(q, \dot{q}, t)$ and Hamiltonian $H(q, p, t)$
• Define generalized momentum $p = \frac{\partial L}{\partial \dot{q}}$
• Hamiltonian given by Legendre transform of Lagrangian: $H(q, p, t) = p\dot{q} - L(q, \dot{q}, t)$
• Hamilton's equations of motion: $\dot{q} = \frac{\partial H}{\partial p}$ and $\dot{p} = -\frac{\partial H}{\partial q}$
• Advantages: Symmetric treatment of coordinates and momenta, rich mathematical structure (symplectic geometry, Poisson brackets)

Optimality conditions for variations

• Ensure solution to Euler-Lagrange equations is minimum or maximum of functional
• Second variation $\delta^2 J[y]$: Second-order term in Taylor expansion of $J[y + \epsilon \eta]$
  • For minimum (maximum), must be non-negative (non-positive) for all admissible variations $\eta(x)$
• Legendre condition: Necessary condition for second variation to be non-negative (non-positive)
  • For minimum: $\frac{\partial^2 F}{\partial y'^2} \geq 0$
  • For maximum: $\frac{\partial^2 F}{\partial y'^2} \leq 0$
• Jacobi's condition: Strengthened Legendre condition requiring absence of conjugate points along solution curve
  • Conjugate points: Points where solution to Jacobi equation (linearized Euler-Lagrange equation) vanishes
  • Presence indicates solution is not strict minimum or maximum
• Weierstrass-Erdmann corner conditions: Apply when solution curve has corners (discontinuities in first derivative)
  • At corner: $F - y' \frac{\partial F}{\partial y'}$ and $\frac{\partial F}{\partial y'}$ must be continuous