Calculus of Variations

Definitions

Functional

— a mapping from a space of functions to $\mathbb{R}$: $J: \{y : [a,b] \rightarrow \mathbb{R}\} \rightarrow \mathbb{R}$.

First variation

— $\delta J[y^*; \eta] = \frac{d}{d\epsilon}\Big|_{\epsilon=0} J[y^* + \epsilon\eta]$

Main lemma

— if a continuous $g$ is such that $\int g(x)\eta(x)\,dx = 0$ for all smooth $\eta$ with zero endpoints, then $g \equiv 0$.

Euler-Lagrange equation

— $F_y - \frac{d}{dx}(F_{y'}) = 0$

Problem

— Find the shortest curve in $\mathbb{R}^2$ between points $A = (0, 0)$ and $B = (1, 1)$.

Integrate

— $\frac{y'}{\sqrt{1 + (y')^2}} = C$ (constant). Solve: $y'^2 = C^2(1 + y'^2) \rightarrow y'^2(1 - C^2) = C^2 \rightarrow y' = \frac{C}{\sqrt{1 - C^2}} = \text{const}$.

Conclusion

— $y' = \text{constant} \rightarrow y = a x + b$—a straight line!

Higher derivatives

— $J = \int F(x, y, y', y'')\,dx$. EL: $F_y - \frac{d}{dx} F_{y'} + \frac{d^2}{dx^2} F_{y''} = 0$.

Formulas

Ordinary mathematical analysis seeks the extremum of a function of a number or vector: find $x$ that minimizes $f(x)$. Calculus of variations solves a problem of another level: find a function $y(x)$ that minimizes some “function of a function”—a functional. Such problems arise in physics (“along...

The history of the discipline begins in 1696, when Johann Bernoulli proposed the brachistochrone problem: find the shape of a slide along which a ball rolls between two points in the shortest time. The problem astonished contemporaries: the answer is not a straight line (the shortest distance), b...

Functional—a mapping from a space of functions to $\mathbb{R}$: $J: \{y : [a,b] \rightarrow \mathbb{R}\} \rightarrow \mathbb{R}$.

Here, $F$ is a given “Lagrangian function” of three arguments: $F(x, y, p)$, where $x$ is the independent variable, $y$ is the value of the function, $p = y'$ is the derivative. The functional “sums” the contribution of $F$ along the curve.

Definitions

Second variation

— $\delta^2 J[y^*; \eta] = \frac{d^2}{d\varepsilon^2}\Big|_{\varepsilon=0} J[y^* + \varepsilon\eta]$

Necessary condition for a minimum

— $\delta^2 J[y^*; \eta] \geq 0$ for all admissible $\eta$.

Legendre condition (necessary)

— for a minimum, it is necessary that $P(x) = F_{y'y'}(x, y^*(x), y^{*\prime}(x)) \geq 0$ on the entire segment $[x_0, x_1]$.

Meaning

— $F$ must be “convex in $y'$” along the extremal. If $P(x) < 0$ at even a single point, the extremal is not a minimum.

Strengthened Legendre Condition (sufficient)

— $P(x) > 0$ strictly on the entire $[x_0, x_1]$.

Example

— $J[y] = \int \frac{1}{2}(y'^2 - y^2)\, dx$ (the pendulum). $F = \frac{1}{2}(p^2 - y^2)$. $F_{y'y'} = 1 > 0$—the Legendre condition is satisfied. This alone is not sufficient for a minimum; the Jacobi condition must also be checked.

Quadratic functional

— $\delta^2 J = \int (P\, (\eta')^2 + Q\, \eta^2)\, dx$—this is a functional of $\eta$. For which $\eta$ is it minimal?

Jacobi condition for a minimum

— the conjugate point $\bar{x}$ must not lie in the open interval $(x_0, x_1)$. If $\bar{x} \in (x_0, x_1)$, the extremal is not a minimum.

Functional

— $J[y] = \frac{1}{2}\int_0^T (y'^2 - y^2)\, dx$ (small oscillations of the pendulum, $y$—the angle).

Jacobi equation

— $-u'' - u = 0 \Rightarrow u'' + u = 0$. This is the harmonic oscillator equation!

Conjugate points

— $u(\bar{x}) = \sin(\bar{x}) = 0 \Rightarrow \bar{x} = \pi, 2\pi, 3\pi, ...$

Conclusion

— If $T < \pi$, then the conjugate point $\bar{x} = \pi > T$—the Jacobi condition is satisfied, the extremal is a minimum. If $T > \pi$, then $\bar{x} = \pi \in (0, T)$—the Jacobi condition is violated, the extremal is NOT a minimum.

Physical meaning

— when $T < \pi$, the pendulum has not yet completed a quarter period—the trajectory is optimal. When $T > \pi$, the pendulum has “swung over”—there is a shorter path.

Morse theorem

— connects critical points of the functional (extremals) with the topology of the path space. The number of geodesics between two points on a compact manifold is ≥ the sum of the Betti numbers of the path manifold. This gives lower bounds on the num...

Finite element method (FEM)

— a variant of Ritz's method with piecewise linear basis functions. This is the foundation of ANSYS, COMSOL, and other engineering analysis systems. FEM reduces an infinite-dimensional variational problem to a finite-dimensional system of linear equ...

Formulas

The Euler-Lagrange equation is a first-order condition, the analogue of “derivative equals zero.” Just like in ordinary calculus, a critical point can be a minimum, maximum, or a saddle point. Second-order conditions are needed to determine the type of extremal. This is especially important in op...

Consider the functional $J[y] = \int F(x, y, y')\, dx$ and its second derivative with respect to the parameter $\varepsilon$ under the perturbation $y^* + \varepsilon\eta$:

Second variation: $\delta^2 J[y^*; \eta] = \frac{d^2}{d\varepsilon^2}\Big|_{\varepsilon=0} J[y^* + \varepsilon\eta]$

Calculation (Taylor expansion in $\varepsilon$ up to $\varepsilon^2$ inclusive):

Definitions

Classical formulation

— among all closed curves of a given length L, find the one that bounds the largest area.

Theorem

— for the problem $min\, J[y] = \int F(x,y,y')\, dx$ with constraint $J_2[y] = \int G(x,y,y')\, dx = C$, the extremal is an extremal of the auxiliary functional:

Formulation

— a flexible chain of fixed length $L$ hangs between points $A$ and $B$. Under the influence of gravity, it assumes a shape minimizing potential energy. Find the shape of the chain.

Euler-Lagrange equation

— the functional $H$ does not explicitly depend on $x$ → use the first integral:

Answer

— $y = a\cosh(x/a + b) - \lambda$, where $a$, $b$ are constants from boundary conditions. This is the chain line (catenary).

Physical interpretation

— the shape of the chain is not a parabola (as thought until Huygens and Leibniz)—it is a hyperbolic cosine! The difference is clearly visible for large sags.

Problem

— fence the maximum territory along a straight shoreline, using a rope of length $L$.

This is the theorem on the isoperimetric inequality

— for a curve of length $L$, the area $\leq L^2/(4\pi)$, with equality achieved for the circle.

Application

— in construction, all columns, beams, and thin-walled structures are designed with a safety margin according to Euler's formula. This is a direct application of isoperimetric problems in variational calculus.

Formulas

According to legend, the Phoenician princess Dido fled to North Africa and asked the local chief for as much land as could be encompassed by a bull's hide. She cut the hide into thin strips and fenced a plot with them—taking the straight shoreline as one side. What shape did Dido choose? Naturall...

Classical formulation: among all closed curves of a given length L, find the one that bounds the largest area.

In analytical form: max $J_1[y] = \int_0^a y\, dx$ given $J_2[y] = \int_0^a \sqrt{1+y'^2}\, dx = L$.

This is a functional with a constraint-functional—an analogue of a constrained optimization problem in finite-dimensional space.

Definitions

Lagrange problem

— $\min J = \int F\, dx$ subject to differential constraints $G(x, y, y') = 0$. Motion along a curve with additional constraints.

Mayer problem

— $\min J = g(x_0, y(x_0), x_1, y(x_1))$—we minimize only a boundary function, without an integral. This is a problem with free endpoints.

Derivation

— when varying $\delta J = 0$ with an unfixed right endpoint, a “remainder” arises from integration by parts:

Transversality condition

— $F_{y'} \cdot \varphi_x = (F - y' F_{y'}) \cdot \varphi_y$

Geometric meaning for the geodesic problem

— if we seek the shortest distance from point A to a curve C, the transversality condition means that the extremal is perpendicular to C at the endpoint. This is intuitively evident: the shortest path from a point to a curve is the perpendicular to ...

Problem

— find the shortest curve from point $A = (0, 2)$ to the parabola $C: y = x^2/2$.

Euler–Lagrange equation

— extremals are straight lines $y = a x + b$.

Formulas

In the simplest formulation of the calculus of variations, both endpoints of the curve are fixed: $y(x_0) = y_0$, $y(x_1) = y_1$. However, in many real problems, this is not the case. For example, one may need to find the shortest path from point A to a certain curve C (the endpoint is not fixed,...

Lagrange problem: $\min J = \int F\, dx$ subject to differential constraints $G(x, y, y') = 0$. Motion along a curve with additional constraints.

Mayer problem: $\min J = g(x_0, y(x_0), x_1, y(x_1))$—we minimize only a boundary function, without an integral. This is a problem with free endpoints.

Bolza problem (general form): $\min J = \int F\, dx + g(x_0, y_0, x_1, y_1)$ under additional constraints. It unifies the Lagrange and Mayer problems.

Definitions

Method of solution

— Either explicit parameterization (as with the pendulum), or the method of Lagrange multiplier functions.

Obstacle problem

— find the minimal surface lying above a given obstacle $\psi(x, y)$. The solution satisfies:

Problem

— a particle of mass $m$ is attached by a thread of length $l$ to the point $O = (0, 0)$. It moves under gravity. Find the equations of motion.

Variables

— $x(t), y(t)$ — coordinates. Constraint: $\varphi = x^2 + y^2 - l^2 = 0$ (inextensible string).

Equations with multiplier

— the Lagrange equation with reaction force from the constraint:

Transition to generalized coordinate

— $\theta$ (angle). $x = l \sin\theta$, $y = -l \cos\theta$. $L = ml^2 \dot{\theta}^2/2 + mgl \cos\theta$. Euler-Lagrange: $ml^2\ddot{\theta} = -mgl\sin\theta \rightarrow \ddot{\theta} + (g/l)\sin\theta = 0$ — the pendulum equation. The constraint h...

Formulas

In the real world, the motion of bodies is almost always constrained: a pendulum moves along an arc of a circle, a car rolls without slipping, a fluid flows through a pipe. Such constraints are called connections (constraints). The calculus of variations with constraints is a theory that enables ...

A holonomic constraint imposes a condition of the form φ(x, y(x)) = 0 — this is a functional equation on the curves. It is called this from the Greek “holos” (whole) — the constraint fully determines the configuration.

Example: the mathematical pendulum in Cartesian coordinates. A particle moves in ℝ² but is attached to a fixed point by a string of length $l$. The holonomic constraint: $x^2 + y^2 = l^2$. The system has $2 − 1 = 1$ degree of freedom. In the generalized coordinate θ (angle): $x = l \sin \theta$, ...

Method of solution: Either explicit parameterization (as with the pendulum), or the method of Lagrange multiplier functions.

Definitions

Action

— the functional $S[q] = \int_{t_0}^{t_1} L(t, q, \dot{q}) dt$, where $L = T - U$ is the Lagrangian (kinetic − potential energy), $q = (q_1,...,q_n)$ are generalized coordinates.

Hamilton's principle

— the physical trajectory between $q(t_0)$ and $q(t_1)$ is the extremal of the functional $S$.

Generalized momentum

— $p_i = \frac{\partial L}{\partial \dot{q}_i}$ (conjugate to $q_i$)

Hamiltonian

— $H = \sum_i p_i\dot{q}_i - L$ (Legendre transformation)

Physical meaning

— for conservative systems $H = T + U$ is the total energy. Conservation of $H \leftrightarrow$ does not explicitly depend on time: $\frac{\partial H}{\partial t} = 0$.

Poisson bracket

— $\{f, g\} = \sum_i \left( \frac{\partial f}{\partial q_i} \frac{\partial g}{\partial p_i} - \frac{\partial f}{\partial p_i} \frac{\partial g}{\partial q_i} \right)$

Noether’s theorem (mechanical version)

— each continuous symmetry of the action corresponds to a conservation law:

Model

— three masses $m_1$, $m_2$, $m_1$ (O−C−O) on springs with stiffness $k$.

Formulas

In the 17th century, Newton described mechanics through forces: F = ma. This worked, but required explicit accounting for all forces, including reaction forces of constraints. In the 18th century, Lagrange showed: all mechanics can be expressed through a single variational principle—the principle...

Action: the functional $S[q] = \int_{t_0}^{t_1} L(t, q, \dot{q}) dt$, where $L = T - U$ is the Lagrangian (kinetic − potential energy), $q = (q_1,...,q_n)$ are generalized coordinates.

Hamilton's principle: the physical trajectory between $q(t_0)$ and $q(t_1)$ is the extremal of the functional $S$.

Lagrange equations (from Euler-Lagrange for S): $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0$, $i=1,...,n$

Definitions

Generalized momentum

— $p = \partial L / \partial y'$—the variable conjugate to $y'$.

Hamiltonian

— $H(x, y, p) = p\cdot y' - L(x, y, y')$, where $y'$ is expressed through $p$ from $p = \partial L/\partial y'$.

Canonical equations

— $\dot{y} = \partial H / \partial p$, $\dot{p} = -\partial H / \partial y$.

Example

— $L = (1/2)y'^2 - U(y)$ (one-dimensional mechanics). $p = y'$. $H = p\cdot y' - (1/2)y'^2 + U = (1/2)p^2 + U$. Canonical equations: $\dot{y} = p$, $\dot{p} = -U'(y)$. This is Newton's equation!

Jacobi’s Theorem

— along an extremal, $p = \partial S / \partial y$ (momentum = partial derivative of the action with respect to the configuration).

Geometric meaning

— level surfaces $S(x, y) = C$ are called "wavefronts." Extremals ("trajectories," "rays") are perpendicular to the wavefronts—they point along $\nabla S$.

Fermat’s principle

— light travels along the path that minimizes travel time. The functional: $J = \int n(x, y)\, ds$, where $n$ is the refractive index, $ds$ is the element of length.

Eikonal equation

— $|\nabla S|^2 = n^2(x, y)$. This is a special case of HJ for optics.

Quantum mechanics

— the wave function $\psi = \exp(iS/\hbar)$. Schrödinger’s equation as $\hbar\to0$: $[(\partial S/\partial t) + H(x, \nabla S)]\,\psi \to 0$. This is the HJ equation! Quantum interference effects are a consequence of finite $\hbar$.

Problem

— free oscillations of a pendulum. $H = p^2/(2ml^2) + mgl(1 - \cos\theta)$.

Period of oscillations

— $T = \partial S/\partial E = 2 \oint d\theta / |p/ml^2| = 2ml^2 \oint d\theta/\sqrt{2ml^2(E-mgl(1-\cos\theta))}$.

Formulas

In the 1820s, William Hamilton noticed a striking analogy: the mathematics of geometrical optics (the path of a light ray) formally coincides with that of classical mechanics. This led him to create a unified formalism—the Hamilton–Jacobi equation. Later, in 1926, Schrödinger showed that quantum ...

Generalized momentum: $p = \partial L / \partial y'$—the variable conjugate to $y'$.

Hamiltonian: $H(x, y, p) = p\cdot y' - L(x, y, y')$, where $y'$ is expressed through $p$ from $p = \partial L/\partial y'$.

Canonical equations: $\dot{y} = \partial H / \partial p$, $\dot{p} = -\partial H / \partial y$.

Definitions

Monge’s problem

— Find a mapping T: X → X (“transport plan”), mapping μ to ν (that is, T_#μ = ν — the “pushforward” of μ under T equals ν), minimizing the total cost:

The formulation problem of Monge

— the mapping T is a “deterministic” plan, but sometimes it is optimal to “split” a unit of material. For example, part of the earth from location x₁ goes to hole y₁, part — to y₂. Monge’s plan does not allow for this.

Kantorovich's dual problem

— by the LP duality theorem:

Properties

— W_p is a true metric on the space of probability measures. Convergence in W_p is equivalent to weak convergence + convergence of the p-th moment.

Wasserstein space

— P_p(ℝⁿ) with metric W_p is the “space of shapes”. Interpolation between μ and ν by W₂ is “optimal morphing” between two distributions.

Meaning

— the optimal transport plan is a “shift along the gradient of a convex function”. This is striking: from the earth-moving problem, convexity arises!

Brenier-McCann polar decomposition

— any mapping T: ℝⁿ → ℝⁿ that preserves measures (T_#μ = ν) decomposes as T = R ∘ ∇φ, where R is a rotation (distance-preserving), ∇φ is optimal transport.

WGAN (Wasserstein GAN)

— training generative neural networks. The regular GAN uses KL-divergence, which is unstable for disjoint distributions. WGAN replaces it with W₁:

Point Cloud Matching

— to match two sets of points {xᵢ} ~ μ and {yⱼ} ~ ν with minimal cost. The Sinkhorn algorithm is a fast iterative method for approximate solution of the Kantorovich problem with entropic regularization.

Morph between images

— averaging images in Wasserstein space. Wasserstein barycenter: min_{ν} Σᵢ wᵢ W₂²(μᵢ, ν). The result is a “mean” image preserving geometry.

Problem

— μ = 0.5 δ₀ + 0.5 δ₂ (mass 0.5 at points 0 and 2), ν = 0.5 δ₁ + 0.5 δ₃ (mass 0.5 at points 1 and 3). Cost c(x,y) = |x−y|².

Admissible plans

— γ is defined by a 2×2 matrix with row sums (0.5, 0.5) and column sums (0.5, 0.5):

Cost

— C(γ) = a|0−1|² + (0.5−a)|0−3|² + (0.5−a)|2−1|² + a|2−3|² = a + 9(0.5−a) + (0.5−a) + a = 4.5 − 8a.

W₂

— √C = √0.5 ≈ 0.707.

Formulas

In 1781, the French mathematician Gaspard Monge posed the question: how can a pile of earth (density μ) be moved into a hole (density ν) with minimal labor costs? The problem turned out to be incredibly difficult and remained unsolved for 160 years. In 1942, the Soviet mathematician Leonid Kantor...

Given two measures μ and ν on a space X (for example, probability distributions on ℝⁿ) and a cost function c(x, y) ≥ 0 (the cost of moving a “unit of material” from x to y).

Monge’s problem: Find a mapping T: X → X (“transport plan”), mapping μ to ν (that is, T_#μ = ν — the “pushforward” of μ under T equals ν), minimizing the total cost:

The formulation problem of Monge: the mapping T is a “deterministic” plan, but sometimes it is optimal to “split” a unit of material. For example, part of the earth from location x₁ goes to hole y₁, part — to y₂. Monge’s plan does not allow for this.

Definitions

On the plane

— $g_{ij} = \delta_{ij} \rightarrow \Gamma^k_{ij} = 0 \rightarrow$ equation: $d^2 x^k/ds^2 = 0 \rightarrow x^k = a^k s + b^k$ (straight lines!).

Covariant acceleration

— $\nabla_{\gamma'} \gamma' = \frac{d^2 x^k}{ds^2} + \Gamma^k_{ij} \dot{x}^i \dot{x}^j$.

Intuition

— on a curved surface, a “straight line” is a curve along which you “walk straight” (not turning). If you draw a straight line on a sheet of paper and roll the paper into a cylinder, the drawn straight line becomes a geodesic (helical curve or strai...

Second variation of length

— $\delta^2 J = \int [|J'|^2 - K(J, \gamma', J, \gamma')] ds$, where $K$ is Gaussian curvature, $J$ is the Jacobi field.

Jacobi–Bonnet theorem

— a geodesic is the shortest curve up to the first conjugate point.

Example

— on a sphere of radius $R$, geodesics are great circles. For the point $A = $“north pole,” the conjugate point is the “south pole” (at distance $\pi R$). The arc of a great circle from the north pole to the south is the shortest. But if you continu...

Physical meaning

— a soap film assumes a minimal surface shape—when there is zero pressure difference, surface tension forces balance at every point. This is indeed $H = 0$.

Plateau's problem

— does a minimal surface exist with a given boundary curve $\Gamma$?

Problem

— find a surface of revolution of minimal area connecting two circles $x^2 + z^2 = r_0^2$ at $y = \pm h$.

Functional

— $\text{Area} = 2\pi \int_{-h}^h r(y)\sqrt{1 + r'(y)^2}\, dy$.

Euler–Lagrange equation

— $\frac{d}{dy}\left[\frac{r\, r'}{\sqrt{1 + r'^2}}\right] - \sqrt{1 + r'^2} = 0$.

Physical application

— a soap film between two rings assumes the shape of a catenoid. When separating the rings: if the distance exceeds a critical value, the film “breaks” into two disks. This is a bifurcation of minimal surfaces!

Formulas

Differential geometry and variational calculus are closely intertwined: the key geometric objects are defined as extremals of functionals. Geodesics—the shortest curves—are extremals of the length functional. Minimal surfaces are extremals of the area functional. This connection is profound: unde...

A geodesic on a surface is a curve minimizing the length between two points (locally). Formally: an extremal of the functional $J[\gamma] = \int |\gamma'(t)| dt$.

where $s$ is arc length, $\Gamma^k_{ij}$ are Christoffel symbols—computed from the metric tensor $g_{ij}$:

$ \Gamma^k_{ij} = \frac{1}{2} g^{kl} (\partial_i g_{jl} + \partial_j g_{il} - \partial_l g_{ij}) $

Definitions

Optimality condition

— for the optimal shape, $j(x) = \text{const}$ on $\partial\Omega$ (or equal to the Lagrange multiplier under an isoperimetric constraint).

Example

— For a diffusion problem ($-\Delta u = f$ in $\Omega$, $u = 0$ on $\partial\Omega$), $J = \int_\Omega u \, dx$: the shape derivative $\frac{dJ}{dt} = -\int_{\partial\Omega} \left( \frac{\partial u}{\partial n} \right)^2 V \cdot n \, dS$. Optimality...

Density approach

— $\rho(x) \in [0, 1]$—"material density" at each point. $\rho = 1$: material present. $\rho = 0$: void.

Task

— $\min_{\rho \in [0,1]} \int_\Omega \rho \, d\Omega$ (material volume) subject to compliance constraints (structure stiffness).

Problem

— $J$ depends on $u$, which depends on $\Omega$. The gradient $\frac{dJ}{d\Omega}$ is needed for optimization, but brute-force calculation requires $N$ computations for $N$ shape parameters.

Adjoint method

— introduce an adjoint field $p$ (solution of the "reverse" problem), then:

Application

— Airbus uses the adjoint method to optimize A380 wing profiles. NASA optimizes turbine blades. Fuel savings—2–5% due to optimal shaping.

Initial state

— $\rho = 0.5$ everywhere. Stiffness is uniform.

After 5 iterations

— "struts" appear from the supports to the loaded point, material flows out of unloaded regions.

After 50 iterations

— black-and-white structure. Triangular struts stretching from the load center to the anchor points. Looks like a bridge truss!

Result

— stiffness increased by 3.5 times at the same mass compared to uniform distribution. Such a design is impossible without topological optimization.

Formulas

An engineer is designing a load-bearing beam of an aircraft wing. Requirements: withstand a given load and weigh as little as possible. Where to remove material? The shape of the wing is optimized precisely as the solution of a variational problem—to minimize the volume (mass) under strength cons...

Given a domain $\Omega \subset \mathbb{R}^n$—the "body"—with boundary $\Gamma = \partial\Omega$. A partial differential equation (deformation, flow, heat) is solved on $\Omega$. The problem: find a shape $\Omega$ minimizing a functional $J(\Omega)$.

A challenge: the region $\Omega$ is an infinite-dimensional object. How to take a "derivative" with respect to the shape?

A family of domains: $\Omega_t$, deformed along a velocity vector $V(x)$ on the boundary:

Definitions

Statement

— a body Ω is in equilibrium under the action of body forces f and surface loads t on a part ∂Ω_t.

Principle of virtual displacements

— a body is in equilibrium if and only if the total virtual work is zero for any admissible virtual displacement δu:

Total potential energy

— Π[u] = ∫_Ω W(ε(u)) dV − ∫_Ω f·u dV − ∫_{∂Ω_t} t·u dS

Principle

— among all admissible displacement fields (satisfying kinematic boundary conditions), the equilibrium field minimizes Π.

Idea

— partition Ω into finite elements Ωₑ (triangles, tetrahedra). Within each element, approximate u by polynomial shape functions Nᵢ(x): u ≈ Σᵢ uᵢ Nᵢ(x), where uᵢ are nodal displacements.

Substitute into Π

— Π[u] = Π(u₁,...,uN)—a function of N real numbers.

Dimension

— N = 3 × (number of nodes). For an aircraft model—millions of degrees of freedom. ANSYS solves such systems within hours.

Castigliano's principle

— for structures with known forces—the displacement at the point of force application equals the derivative of the complementary energy with respect to that force. Simple for trusses and beams.

Problem

— a beam of length L, clamped at x=0 (fixed: u(0)=u'(0)=0), the free end loaded by a force P at x=L. Find the deflected shape of the beam.

Functional

— Π[u] = ∫₀ᴸ (EI/2)(u'')² dx − P·u(L)

Euler-Lagrange equation

— ∂Π/∂(δu) = 0 → EI u'''' = 0 on (0, L).

Boundary conditions

— u(0) = u'(0) = 0 (clamping), EI u''(L) = 0, EI u'''(L) = P.

Solution

— u'''' = 0 → u = ax³ + bx² + cx + d. From BC: d = 0, c = 0, 6aL + 2b = 0, 6aEI = P.

Deflection at the end

— u(L) = PL³/(3EI)—the standard formula of structural mechanics!

Models

— neo-Hookean (W = μ(I₁−3)/2), Mooney–Rivlin (W = C₁(I₁−3) + C₂(I₂−3)). Used in biomechanics (heart, blood vessel, skin modeling), rubber products, soft robotics.

Formulas

Continuum mechanics is the theory of deformable bodies: beams, plates, fluids, rubbers. Its equations (of equilibrium, motion) are derived from variational principles. This is not merely a mathematical convenience: the variational formulation directly generates the finite element method—the main ...

Statement: a body Ω is in equilibrium under the action of body forces f and surface loads t on a part ∂Ω_t.

Principle of virtual displacements: a body is in equilibrium if and only if the total virtual work is zero for any admissible virtual displacement δu:

This is the "weak" formulation of the equilibrium equations. The "strong" form follows from integration by parts: ∂ⱼσᵢⱼ + fᵢ = 0 in Ω, σᵢⱼnⱼ = tᵢ on ∂Ω_t.

Definitions

Invariance

— $S[\bar{y}] = S[y]$ for all admissible $y$ — a condition on $L$ and $(\xi, \eta)$.

Formulas

In 1915, Emmy Noether proved a theorem that physicists call one of the greatest achievements of mathematical physics of the 20th century. The theorem asserts: every continuous symmetry of a physical system corresponds to a conservation law. Is space homogeneous? → momentum is conserved. Is time h...

A symmetry of the action $S[y] = \int L(x, y, y')\,dx$ is a one-parameter group of transformations $(x, y) \to (\bar{x}(x, y, \varepsilon), \bar{y}(x, y, \varepsilon))$ as $\varepsilon \to 0$, not changing the value of $S$.

Infinitesimal transformation: $\bar{x} = x + \varepsilon\xi(x, y) + O(\varepsilon^2)$, $\bar{y} = y + \varepsilon\eta(x, y) + O(\varepsilon^2)$.

Invariance: $S[\bar{y}] = S[y]$ for all admissible $y$ — a condition on $L$ and $(\xi, \eta)$.