Variational Problems with Constraints and Connections

Constrained Systems in Physics and Mechanics

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 one to systematically work with such problems. The distinction between holonomic (final) and non-holonomic (differential) constraints is fundamental and has deep physical consequences.

Holonomic Constraints

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): $x = l \sin \theta$, $y = −l \cos \theta$; the constraint disappears!

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

Non-holonomic Constraints

A non-holonomic constraint imposes a differential condition: $g(x, y, y') = 0$. It cannot be “integrated” to a final relation — it imposes a restriction on velocities, not on configurations.

Classical example: rolling a coin without slipping. A coin rolls on a plane without slipping. The velocity of the contact point $= 0$: $\dot{x} - r \dot{\theta} \cos\varphi = 0$, $\dot{y} - r \dot{\theta} \sin\varphi = 0$. These are two differential conditions on four variables. They cannot be reduced to a smaller number of configurational variables!

For the problem $\min J = \int F , dx$ under $g(x, y, y') = 0$, we introduce a Lagrange multiplier function $\lambda(x)$:

$ \int [F + \lambda(x)g(x, y, y')] , dx $

and the Euler-Lagrange equation for variation with respect to $y$ gives a system determining $y(x)$ and $\lambda(x)$.

Carathéodory’s Theorem on Complete Variational Problems

General formulation with $n$ variables $y_i(x)$ and $m$ constraints $G_j(x, y, y') = 0$:

Necessary conditions for optimality:

Euler-Lagrange equations (modified): $F_{y_i} - \frac{d}{dx} F_{y_i'} + \sum_j \lambda_j(x) (G_j){y_i} - \frac{d}{dx} \sum_j \lambda_j (G_j){y_i'} = 0$
Constraints: $G_j(x, y, y') = 0$ along the trajectory
Transversality condition at free endpoints
Legendre-Jacobi conditions for sufficiency

Variational Inequality

In problems with one-sided constraints ($y \geq 0$ — “obstacle” from below), the admissible set is not a linear space but a convex cone.

The extremality condition: the variational inequality:

$ \delta J[y^;\eta - y^] \geq 0 \quad \text{for all admissible } \eta $

If $y^$ is in the interior → $\delta J = 0$ (usual Euler-Lagrange equation). If $y^$ is on the boundary ($y^* = 0$) → the inequality is strict.

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

$ -\Delta u \geq 0 \text{ in } \Omega, \quad u \geq \psi \text{ in } \Omega, \quad (-\Delta u)(u - \psi) = 0 $

This is a system of nonlinear constraints — a strong inequality. It is applied in financial mathematics (the price of an American option) and the theory of elastoplasticity.

Complete Analysis: The Pendulum via the Lagrange Multiplier Method

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).

Lagrangian: $L = T - U = m(\dot{x}^2 + \dot{y}^2)/2 - mgy$. Action: $S = \int L, dt$.

Equations with multiplier: the Lagrange equation with reaction force from the constraint:

$ m\ddot{x} = 2\lambda x, \quad m\ddot{y} = 2\lambda y - mg, \quad x^2 + y^2 = l^2 $

Twice differentiating the constraint: $x\ddot{x} + y\ddot{y} + \dot{x}^2 + \dot{y}^2 = 0$.

Substituting: $x(2\lambda x/m) + y(2\lambda y/m - g) + v^2 = 0 \rightarrow 2\lambda l^2/m - g y + v^2 = 0 \rightarrow \lambda = m(gy - v^2)/(2l^2)$.

Reaction from the string: $T = -2\lambda = m(v^2 - g y)/l^2$ (centripetal acceleration plus normal weight).

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 has disappeared, $\lambda$ is eliminated!

Lagrange Constraints vs Non-holonomic Constraints

In variational problems with constraints, two types of connections are distinguished:

Holonomic: $\varphi(x, y) = 0$ — a constraint only on coordinates. It can be explicitly solved and substituted, reducing the dimensionality.
Non-holonomic: $\varphi(x, y, y') = 0$ — the constraint includes derivatives and cannot, in general, be integrated. Classic example: rolling of a sphere without slipping (5 holonomic + 2 non-holonomic constraints). These constraints require fundamentally different methods: Lagrange multipliers are applicable, but must be used with caution.

Hamilton-Pontryagin Principle

For problems with constraints on the control $u \in U$, Pontryagin (1956) formulated the maximum principle: the optimal control at each instant must maximize the “Hamiltonian” $H = p^\top f - F$. This is a relaxed version of the Euler-Lagrange equation, allowing discontinuous control (“bang-bang”). The maximum principle is the standard tool in optimal control, replacing the calculus of variations in the presence of control constraints.

Numerical Methods

Problems with constraints require specialized numerical methods:

Direct NLP transformation: discretization of states and controls, adding constraints as nonlinear conditions, solution via IPOPT, SNOPT
Projection method: after each gradient descent step, project the solution onto the set of admissible functions
Augmented Lagrangian: penalty function plus multipliers — robust solution of problems with active constraints

Applications

Robotics: motion of manipulators with mechanical connections (joints, contacts)
Transportation planning: accounting for constraints on acceleration, jerk, turning angle
Aerospace industry: trajectory optimization with constraints on thrust, heating, overloads
Biomechanics: modeling human gait as energy optimization under anatomical joint constraints
Chemical process control: constraints on temperature, pressure, reagent concentration — a standard formulation of a variational problem with active connections