Constrained Variational Principle

• Constrained variational principle is a generalization of classical variational methods, requiring variations to satisfy physical, geometric, or control constraints.
• It leverages geometric structures like fiber and jet bundles to naturally encode constraints and derive Euler–Lagrange equations with Lagrange multipliers.
• The principle underpins advanced analyses in optimal control, geometric mechanics, and numerical integrators by unifying continuous and discrete formulations.

A constrained variational principle is a generalization of the classical variational principle in which admissible variations (or curves, functions, measures) are required to satisfy explicit constraints—typically expressing physical, geometric, or control-theoretic restrictions on the admissible configurations. The modern mathematical theory encompasses classical holonomic and non-holonomic constraints, pointwise or integral restrictions, and extends to non-smooth, measure-theoretic, and higher-order situations. Constrained variational principles are foundational for the analysis and optimal control of differential systems, geometric mechanics, field theories with gauge symmetries, and numerous applications in mathematical physics, engineering, and optimization.

1. Geometric and Bundle-Theoretic Foundations

Constrained variational problems are naturally formulated in a geometric setting using fiber bundles, jet bundles, and their associated structures. The configuration or "event" space is typically modeled as a fiber bundle $V_{n+1} \rightarrow \mathbb{R}$ with local coordinates $(t, q^1,\ldots,q^n)$; the first jet bundle $j_1(V_{n+1})$ provides the velocity (or tangent) data. Constraints are imposed by selecting a submanifold $A \subset j_1(V_{n+1})$, often defined by local relations

$\dot{q}^i = \psi^i(t, q, z)$

where $z^A$ are additional variables parametrizing the constrained velocities.

The admissible class consists of curves whose first jet prolongations factor through $A$. Jet bundle geometry, contact subbundles, and vertical and horizontal splittings (sometimes via a choice of "infinitesimal control" or discrete connection in the reduction context) encode both the constraints and the allowed variations in an intrinsic, coordinate-free manner (Massa et al., 2010, Massa et al., 2015, Bloch et al., 2018).

2. Formulation and Classification of Constraints

Constraints are classified as:

• Holonomic: constraints on configuration variables, integrable to constraints on the admissible paths;
• Nonholonomic: typically constraints on the velocities or higher derivatives that are not integrable to position constraints (e.g., rolling without slipping, non-integrable Pfaffian conditions);
• Pointwise (local) or global (integral) constraints;
• Equality or inequality constraints.

Nonholonomic constraints, especially those affine in velocities, play a central role in geometric mechanics and are naturally encoded as distributions or subbundles of the tangent (or jet) bundle. In variational formulations, admissible variations must satisfy linearized versions of these constraints, often leading to a linear variational equation along extremal curves (Massa et al., 2015, Massa et al., 2010).

3. Lagrangian and Hamiltonian Approaches with Constraints

Lagrangian Setting

The action functional is defined as the integral of a Lagrangian $L$ over admissible curves respecting the constraints,

$I[\gamma] = \int_{t_0}^{t_1} L(h(t))\,dt$

with $h$ a lift of $\gamma$ to the constraint set $A$. The first variation yields constrained Euler–Lagrange equations, incorporating Lagrange multipliers when the constraints are enforced via analytic relations.

A central geometric advance is the explicit tensorial (covariant) representation of the second variation using adapted Lagrangians obtained via gauge transformations: $L' = L - \frac{dS}{dt}$ for a suitable function $S$, ensuring the essential Hessian (second variation bilinear form) is tensorial and gauges out non-invariant coordinate artifacts (Massa et al., 2010).

Hamiltonian and Contact Bundle Reformulation

Through Legendre transformation and the construction of Pontryagin–Poincaré–Cartan forms, constrained variational problems can be reformulated as (possibly free) Hamiltonian systems on suitable contact bundles, often after reduction via connections or principal bundle techniques: $\Theta = p_i (dq^i - \psi^i dt)$

$H = p_i \psi^i - L$

Stationarity yields Hamilton’s equations with constraints, sometimes involving additional (momentum map) conditions that characterize gauge or diffeomorphism constraints in field theories (Massa et al., 2015, Diez et al., 2018).

4. Minimality, Second Variation, and Covariant Criteria

A central concern is establishing criteria for a constrained extremal (stationary solution) to be a local minimizer. The main results can be summarized:

• Legendre Condition: The matrix $G_{AB}(t)$ of second derivatives of the adapted Lagrangian with respect to the "controlled" (vertical) variables must be positive semi-definite (necessary), or definite (sufficient and necessary for regular extremals).
• Absence of Conjugate Points: By interpreting the second variation in terms of covariant Jacobi fields (solutions to the linearized constrained Euler–Lagrange system), the presence of conjugate points along the interval signals the failure of minimality. Absence of such points is both necessary and sufficient for the constrained minimum (Massa et al., 2010).
• Gauge-Covariance and Essential Hessian: By suitable gauge transformation of the Lagrangian, one obtains canonical, coordinate-independent forms for the second variation that clarify and generalize classical results.

5. Gauge Structure, Reduction, and Pontryagin’s Maximum Principle

Constrained variational principles admit gauge-theoretic reformulations. A gauge-invariant approach replaces the Lagrangian by a section of an affine scalar bundle over the velocity space, absorbing gauge ambiguities (i.e., the invariance under addition of total derivatives) at the bundle level (Bruno et al., 2011). This permits a geometrically natural lifting to a principal bundle and leads to a clean geometric proof of equivalence: $$\text{Constrained Problem in State Space} \quad \Leftrightarrow \quad \text{Free Variational Problem in Extended Affine Bundle}$$ Pontryagin’s maximum principle emerges as a geometric projection: extremals of the augmented (Pontryagin–Poincaré–Cartan) action on the higher-affine bundle project down to extremals of the constrained problem, with a one-to-one correspondence under normality conditions (Bruno et al., 2011, Massa et al., 2015).

This formalism naturally incorporates both holonomic and nonholonomic constraints and unifies variational and optimal control perspectives.

6. Discretization, Numerical Integrators, and Applications

Variational integrators for constrained and higher-order systems are constructed by discretizing the variational principle itself rather than the equations of motion. The discrete action sum

$S_d = \sum_n L_d(q_n, q_{n+1})$

with a discrete Lagrangian $L_d$ approximating the action over intervals, leads (by discrete variation plus suitable handling of constraints and reductions) to symplectic numerical schemes that inherit conservation properties (momentum maps, symplectic structure, energy behavior) from the continuous system (Colombo et al., 2013, Bloch et al., 2018).

The framework extends to higher-order systems, systems with nontrivial principal bundle symmetry (using discrete connections and reduction), and optimal control of underactuated mechanical systems—where control inputs act on a reduced configuration or shape space.

Typical examples include energy-minimum control of electrons in magnetic fields, coupled rigid bodies, cubic spline interpolation on manifolds, and underactuated robotic control (Bloch et al., 2018, Colombo et al., 2013).

7. Broader Implications and Mathematical Developments

The geometric, covariant theory of constrained variational principles achieves several goals:

• It generalizes classical calculus of variations to systems with differential and algebraic constraints, including nonintegrable (nonholonomic) constraints.
• It provides canonical, invariant formulations of the first and second variations, clarifies the global structure of minimizers (via Jacobi fields), and gives precise minimality conditions.
• It connects naturally to geometric control theory, gauge field theory (via covariant momentum maps), and optimal control.
• The discrete extension enables consistent, structure-preserving numerical schemes for complex constrained dynamical systems.
• The methods are readily adaptable to systems with symmetries, reductions, and higher-order variational problems.

This body of theory underpins a rigorous approach to a wide spectrum of control, mechanical, and field-theoretic models, furnishing both the mathematical language and computational tools for modern constrained dynamical analysis (Massa et al., 2010, Bruno et al., 2011, Massa et al., 2015, Bloch et al., 2018).