From variational principles to geometry (2509.22171v1)

Published 26 Sep 2025 in math-ph and math.MP

Abstract: A method to construct a geometric structure with the same solutions as a given variational principle is presented. The method applies to large families of variational principles. In particular, the known results that assign cosymplectic geometry to Hamilton's principle and cocontact geometry to Herglotz's principle for regular Lagrangians are recovered. The unified Lagrangian-Hamiltonian formalism is also recovered via the absorption of the holonomy conditions. The method is applied to singular time-dependent Lagrangians, proving that they can always be described with a (pre)cosymplectic structure, although it is not always given by the Lagrangian $2$-form. When applied to singular action-dependent Lagrangians, the method does not always lead to (pre)cocontact geometry. In these cases, the resulting geometry associated with the Herglotz's variational principle is new.

Summary

The paper introduces a unified framework that associates geometric structures to a wide range of variational problems, including those with constraints and singular Lagrangians.
It recovers established correspondences like cosymplectic and cocontact geometries while extending to cases where traditional frameworks are inadequate.
It proposes novel methods for dynamical equivalence and classification, with implications for nonholonomic mechanics and field theories.

Geometric Structures from Variational Principles: A Unified Framework

Introduction

This paper develops a systematic method for associating geometric structures to variational principles, extending the reach of geometric mechanics beyond classical symplectic and contact frameworks. The approach is designed to handle a broad class of variational problems, including those with constraints and singular Lagrangians, and provides a unified language for comparing and classifying dynamical systems via their underlying geometry. The method recovers known correspondences—such as cosymplectic geometry for time-dependent Hamiltonian systems and cocontact geometry for Herglotz-type action-dependent systems—and generalizes to cases where standard geometric structures fail, notably for singular Lagrangians.

Geometric Variational Problems: Formalization

The paper introduces the notion of a geometric variational problem (GVP), defined by a triple $(\Omega, J, \text{Adm})$ on a fiber bundle $\pi: E \to \mathbb{R}$ , where $\Omega$ is a (not necessarily closed) 2-form, $J$ is a set of constraints (functions or forms), and $\text{Adm}$ is a set of admissible variations (vector fields). The solutions are sections $\gamma$ of $\pi$ satisfying both the constraints and the vanishing of the pullback of $i_\xi \Omega$ for all $\xi \in \text{Adm}$ .

A key technical result is that, under $B(\pi)$ -linearity of $\text{Adm}$ , the variational equations can be reformulated in terms of geometric data, allowing the use of the fundamental lemma of the calculus of variations to pass from integral to pointwise conditions.

Reduction and Absorption of Constraints

The method proceeds in two main steps:

Reduction of Admissible Variations: For variational problems with constraints (transversality, nonholonomic, vakonomic, or mixed), the admissible variations are systematically reduced to vertical vector fields, and the constraints are encoded into the geometric structure via modified 2-forms. For example, nonholonomic constraints are incorporated by adding terms of the form $\sigma_\alpha \wedge \eta^\alpha$ to the original 2-form, where $\eta^\alpha$ are constraint 1-forms and $\sigma_\alpha = i_{R_\alpha} \Omega$ for suitable vector fields $R_\alpha$ .
Absorption of Constraints: Constraints are further absorbed into the geometry by extending the phase space with auxiliary variables (e.g., Lagrange multipliers or momenta), yielding an unconstrained GVP where all dynamical information is encoded in the 2-form. This process generalizes the Skinner-Rusk unified formalism and is applicable to both holonomic and nonholonomic constraints.

Recovery of Known Geometric Structures

The framework recovers standard geometric structures in classical mechanics:

Symplectic and Cosymplectic Geometry: For regular time-dependent Lagrangians, the method yields cosymplectic structures $(\omega, \tau)$ , with $\omega$ derived from the Lagrangian and $\tau$ the time 1-form. The equations of motion correspond to those of cosymplectic Hamiltonian systems.
Contact and Cocontact Geometry: For action-dependent (Herglotz-type) Lagrangians, the method produces cocontact structures, with the dynamics governed by cocontact Hamiltonian equations. The energy dissipation law is naturally encoded via the geometric formalism.
Unified Lagrangian-Hamiltonian Formalism: Absorbing holonomy constraints leads to the Skinner-Rusk formalism, where the phase space is extended to include both configuration and momentum variables, and the dynamics are governed by a single 2-form.

Singular Lagrangians and Generalized Geometries

A significant contribution is the treatment of singular Lagrangians, where standard geometric structures (precosymplectic, precontact) may not exist or fail to capture the dynamics. The method constructs alternative 2-forms that always admit a Reeb vector field, ensuring the existence of a geometric description for any Lagrangian. For singular action-dependent Lagrangians, the resulting geometry may not be precontact or precocontact; instead, the dynamics are encoded in a more general structure, often locally conformally symplectic (l.c.s.) or multicontact, depending on the regularity and the nature of the constraints.

The paper provides explicit constructions and local coordinate expressions for these generalized structures, demonstrating that the equations of motion (e.g., Herglotz-Euler-Lagrange equations) can always be derived geometrically, even when the standard contact or cosymplectic framework fails.

Equivalence and Classification of Dynamical Systems

The framework introduces precise notions of dynamical equivalence, subsystems, extensions, and restrictions between variational problems, formalized via bundle maps and the correspondence of solution sets. This enables a rigorous comparison and classification of dynamical systems based on their geometric structures, and clarifies the relationships between different formulations (e.g., Lepage equivalence, natural transformations).

Implications and Future Directions

The method provides a powerful tool for the geometric analysis of dynamical systems, with several implications:

Generalization to Field Theories: The approach is compatible with multisymplectic and multicontact extensions, suggesting applicability to classical field theories and nonconservative systems.
Constraint Algorithms and Singular Systems: The framework accommodates constraint algorithms for singular systems, ensuring that the correct geometric structure is identified even when standard forms are degenerate.
Nonholonomic Mechanics and Distributional Approaches: The method highlights the role of distributions and suggests further generalization to $k$ -contact and multicontact geometries, potentially unifying the treatment of nonholonomic and vakonomic constraints.
Locally Conformally Symplectic and Multicontact Structures: The identification of l.c.s. and multicontact structures in the context of singular action-dependent Lagrangians opens new avenues for the geometric paper of dissipative and nonconservative systems.
Uniqueness and Classification: The lack of uniqueness in the choice of geometric structure (e.g., dependence on the choice of Reeb vector field) points to deeper questions about the intrinsic geometry of dynamical systems and the classification of equivalent structures.

Conclusion

This paper presents a comprehensive and constructive method for associating geometric structures to variational principles, encompassing a wide range of dynamical systems, including those with constraints and singularities. The approach unifies and generalizes existing geometric frameworks, recovers known results, and provides new insights into the geometry of singular and action-dependent systems. The implications for the classification, analysis, and extension of geometric mechanics are substantial, and the method lays the groundwork for further developments in the geometric paper of dynamical systems, including field theories and nonholonomic mechanics.

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From Variational Principles to Geometry — A Simple Guide

What this paper is about

This paper shows a general way to turn “how things move” problems (described by a variational principle) into a clean geometric picture. Think of it as converting instructions like “the path that makes this quantity smallest” into a geometric toolkit (a specific kind of mathematical structure) that completely captures the same motions. The method works for many kinds of systems, including those that change in time or lose energy, and it unifies several known theories under one roof.

1) Big idea and purpose

In physics and math, a “variational principle” says that a system follows the path that makes some number (the action) as small or as large as possible. This is how we get famous rules like the Euler–Lagrange or Hamilton’s equations.

But there are many different geometric toolkits to describe motion: symplectic (for conservative systems), cosymplectic (for time-dependent systems), contact and cocontact (for systems with energy loss or gain), and more. Each toolkit comes with its own powerful methods.

The purpose of this paper is to give a general recipe for:

Starting from any variational principle (possibly with constraints),
Building the right geometric structure that has exactly the same solutions (motions),
And recovering or extending known geometric descriptions (like cosymplectic for Hamilton’s principle and cocontact for Herglotz’s principle).

2) The main questions, in simple terms

The paper asks:

Given a “least action” type problem, can we systematically find the best geometric toolkit that describes the same motions?
How do different types of constraints (rules the paths must follow) affect which geometry we should use?
Can we recover well-known geometries (cosymplectic and cocontact) from standard principles (Hamilton’s and Herglotz’s)?
What happens for tricky (singular) systems where the usual tools fail?

3) What the authors do (the method), with simple analogies

The method has two main steps. First, some key ideas:

Variational principle: Choose a path so some number (the action) is optimal.
Admissible variations: The tiny “allowed” wiggles of the path you’re allowed to consider when optimizing.
Constraints: Extra rules the path must obey (for example, “stick to this surface,” or “your speed relates to your position in this way”).
Geometric structure: A toolkit (like symplectic, cosymplectic, contact, or cocontact geometry) that encodes the system’s physics into shapes and forms, so solving the geometry gives the motion.

Now the steps:

Step 1: Convert the variational principle into a geometric version with a 2-form (think of a special “measuring tool” on the space of states) plus constraints and admissible variations.
- The clever part: modify this 2-form so that you no longer have to restrict to special “allowed” variations — you can allow all variations and still get exactly the same solutions. This is like adjusting your measurement so that the rules are built into it.
Step 2: Absorb the remaining constraints into the geometry itself, sometimes by adding extra variables. Then the geometry alone encodes all the dynamics. This final geometric structure is the one that “is” your system.

Along the way, the paper treats several kinds of constraints:

Transversality (you really are a time-graph: time flows forward).
Nonholonomic constraints (rules on velocities that you must obey, like rolling without slipping).
Vakonomic constraints (keep the constraints true under variations; typical for “holonomy” in Lagrangians).
Holonomy for Lagrangian mechanics (paths must be consistent with velocity = time derivative of position). The Poincaré–Cartan 1-form is used here as the right “action” representative.

In everyday terms: they reshape the measuring tools so all the rules are embedded in the geometry, and the set of solutions doesn’t change.

4) Main findings and why they matter

Here is what the method achieves:

It recovers known correspondences:
- Hamilton’s principle (the usual Lagrangian/Hamiltonian story) naturally leads to cosymplectic geometry when time appears explicitly. This matches the well-known geometric description of time-dependent systems.
- Herglotz’s principle (for systems where the action itself depends on the evolving “action variable,” modeling dissipation) leads to cocontact geometry in the regular case. That’s the right geometry for systems that can lose or gain energy.
- The unified Lagrangian–Hamiltonian framework (sometimes called the Skinner–Rusk type approach) also emerges by “absorbing” the holonomy conditions.
It handles singular systems (where the usual equations are not nicely solvable in the standard way):
- For singular time-dependent Lagrangians: you can still describe them with a (pre)cosymplectic structure. “Pre-” here means the structure may be degenerate, but it still works to encode the dynamics. Sometimes the standard “Lagrangian 2-form” is not the correct one — the method tells you how to fix it.
- For singular action-dependent Lagrangians (Herglotz type): you don’t always get (pre)cocontact geometry. In those cases, the method produces a new kind of geometric structure (a fresh result).
It gives a clean geometric route to the evolution laws:
- For example, it naturally exposes energy conservation (in cosymplectic systems) and energy dissipation laws (in cocontact systems), which are core physical principles.

Why this matters:

It unifies different “compartments” of geometric mechanics under a single procedure. Instead of guessing the right geometry and hoping it fits, you derive it from the variational principle.
It keeps all the strong tools of geometry available: once you have the right structure, you can use the matching theorems, coordinate choices, and symmetry methods.

5) What this could mean going forward

A systematic “geometry-from-variations” pipeline: Given any variational problem (including constrained or dissipative ones), you can build the matching geometry that fully captures its dynamics.
Better classification of systems: Helps decide whether a system should be treated as symplectic, cosymplectic, contact, cocontact, or something new.
Stronger methods for challenging cases: The approach works even when standard formulas break down (singular cases), pointing to the correct modified or entirely new geometric structures.
Broader impact: Once the right geometry is known, powerful geometric tools (like reduction by symmetries, integrability results, and energy laws) can be adapted and applied more widely, including to time-dependent and dissipative systems.

In short, this paper provides a general, practical way to map “optimize a path” problems to the best-fitting geometric language, recovers classic results, and opens the door to new ones when the usual tools aren’t enough.

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Knowledge Gaps

Knowledge gaps, limitations, and open questions

The paper proposes a general method to associate geometric structures to variational problems, recovering known cases and producing new ones. The following points summarize what remains missing, uncertain, or unexplored, framed as actionable directions for future work:

Clarify uniqueness and canonicality of the constructed geometry:
- The method does not prove uniqueness of the associated geometric structure. Provide criteria for when the resulting 2-form (or structure) is unique up to a natural equivalence (e.g., Lepage-equivalence, exact terms, or gauge transformations), and define a canonical representative if possible.
- Quantify the dependence on auxiliary choices (e.g., choice of $R_t$ , Reeb-like $R_\alpha$ in nonholonomic reduction, or trivialization of the bundle), and determine whether there is a canonical, coordinate-free prescription minimizing this ambiguity.
Existence of vector field solutions and solvability conditions:
- The key transversality reduction (Theorem 3.1) requires existence of $X\in\Gamma(V(\pi))$ solving $i_X\omega=\sigma_t$ and satisfying $i_{R_t-X}\alpha=0$ for constraints. Provide intrinsic sufficient and necessary conditions (e.g., rank conditions on $\omega$ , compatibility conditions with $J$ ) ensuring existence and uniqueness of such $X$ globally.
- Develop an explicit algorithm to test solvability of these equations and to construct $X$ (or diagnose obstruction) for a given variational problem.
Co-orientation hypothesis in nonholonomic constraints:
- The construction in Section 4 requires co-oriented constraints (existence of forms $\eta^\alpha$ independent with $\tau$ ). Extend the method to non–co-oriented or singular constraint distributions, and characterize when a suitable $\bar\Omega$ exists without this hypothesis.
Vakonomic constraints beyond special cases:
- The vakonomic treatment is handled only for constraints generated by functions and for the holonomy (Cartan) constraints of first-order Lagrangians. Generalize to arbitrary vakonomic constraints (e.g., differential-form–generated ideals not arising from functions or Cartan forms) and identify conditions under which $B(\pi)$ -linearity can be recovered or circumvented.
- Provide a constructive procedure for the “absorption” of vakonomic constraints (Lepage-like replacements) in more general settings and characterize the ambiguity of such replacements.
Absorption step yields extensions rather than equivalences:
- The final “constraint absorption” step (Section 7, per the abstract) produces only extensions (not dynamic equivalences). Analyze the extent and nature of the mismatch: what properties of the original dynamics are preserved or potentially altered? Can one quantify or minimize the enlargement of the solution set, or identify additional constraints restoring equivalence?
Singular Lagrangians: characterization of when standard geometry fails:
- For singular time-dependent Lagrangians, the paper claims one always gets a (pre)cosymplectic structure, but not necessarily that of the Lagrangian 2-form. Give necessary and sufficient conditions for when the Lagrangian 2-form suffices versus when a modified structure is needed, and describe how to construct the latter canonically.
- For singular action-dependent Lagrangians (Herglotz type), the resulting geometry is not always (pre)cocontact, and is said to be “new.” Classify these new structures: are they instances of Jacobi, Dirac-Jacobi, or another known generalized geometry? Provide intrinsic invariants and normal forms distinguishing the cases.
Global and topological issues:
- Many constructions rely on a global time coordinate and a trivial bundle $E\cong\mathbb{R}\times N$ . Generalize to nontrivial time bundles (e.g., fibrations over $S^1$ ), nontrivial base manifolds, and settings without a global $R_t$ with $i_{R_t}\tau=1$ .
- Address global obstructions to defining $(\omega,\sigma_t)$ or $\bar\Omega$ and specify patching/gluing conditions on overlaps for global well-definedness.
Relation to established generalized frameworks:
- Make precise the relationship between the proposed method and Dirac, Courant, Jacobi, and Dirac–Jacobi structures used to encode constraints and dissipation. When does the constructed $(E,\Omega,J)$ correspond to a Dirac structure, and when does it define a (co)contact or Jacobi pair?
- Compare the nonholonomic construction here with the standard nonholonomic Dirac formalism; identify conditions under which they agree or differ.
Symmetries, reduction, and integrability:
- The method aims to bridge compartments in geometric mechanics but does not develop symmetry/reduction theory for the newly assigned structures. Establish momentum maps, reduction procedures, and integrability criteria compatible with the constructed geometries, including for the “new” geometries beyond (pre)cosymplectic/(pre)cocontact.
- Determine whether the construction commutes with symmetry reduction: if a VP is symmetric, is the associated geometry automatically reducible, and does the reduced geometric structure coincide with the one assigned to the reduced VP?
Boundary terms and admissible variations:
- The analysis intentionally ignores boundary terms and restricts variations to vector fields on $E$ (rather than vector fields along the curve or variational fields with boundary behavior). Extend the framework to include boundary contributions, natural boundary conditions, and corner conditions, and analyze how these affect the assigned geometry and equations.
- Clarify how Noether-type results and conserved quantities emerge (or fail to) in the non-closed $\Omega$ and constrained settings.
Closedness of the 2-form and classification of “nonstandard” geometries:
- The method does not require $\Omega$ or $\bar\Omega$ to be closed, which falls outside classical symplectic/cosymplectic settings. Provide a systematic classification of the non-closed cases, their associated Reeb-like dynamics, and interpret conservation/dissipation laws (e.g., generalizing $i_Z\sigma_t=0$ ) in this broader context.
Higher-order systems and field theories:
- The paper restricts to first-order, time-dependent mechanics. Extend the method to higher-order Lagrangians and to field theories (higher-dimensional bases), and compare with $k$ -symplectic, multisymplectic, and contact/multicontact analogues.
Well-posedness and completeness:
- Analyze well-posedness of the initial value problem under the constructed geometric structures: existence, uniqueness, and completeness of solution vector fields, especially in the singular and non-closed cases.
External/forced and nonconservative systems beyond Herglotz:
- Herglotz’s principle captures certain dissipative systems; generalize the method to broader classes of forced, nonconservative, or memory-dependent systems, and identify the corresponding geometries.
Computational and algorithmic aspects:
- Provide implementable procedures (symbolic/numeric) to construct $\Omega_1$ and $\Omega_2$ from a given VP, choose $R_t$ and $R_\alpha$ , test regularity/co-orientation, and verify dynamic equivalence/extension. Benchmark on nontrivial examples beyond those in the paper.
Reparametrization-invariant problems:
- The framework presumes a fixed base time with $i_Z\tau=1$ . Extend to reparametrization-invariant (e.g., relativistic) variational problems where no preferred time variable exists.
Mixed constraints (nonholonomic + vakonomic):
- Although the abstract claims treatment of mixed constraints, a general, unified theorem for combining transversality, nonholonomic, and vakonomic constraints is not presented in the provided text. Formulate and prove a comprehensive mixed-constraint reduction with clear hypotheses and outputs.
Equivalence classes of variational 1-forms:
- Precisely characterize how replacing a variational 1-form $\theta$ by Lepage-equivalent forms affects the constructed geometry, and define the equivalence class of geometries associated to a given VP under such replacements (beyond the Euler–Lagrange equivalence).
Physical interpretation and observables under extension:
- When absorbing constraints yields only an extension (not equivalence), clarify the physical interpretation of new variables and trajectories, and specify how to project back observables and conserved quantities to the original system without ambiguity.

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Practical Applications

Immediate Applications

The paper provides a constructive pipeline to map a variational principle (data: θ, constraints J, admissible variations Adm) into a geometric structure that fully encodes the dynamics (via a 2-form Ω and possibly absorbed constraints). The following are actionable, near-term uses:

Automated geometry assignment for dynamical models
- Sector(s): software tooling, academia
- Use case: Given a user-specified variational problem (θ, J, Adm), automatically construct an equivalent geometric variational problem (Ω, J', Adm') where all equations of motion are encoded in Ω (and possibly absorbed constraints), and classify the geometry (symplectic, presymplectic, cosymplectic, pre-cosymplectic, contact, cocontact, new).
- Tools/products/workflows: Python/Sage/SymPy library that ingests θ, J, Adm and outputs Ω, σ, ω, Reeb-like vector fields (R_t, R_α), and the resulting structure; integration with differential-geometry packages (SageManifolds), CAS notebooks, and model-based design pipelines (e.g., Modelica).
- Assumptions/dependencies:
- B(π)-linearity of Adm (or use provided reductions for nonholonomic/vakonomic cases).
- Co-oriented 1-form constraints for nonholonomic handling.
- Availability of R_t with i_{R_t}τ = 1 and, when needed, fields R_α satisfying i_{R_α}η^β = δ_α^β.
- Trivial bundle E ≅ ℝ × N for some analyses (autonomous reductions).
Structure-preserving simulation selection
- Sector(s): robotics, aerospace, mechanical engineering, software (physics engines, games/VFX)
- Use case: Use the identified geometry to select appropriate integrators (e.g., symplectic for conservative, contact/cocontact for dissipative/time-action systems; cosymplectic for time-dependent systems), improving long-term stability, energy behavior, and realism.
- Tools/products/workflows: A “geometry-aware integrator picker” that binds the Ω-based classification to a curated suite of integrators (symplectic variational integrators, contact structure-preserving integrators, presymplectic/DAE solvers).
- Assumptions/dependencies: Existence of a vector-field solution (conditions like i_Xω = σt); numerical routines to compute σ_t = i{R_t}Ω and ω = Ω + σ_t ∧ τ; reliability of available structure-preserving integrators for the identified class.
Robust handling of constrained dynamics (nonholonomic/vakonomic)
- Sector(s): robotics (wheeled robots, rolling constraints), automotive (vehicle dynamics), mechanical design, computational graphics
- Use case: Convert nonholonomic constraints (1-forms) and vakonomic function constraints into an equivalent Ω̄ that allows using standard vertical variations and geometric solvers; streamline simulation and control of rolling/slipping constraints and optimality-driven constraints.
- Tools/products/workflows: Implementation of Theorem (nonholonomic co-orientation) to build Ω̄ = Ω + σ_α ∧ η^α and solve with vertical variations; plug-in modules for simulation engines to encode constraints geometrically rather than via penalty methods.
- Assumptions/dependencies: Co-oriented constraint 1-forms with accessible R_α; reliable detection of constraint type (nonholonomic vs vakonomic); careful handling when constraints generate DAEs.
Energy-aware control via cocontact/Herglotz formalism
- Sector(s): robotics, automotive/transport, energy systems
- Use case: Design controllers leveraging the energy dissipation law derived from the geometry (e.g., i_Zσ_t = 0 and Z(H) = −H R_s(H) + R_t(H) in cocontact systems), enabling principled handling of friction, damping, or actuator limits.
- Tools/products/workflows: Control design templates that enforce geometric energy/dissipation laws; economic MPC or energy management controllers that use action-dependent (Herglotz) models.
- Assumptions/dependencies: A physically meaningful H; validity of action-dependent modeling; parameter identification for R_s(H), R_t(H).
Unified Lagrangian–Hamiltonian workflows (with holonomy absorption)
- Sector(s): software tooling, academia
- Use case: Automate switching between Euler–Lagrange and (co)symplectic/cocontact Hamiltonian formulations by absorbing holonomy constraints via the Poincaré–Cartan form Θ_L; facilitate symbolic derivations and cross-checks.
- Tools/products/workflows: CAS routines that generate ΘL, dΘ_L, and the corresponding equations; automatic derivation of Euler–Lagrange equations as iξ dΘ_L = 0 with admissible/vertical variations.
- Assumptions/dependencies: Regular Lagrangian for full equivalence to all-vector-fields formulations; for singular L, rely on pre-(co)symplectic treatment.
Singular time-dependent Lagrangians: guaranteed pre-cosymplectic description
- Sector(s): simulation software, control and systems engineering
- Use case: For models with singular L (non-invertible mass matrices, control-affine mechanics), apply the method to obtain a pre-cosymplectic structure even when the “Lagrangian 2-form” is inadequate, enabling consistent constraint algorithms (Dirac–Bergmann–Gotay–Nester style).
- Tools/products/workflows: Geometric DAE solvers that exploit pre-cosymplectic structure; diagnostics to separate primary/secondary constraints without ad hoc regularization.
- Assumptions/dependencies: Availability of constraint algorithms compatible with pre-cosymplectic geometry; careful handling of gauge freedoms.
Ready-to-use benchmark models demonstrating edge cases
- Sector(s): academia, education, software testing
- Use case: Use L(t,q,v) = t v and L(q,v,s) = s v to test detection of nonstandard geometries where standard (pre)cosymplectic/(pre)contact fail; include them in unit/regression test suites for modeling tools.
- Tools/products/workflows: Open-source example repository with scripts to compute Ω, Ω̄, σ_t, ω, and solved vector fields Z.
- Assumptions/dependencies: Exact symbolic manipulations or validated numerical approximations.
Curriculum enhancement and researcher onboarding
- Sector(s): education, academia
- Use case: Teach “from variational principle to geometry” as a unifying lens for mechanics, explicitly covering transversality, nonholonomic, vakonomic, and action-dependent cases; clarify when and why contact/cocontact vs (pre)cosymplectic applies.
- Tools/products/workflows: Lecture notes, worked examples, Jupyter notebooks illustrating the pipeline and theorems; interactive visualizations.
Improved physics engines and modeling guidelines
- Sector(s): software (game engines, CAD/CAE), digital twins
- Use case: Replace heuristic damping/constraint handling with geometry-derived forms (contact/cocontact for dissipation, Ω̄ for constraints), improving realism and numerical stability.
- Tools/products/workflows: Engine plugins that accept (θ, J, Adm) and internally generate Ω-based solvers; developer documentation mapping constraint types to geometric encodings.
- Assumptions/dependencies: Developers adopt the VP input layer; performance-optimized implementations of geometry-aware solvers.

Long-Term Applications

These build on the method’s unification and new geometries, requiring further research, scaling, or development:

End-to-end code generation from variational specification
- Sector(s): robotics, automotive, aerospace
- Use case: From a high-level VP (θ, J, Adm), auto-generate structure-preserving solvers and controller code (C++/Rust) with formal contracts (invariants/dissipation laws) derived from Ω.
- Dependencies: Verified code gen, certified numerical kernels, toolchain integration (ROS, AUTOSAR, DO-178C compliance).
New integrators for cocontact and pre-(co)symplectic singular systems
- Sector(s): computational physics, software
- Use case: Develop high-order, structure-preserving schemes tailored to cocontact and singular pre-(co)symplectic systems with guaranteed stability/long-time behavior.
- Dependencies: Theoretical advances in error/stability analysis; benchmarks and standardized test suites.
Symmetry, reduction, and momentum maps beyond symplectic
- Sector(s): academia, robotics and multi-body dynamics
- Use case: Extend reduction theorems and momentum map machinery to the new geometries uncovered for action-dependent/singular systems; enable principled model reduction for open, time-dependent systems.
- Dependencies: New theorems, computational analogs, and robust numerical reduction pipelines.
Standardized VP interface for model-based engineering
- Sector(s): systems engineering, digital twins
- Use case: Adopt (θ, J, Adm) as a portable modeling interface across tools (Modelica, FMI, Acados), enabling consistent translation to geometry-aware solvers and controllers.
- Dependencies: Community consensus, standardization efforts, open formats.
Energy systems and nonequilibrium thermodynamics modeling
- Sector(s): energy, automotive (EVs), industrial control
- Use case: Use Herglotz/action-dependent frameworks and cocontact geometry to model dissipation, path-dependence, and energy exchange (batteries, heat engines, regenerative braking) in a principled way; inform policy via vetted models of efficiency limits.
- Policy impact: Better-informed efficiency standards and lifecycle assessments based on validated geometric models.
- Dependencies: Experimental validation, parameter identification, domain-specific coupling (electrochemistry/thermal models).
Hybrid/switching systems with time/action dependence
- Sector(s): autonomous systems, power electronics
- Use case: Geometry-guided control and verification for hybrid dynamics that include dissipative and time-dependent phases; safe switching laws preserving required invariants/inequalities.
- Dependencies: Hybrid extensions of the geometric method; verification tooling.
Field-theoretic generalizations for multi-physics PDEs
- Sector(s): computational engineering, materials, geophysics
- Use case: Extend the method to k-symplectic, multisymplectic, and contact analogues for PDE models with dissipation; unlock structure-preserving PDE solvers in nonequilibrium settings.
- Dependencies: Theoretical development, scalable discretizations, HPC implementations.
Geometric priors in physics-informed ML
- Sector(s): ML for engineering, science
- Use case: Enforce Ω-structure, constraints, and derived conservation/dissipation laws as hard priors/regularizers in PINNs or operator learning; improve extrapolation and stability.
- Dependencies: Differentiable geometry toolkits; ML libraries supporting constrained training.
Formal verification and certification of safety-critical systems
- Sector(s): aerospace, medical devices, nuclear
- Use case: Encode geometric invariants (e.g., i_Zσ_t = 0) and constraint adherence into proof obligations for theorem provers; certify long-run behavior of controllers and simulators.
- Dependencies: Formalization of these geometries in proof assistants; tool interoperability.
Taxonomy and repository of geometric archetypes
- Sector(s): academia, education, software tooling
- Use case: Curate a public database linking constraint types and variational forms to the resulting geometries and solver templates, including edge cases (e.g., singular action-dependent Lagrangians).
- Dependencies: Community contributions, continuous integration with test problems.

Cross-cutting assumptions and limitations

Existence and computation of Reeb-like fields (R_t, R_α) and co-orientation of 1-form constraints are required in several constructions.
For vakonomic constraints not generated by functions or the jet holonomy case, B(π)-linearity may fail and additional care is needed.
Regularity of the Lagrangian enables stronger equivalences; singular cases require pre-(co)symplectic treatments and constraint algorithms.
Absorbing constraints may yield extensions (not strict equivalences), sometimes introducing new variables.
The identified geometric structure is not proven unique; different construction choices may yield dynamically equivalent but distinct formulations.
Some reductions assume trivial bundles E ≅ ℝ × N and autonomy (time-invariance of projected forms).

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Glossary

b-symplectic geometry: A variant of symplectic geometry adapted to manifolds with a distinguished hypersurface where the symplectic form degenerates in a controlled way. Example: "it is natural to use $b$ -symplectic geometry \cite{GMPS-2015}."
Cartan codistribution: The collection of contact (Cartan) 1-forms on a jet bundle that encode holonomy of sections. Example: "The holonomy condition can be imposed by implementing the Cartan codistribution of $J^1\rho$ as vakonomic constraints."
cocontact geometry: A geometric structure combining time and action dependence that generalizes contact geometry for non-conservative systems. Example: "Thus, the Hamiltonian version of the Herglotz variational problem leads to cocontact geometry."
cocontact Hamiltonian equations: The dynamical equations governing Hamiltonian systems in cocontact geometry. Example: "it becomes apparent that they are the cocontact Hamiltonian equations:"
contact geometry: An odd-dimensional counterpart to symplectic geometry, modeling dissipative or open systems via a contact form and Reeb vector field. Example: "In particular, contact geometry has recently received a lot of attention \cite{Bravetti2017, LL-2018, GGMRR-2019b}."
contactification: The process of embedding a symplectic system into a contact system by adding a variable. Example: "There also exist the processes of contactification and symplectification, where one can embed a contact system into a symplectic system, and viceversa, by adding or eliminating a variable \cite{GGKM-2024,Sloan:2024kzb}"
co-symplectic Hamiltonian dynamics: Hamiltonian dynamics formulated on cosymplectic manifolds (time-dependent setting). Example: "They correspond to co-symplectic Hamiltonian dynamics \cite{munoz-lecanda_geometry_2024}."
cosymplectic geometry: A geometric framework for explicitly time-dependent systems, given by a closed 1-form and a closed 2-form. Example: "Another natural generalization of symplectic geometry is cosymplectic geometry, which is the natural framework when the system is explicitly time-dependent."
differential ideal: An ideal in the algebra of differential forms that is closed under exterior differentiation, central in the study of exterior differential systems. Example: "Consequently, in the Exterior Differential Systems approach, the central objects of study are differential ideals."
dynamically equivalent: A relation between variational problems that have exactly the same set of solutions. Example: "V_1 is dynamically equivalent to V_2 if Sol_1=Sol_2."
dynamically μ-equivalent: A relation where solutions of two variational problems are connected via a bundle map μ, possibly not a diffeomorphism. Example: "V_1 is dynamically $\mu$ -equivalent to V_2 if there exists a smooth bundle map $\mu:E_1\rightarrow E_2$ such that $\tilde{\mu}(Sol_1)=Sol_2$ and $\tilde{\mu}$ is injective on $Sol_1$ ."
Exterior Differential Systems: A geometric framework that studies systems of differential forms and their integral manifolds using differential ideals. Example: "An example of this idea is the Exterior Differential Systems approach \cite{Griffiths}."
fiber bundle: A space locally resembling a product of a base and a fiber, used to model time-dependent configuration spaces in mechanics. Example: "Consider the fiber bundle $\pi:E\rightarrow \mathbb{R}$ ."
geometric mechanics: The study of dynamical systems using differential-geometric structures rather than coordinate-based formulations. Example: "Geometric mechanics studies dynamical systems with geometric objects."
geometric variational problem (GVP): A variational formulation specified by a 2-form, constraints, and admissible variations. Example: "A geometric variational problem or principle (GVP) on a fiber bundle $\pi:E\rightarrow \mathbb{R}$ is given by the triple $(\Omega,J,\textup{Adm})$ "
Herglotz's variational principle: A generalization of Hamilton’s principle allowing the action to depend on state variables, suited for dissipative systems. Example: "In these cases, the resulting geometry associated with the Herglotz's variational principle is new."
holonomy condition: The requirement that sections of a jet bundle represent genuine derivatives of curves, enforced by Cartan forms. Example: "The holonomy condition can be imposed by implementing the Cartan codistribution of $J^1\rho$ as vakonomic constraints."
Jacobi structure: A geometric structure generalizing Poisson, symplectic, and contact geometries via a bivector and vector field. Example: "A result in this line of thought is the Jacobi structure introduced by Lichnerowicz \cite{Lichnerowicz}, which encompasses several structures, including symplectic, contact and Poisson."
jet bundle: A bundle whose points encode derivatives of sections up to a given order; the first jet bundle includes positions and velocities. Example: "Consider the first jet bundle $\pi:E=J^1\rho\rightarrow \mathbb{R}$ "
k-symplectic: An extension of symplectic geometry for field theories with k independent variables. Example: "like $k$ -symplectic and multisymplectic (see \cite{LeSaVi2016, Roman-Roy:2005vwe} and references therein, respectively)."
Lagrangian energy: The energy associated to a Lagrangian, typically E_L = v·∂L/∂v − L. Example: "The Lagrangian energy is"
Lepage equivalent theory: A theory identifying differential forms that yield the same Euler–Lagrange equations under holonomy constraints. Example: "The Lepage equivalent theory \cite{Krupka} studies the different differential forms leading to the same differential equations in the presence of holonomy constraints generated by the Cartan codistribution."
Liouville dynamics: The component of contact dynamics associated with the Liouville (scaling) vector field. Example: "the dynamics of a contact system can be split into Reeb dynamics and Liouville dynamics \cite{BLMP-2020}."
Liouville-Mineur-Arnold theorem: A theorem guaranteeing canonical (action-angle) coordinates for integrable Hamiltonian systems. Example: "the Liouville-Mineur-Arnold theorem \cite{AM-78} states that one can find canonical coordinates adapted to the dynamics."
Meyer-Marsden-Weinstein reduction theorem: A symplectic reduction result using symmetries and momentum maps to lower the system’s dimension. Example: "the Meyer-Marsden-Weinstein reduction theorem \cite{MW} shows how to reduce a system to a lower-dimensional one using its symmetries."
momentum map: A map from phase space to the dual of a Lie algebra capturing conserved quantities from symmetries. Example: "a complete classification of integrable dynamical systems is possible through the use of the momentum map and polytopes \cite{Atiyah}."
multisymplectic: A generalization of symplectic geometry to field theories using closed forms of degree greater than two. Example: "like $k$ -symplectic and multisymplectic (see \cite{LeSaVi2016, Roman-Roy:2005vwe} and references therein, respectively)."
nonholonomic constraints: Velocity-level constraints not integrable to configuration constraints, imposed directly on admissible variations. Example: "Nonholonomic Constraints"
Poincaré-Cartan form: A 1-form associated to a Lagrangian that encodes the Euler–Lagrange equations geometrically. Example: "The $1$-form $\Theta_L$ is the PoincarÃ©-Cartan form"
precontact geometry: A degenerate version of contact geometry suitable for singular action-dependent Lagrangians. Example: "It has been extended to precontact geometry \cite{LL-2019} for singular action-dependent Lagrangians"
presymplectic geometry: A degenerate symplectic structure (closed but possibly singular 2-form) used for systems with non-regular Lagrangians. Example: "When the Lagrangian is not regular, the corresponding structure is presymplectic geometry."
Reeb dynamics: Dynamics along the Reeb vector field inherent to contact-type structures. Example: "the dynamics of a contact system can be split into Reeb dynamics and Liouville dynamics \cite{BLMP-2020}."
symplectic geometry: The geometric framework of Hamiltonian mechanics based on a non-degenerate closed 2-form. Example: "The prototypical theory in geometric mechanics is symplectic geometry."
symplectification: The process of embedding a contact system into a symplectic one (or vice versa) by adding/eliminating a variable. Example: "There also exist the processes of contactification and symplectification, where one can embed a contact system into a symplectic system, and viceversa, by adding or eliminating a variable \cite{GGKM-2024,Sloan:2024kzb}"
total derivative: The derivative operator along time that includes contributions from dependent variables in jet coordinates. Example: "where $D_t=\frac{\partial}{\partial t}+v^i\frac{\partial}{\partial q^i}$ is the total derivative."
transversality constraint: The restriction that admissible variations preserve the section property (typically vertical with respect to the base). Example: "Together with the transversality constraint, we have that,"
vakonomic constraints: Constraints implemented by restricting admissible variations to those that preserve a given differential ideal. Example: "A set of differential forms $J\subset\Omega^\bullet{E}$ is implemented as vakonomic constraints"
variational problem (VP): A formulation specifying an action (often via a 1-form), constraints, and admissible variations to determine solutions. Example: "A variational problem or principle (VP) on a fiber bundle $\pi:E\rightarrow M$ is a triple $(\theta,J,\textup{Adm})$ "
vertical distribution: The subbundle of the tangent bundle consisting of vectors tangent to the fibers (kernel of the projection). Example: "V(\pi): vertical distribution of $\pi$ ."

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From variational principles to geometry (3 likes, 0 questions)

From variational principles to geometry (2509.22171v1)

Summary

Geometric Structures from Variational Principles: A Unified Framework

Introduction

Geometric Variational Problems: Formalization

Reduction and Absorption of Constraints

Recovery of Known Geometric Structures

Singular Lagrangians and Generalized Geometries

Equivalence and Classification of Dynamical Systems

Implications and Future Directions

Conclusion

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