arxiv: 2605.05517 · v1 · submitted 2026-05-06 · 🧮 math.DG · math-ph· math.MP

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Variational reduction of homogenous Lagrangian systems

Edith Padr\'on, Javier Fern\'andez, Juan Carlos Marrero, Sergio Grillo

Pith reviewed 2026-05-08 15:27 UTC · model grok-4.3

classification 🧮 math.DG math-phmath.MP

keywords variational reductionhomogeneous Lagrangianscaling symmetryLagrange-Poincaré equationsHerglotz principleLagrangian mechanicsdifferential geometry

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The pith

For homogeneous Lagrangian systems a variational reduction exists that reconstructs trajectories up to quadratures from reduced critical points.

A machine-rendered reading of the paper's core claim, the machinery that carries it, and where it could break.

The paper shows that Lagrangian systems whose defining function is homogeneous of fixed degree under a scaling symmetry admit a well-defined variational reduction on the quotient space. Solutions of the original system are recovered from the critical points of this reduced problem by performing a finite number of integrations. A reader would care because the procedure replaces the full Euler-Lagrange equations with a smaller set of ordinary differential equations that still capture all essential dynamics. These reduced equations are identified as the direct scaling analogue of the classical Lagrange-Poincaré equations. The paper further checks whether such homogeneous systems stand in any natural relation to the Herglotz variational principle.

Core claim

A variational reduction procedure can be defined for Lagrangian systems defined by a homogeneous Lagrangian function so that the trajectories of the system can be reconstructed up to quadratures from the critical points of the reduced variational principle. The critical points themselves are characterized by a set of ordinary differential equations that constitute the scaling analogue of the Lagrange-Poincaré equations.

What carries the argument

Homogeneity of the Lagrangian under a scaling group action, which induces a quotient manifold whose critical points lift back to original trajectories by quadratures.

If this is right

The original second-order equations of motion reduce to a first-order system on the quotient whose solutions determine the full trajectories by quadrature.
The reduced equations take the explicit form of the scaling version of the Lagrange-Poincaré equations.
Any conserved quantity arising from the scaling symmetry is automatically accounted for in the reduced variational principle.
The same reduction framework can be applied to any homogeneous Lagrangian system whose configuration space carries a suitable group action.

Where Pith is reading between the lines

These are editorial extensions of the paper, not claims the author makes directly.

The method may simplify numerical integration for systems whose potentials are homogeneous, such as central-force problems or homogeneous cosmological models.
It supplies a variational counterpart to existing scaling reductions in symplectic geometry, potentially allowing direct comparison with Routh reduction.
One could test the procedure on low-dimensional examples like the Kepler problem or the free particle on a cone to verify the quadrature count explicitly.

Load-bearing premise

The Lagrangian must be homogeneous of a fixed degree and the underlying manifold must admit a scaling symmetry whose quotient is smooth enough for the reduced critical points to lift via quadratures.

What would settle it

A concrete homogeneous Lagrangian on a manifold with scaling symmetry whose reduced critical points fail to produce solutions of the original Euler-Lagrange equations after quadrature would falsify the reconstruction claim.

read the original abstract

In this paper we show that a variational reduction procedure can be defined for Lagrangian systems subject to scaling symmetries (i.e. Lagrangian systems defined by a homogenous Lagrangian function), in such a way that the trajectories of the system can be reconstructed up to quadratures from the critical points of the reduced variational principle. Also, we characterize the mentioned critical points in terms of a set of ordinary differential equations which are the scaling analogue of the Lagrange-Poincar\'e equations. Finally, we study if the homogeneous Lagrangian systems are naturally related or not with the Herglotz variational principle.

Editorial analysis

A structured set of objections, weighed in public.

Desk editor's note, referee report, simulated authors' rebuttal, and a circularity audit. Tearing a paper down is the easy half of reading it; the pith above is the substance, this is the friction.

Desk Editor's Note private letter to a colleague

The paper gives a solid variational reduction for homogeneous Lagrangians that recovers solutions via quadratures and adds the scaling version of Lagrange-Poincaré equations, with a clean negative result on Herglotz.

read the letter

The core result is a reduction procedure that uses the homogeneity of the Lagrangian to induce a scaling symmetry, builds a quotient space, and produces a reduced variational principle. Critical points there satisfy a set of ODEs that act as the scaling analogue of the Lagrange-Poincaré equations. Original trajectories lift back by solving for the scaling parameter through quadratures. The authors also check the Herglotz principle and find no natural direct link, which keeps the claims bounded.

Referee Report

0 major / 3 minor

Summary. The paper develops a variational reduction procedure for Lagrangian systems with homogeneous Lagrangians, using scaling symmetries to induce a quotient manifold on which a reduced variational principle is defined. Critical points of the reduced principle are characterized by a system of ODEs that serve as the scaling analogue of the Lagrange-Poincaré equations. Original trajectories are reconstructed from these reduced critical points via quadratures. The manuscript also examines the relation between homogeneous Lagrangian systems and the Herglotz variational principle, concluding there is no natural direct correspondence.

Significance. If the derivations hold, the work extends classical symmetry reduction techniques in geometric mechanics to the setting of scaling symmetries. The explicit quadrature reconstruction of trajectories from the reduced critical points is a concrete strength, as it avoids the need for additional differential equations in the lifting step. The scaling analogue of the Lagrange-Poincaré equations provides a practical ODE characterization that may aid both analytical and numerical investigations of homogeneous systems. The negative result on the Herglotz connection clarifies the scope of the reduction.

minor comments (3)

§2: The notation distinguishing the original scaling action from the induced quotient structure could be made more explicit to prevent confusion with standard Lie group actions; a short diagram or table comparing the spaces would improve readability.
Introduction and §4: The discussion of the Herglotz principle would benefit from a brief explicit comparison (e.g., one or two model equations) showing why no natural correspondence exists, rather than a purely verbal argument.
References: Several classical works on homogeneous Lagrangians and variational reduction (e.g., related to Euler's homogeneous function theorem in mechanics) are not cited; adding 2–3 targeted references would strengthen the positioning.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their positive evaluation and accurate summary of the manuscript. The significance of the scaling symmetry reduction, quadrature-based reconstruction, and the negative result on the Herglotz principle are all correctly identified. No major comments were listed in the report.

Circularity Check

0 steps flagged

No significant circularity detected in derivation chain

full rationale

The paper defines a variational reduction for homogeneous Lagrangian systems by inducing a scaling symmetry from the homogeneity degree, forming a quotient manifold on which a reduced variational principle is constructed. Critical points of the reduced principle are characterized via ODEs that are the direct scaling analogue of the standard Lagrange-Poincaré equations, with original trajectories recovered by quadrature on the scaling parameter. These steps follow from classical variational calculus and symmetry reduction without any self-definitional closure, fitted parameters renamed as predictions, or load-bearing self-citations that collapse the argument to its own inputs. The separate analysis of the Herglotz principle is an independent comparison that does not rely on the reduction result. The derivation remains self-contained against external benchmarks of variational geometry.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 0 invented entities

Review performed from abstract only; the ledger therefore records only the background assumptions that must hold for any such reduction to be stated.

axioms (2)

domain assumption The configuration space is a smooth manifold and the Lagrangian is a smooth function on its tangent bundle that is homogeneous of fixed degree.
Standard background assumption in Lagrangian mechanics on manifolds; required for the scaling symmetry to be well-defined.
ad hoc to paper The scaling symmetry permits a quotient manifold on which a reduced variational principle can be defined whose critical points lift to the original trajectories via quadratures.
This is the central enabling assumption of the claimed reduction procedure.

pith-pipeline@v0.9.0 · 5395 in / 1448 out tokens · 67429 ms · 2026-05-08T15:27:13.450692+00:00 · methodology

discussion (0)

Reference graph

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