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arxiv: 2604.19655 · v1 · submitted 2026-04-21 · 🌊 nlin.SI

Duality of Hamiltonian and Lagrangian formulations for integrable systems

Pierandrea Vergallo , Mats Vermeeren This is my paper

Pith reviewed 2026-05-10 00:42 UTC · model grok-4.3

classification 🌊 nlin.SI
keywords Hamiltonian potential variablesLagrangian multiformsbi-Hamiltonian systemsKdV equationdispersionless limitspolytropic gas dynamicsconstant astigmatism equationsymplectic operators
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The pith

Hamiltonian potential variables map bi-Hamiltonian operators to symplectic ones, yielding Lagrangian multiforms for integrable systems.

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

The paper introduces Hamiltonian potential variables that convert Hamiltonian operators into symplectic operators in a dual space. This generalizes the substitution of a potential variable used to obtain a Lagrangian density for the KdV equation. With this mapping the authors construct Lagrangian structures for bi-Hamiltonian systems, reformulate the Lenard scheme in symplectic terms, and obtain pairs of Lagrangian multiforms for the KdV equation, its dispersionless limits, polytropic gas dynamics, and the constant astigmatism equation. A sympathetic reader would care because the duality supplies a systematic route from known Hamiltonian descriptions to variational ones, including cases where Lagrangian multiforms were previously unavailable.

Core claim

We introduce the concept of Hamiltonian potential variables to map Hamiltonian operators into symplectic operators in a dual space. This generalises the classical trick of switching to a potential variable to obtain a Lagrangian density for the Korteweg-de Vries (KdV) equation. Building on this concept, we present the Lagrangian structure for bi-Hamiltonian systems, discuss the Lenard scheme in the symplectic formalisms, and apply this to construct pairs of Lagrangian multiforms for the KdV equation and some dispersionless limits of it. We also consider the examples of polytropic gas dynamics and the constant astigmatism equation, for which no Lagrangian multiforms were previously known.

What carries the argument

Hamiltonian potential variables that map Hamiltonian operators into non-degenerate symplectic operators in a dual space.

If this is right

  • Bi-Hamiltonian systems admit corresponding Lagrangian formulations via the dual symplectic operators.
  • The Lenard recursion scheme can be carried out directly in the symplectic formalism.
  • Pairs of Lagrangian multiforms exist for both dispersive and dispersionless versions of the KdV equation.
  • Lagrangian multiforms can be constructed for polytropic gas dynamics and the constant astigmatism equation.
  • The duality supplies a variational description wherever a bi-Hamiltonian operator pair is known.

Where Pith is reading between the lines

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

  • The same substitution technique may generate Lagrangian descriptions for other bi-Hamiltonian integrable equations not treated in the paper.
  • Switching between Hamiltonian and Lagrangian pictures could simplify the search for conserved quantities or symmetries in related systems.
  • The construction suggests that dispersionless limits preserve the Lagrangian multiform property under this duality.

Load-bearing premise

Hamiltonian potential variables can be introduced consistently for the given bi-Hamiltonian systems so that the resulting symplectic operators remain non-degenerate.

What would settle it

An explicit calculation for the constant astigmatism equation that produces a degenerate symplectic operator after the potential-variable substitution would show the mapping fails for at least one claimed case.

Figures

Figures reproduced from arXiv: 2604.19655 by Mats Vermeeren, Pierandrea Vergallo.

Figure 1
Figure 1. Figure 1: Schematic overview of the two Lenard scheme, Hamiltonian and symplectic, linked by [PITH_FULL_IMAGE:figures/full_fig_p012_1.png] view at source ↗
read the original abstract

We introduce the concept of Hamiltonian potential variables to map Hamiltonian operators into symplectic operators in a dual space. This generalises the classical trick of switching to a potential variable to obtain a Lagrangian density for the Korteweg-de Vries (KdV) equation. Building on this concept, we present the Lagrangian structure for bi-Hamiltonian systems, discuss the Lenard scheme in the symplectic formalisms, and apply this to construct pairs of Lagrangian multiforms. We discuss the key model of the KdV equation and some dispersionless limits of it. We present a pair of Lagrangian multiforms for these equations, one of which is new. We also consider the examples of polytropic gas dynamics and the constant astigmatism equation, for which no Lagrangian multiforms were previously known.

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.

Referee Report

0 major / 3 minor

Summary. The paper introduces Hamiltonian potential variables to map Hamiltonian operators to symplectic operators in a dual space, generalizing the classical potential-variable trick for the KdV equation. It develops the Lagrangian structure for bi-Hamiltonian systems, reformulates the Lenard scheme in the symplectic setting, and constructs pairs of Lagrangian multiforms (one new) for the KdV equation, its dispersionless limits, polytropic gas dynamics, and the constant astigmatism equation.

Significance. If the constructions are valid, the work supplies a systematic duality between Hamiltonian and Lagrangian formulations for integrable systems, allowing Lagrangian multiforms to be derived for models where none were previously known. The explicit operator mappings and multiform constructions for multiple concrete examples (including dispersionless KdV and gas dynamics) constitute a clear strength and could facilitate further unification of Hamiltonian and variational approaches in soliton theory.

minor comments (3)
  1. [§4] §4 (KdV and dispersionless limits): the explicit form of the new Lagrangian multiform for the dispersionless case is stated but the verification that it reproduces the correct Euler-Lagrange equations via the multiform variational principle is only sketched; adding the intermediate steps would improve readability.
  2. [§5] §5 (polytropic gas dynamics): the non-degeneracy of the symplectic operator obtained after introducing the Hamiltonian potential variable is asserted but not accompanied by a rank calculation or kernel check; a short appendix entry would confirm this for the claimed range of polytropic indices.
  3. Notation: the symbol for the Hamiltonian potential variable is introduced without a dedicated definition box or table comparing it to the classical KdV potential; a small comparison table would aid readers unfamiliar with the generalization.

Simulated Author's Rebuttal

0 responses · 0 unresolved

We thank the referee for their supportive summary of the manuscript, recognition of its significance in establishing a systematic duality between Hamiltonian and Lagrangian formulations, and recommendation for minor revision. No major comments were provided in the report.

Circularity Check

0 steps flagged

No significant circularity detected

full rationale

The derivation introduces the new concept of Hamiltonian potential variables as a direct generalization of the classical KdV potential substitution, then maps Hamiltonian operators to symplectic ones via explicit operator transformations. Lagrangian multiforms and Lenard-scheme applications are constructed case-by-case for KdV, its dispersionless limits, polytropic gas dynamics, and the constant astigmatism equation using these mappings. All steps rely on algebraic operator identities and explicit verification rather than parameter fitting, self-referential definitions, or load-bearing self-citations; the constructions remain independent of their own outputs and do not reduce by construction to prior inputs.

Axiom & Free-Parameter Ledger

0 free parameters · 2 axioms · 1 invented entities

The work rests on standard properties of Hamiltonian and symplectic operators in the theory of integrable systems. No free parameters are introduced in the abstract. The new entity is the Hamiltonian potential variable itself.

axioms (2)
  • standard math Hamiltonian operators are skew-symmetric and satisfy the Jacobi identity.
    Invoked implicitly when mapping to symplectic operators.
  • domain assumption Bi-Hamiltonian structures admit a Lenard scheme.
    Used to discuss the Lenard scheme in the symplectic formalism.
invented entities (1)
  • Hamiltonian potential variables no independent evidence
    purpose: To map Hamiltonian operators into symplectic operators in a dual space.
    New variables introduced to generalize the potential-variable trick for KdV.

pith-pipeline@v0.9.0 · 5426 in / 1401 out tokens · 19491 ms · 2026-05-10T00:42:47.894331+00:00 · methodology

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