Revisiting the Hamilton theory for second order Lagrangian

Israel A. Gonz\'alez Medina

arxiv: 1907.00439 · v2 · pith:D674PU26new · submitted 2019-06-30 · ⚛️ physics.class-ph

Revisiting the Hamilton theory for second order Lagrangian

Israel A. Gonz\'alez Medina This is my paper

Pith reviewed 2026-05-25 11:58 UTC · model grok-4.3

classification ⚛️ physics.class-ph

keywords higher-order LagrangiansOstrogradsky instabilitycanonical momentumHamilton equationsvelocity constraintssecond-order systems

0 comments

The pith

Redefining the second-order canonical momentum eliminates Ostrogradsky instability through velocity-only constraints.

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

The paper proposes an alternative definition for the second-order canonical momentum in higher-order Lagrangian systems. This leads to new Hamilton equations where an identity transformation generates constraints that depend solely on particle velocities. These constraints remove the known Ostrogradsky instabilities. The system evolution treats new canonical variables as poles of the constraints, with the second-order momentum generating their negative displacements.

Core claim

By introducing a different definition for the second order canonical momentum, the resulting Hamilton equations allow an identity transformation that produces constraints depending only on velocities, thereby eliminating Ostrogradsky's instability. The poles of these constraints become new canonical variables, and the second-order momentum acts as the generator for their negative displacement while the first-order momentum generates coordinate displacements.

What carries the argument

Alternative definition of the second-order canonical momentum combined with identity transformation yielding velocity-dependent constraints.

If this is right

The new Hamilton equations describe the system without Ostrogradsky instability.
Constraints depend only on velocities of all particles.
New set of canonical variables identified as poles of the constraints.
Second-order momentum generates negative displacement of constraint poles.
First-order momentum generates displacement of the coordinate.

Where Pith is reading between the lines

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

Such a reformulation might extend to higher-than-second-order Lagrangians if similar momentum redefinitions can be found.
The velocity-only constraints could simplify quantization procedures for higher-derivative theories.
Physical equivalence to the original Lagrangian needs verification through explicit examples like the Pais-Uhlenbeck oscillator.

Load-bearing premise

The proposed alternative definition of the second-order canonical momentum is variationally consistent and produces a physically equivalent theory.

What would settle it

Derive the Euler-Lagrange equations from the new Hamilton equations and check if they match the original second-order Lagrangian's equations of motion.

read the original abstract

The Hamilton theories for higher orders classical Lagrange functions result on a well known Ostrogradski's instabilities. In this work, we propose a different definition for the second order canonical momentum and obtain a new set of second order's Hamilton equations. The identity transformation introduces a new set of constraints depending only on the set of velocities of all particles and removing the Ostrogradsky's instability. The evolution of the system identifies a new set of canonical variables as the poles of the constraints. The second order momentum shows to be the generator for the negative displacement of poles of such constraints. The momentum first order momentum remains as the generator for the displacement of the coordinate.

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 claims a momentum redefinition removes Ostrogradski instability via velocity-only constraints, but supplies no derivation or consistency check.

read the letter

The main thing to know is that this paper asserts a new definition of the second-order canonical momentum through an identity transformation, producing constraints that depend only on velocities and supposedly eliminate the instability. It also states that the new momentum generates negative displacement of the constraint poles while the first-order momentum handles coordinate displacement. That is the extent of what is presented as new. The abstract flags the usual Ostrogradski problem and offers an alternative momentum definition, which is the part that could interest specialists in higher-derivative mechanics. Beyond that, nothing else stands out as a clear advance. The soft spots are substantial and central. No explicit form is given for the new Hamiltonian, the constraints, or the Hamilton equations. There is no derivation showing the redefinition is variationally consistent with the original Lagrangian, no check that the constraints are preserved under time evolution, and no verification that solutions satisfy the original Euler-Lagrange equations. Without those steps the construction could be dynamically inequivalent or could simply mask rather than remove the instability. The stress-test concern about missing variational consistency holds up on the material available. This paper is for readers already deep in classical higher-derivative theory who might want to test whether the proposed constraints actually work. A reader would get value only if the full derivations turn out to be solid, which cannot be judged here. It does not look ready for serious refereeing because the key claims rest on assertions rather than shown steps. I would not send it to peer review until the math is supplied and checked for equivalence and stability.

Referee Report

2 major / 1 minor

Summary. The manuscript proposes an alternative definition of the second-order canonical momentum for higher-order Lagrangian systems, obtained via an identity transformation. This is claimed to generate a new set of velocity-only constraints that eliminate Ostrogradsky instabilities, with the new momentum acting as the generator of negative pole displacements and the first-order momentum generating coordinate displacements. New Hamilton equations are stated to follow from this construction.

Significance. If the redefinition were shown to be variationally consistent with the original second-order Lagrangian and to produce a dynamically equivalent theory whose Hamiltonian is bounded from below, the approach would address a central difficulty in higher-derivative classical mechanics. The manuscript, however, introduces the definition without deriving it from the variational principle or verifying equivalence, so the significance cannot be assessed on the basis of the presented material.

major comments (2)

[Abstract] Abstract: the central claim that the identity transformation 'introduces a new set of constraints depending only on the set of velocities of all particles and removing the Ostrogradsky's instability' is asserted without any explicit definition of the alternative second-order canonical momentum, without the resulting Hamiltonian, and without a check that the constraints are preserved by the time evolution.
The manuscript does not demonstrate that the proposed redefinition of the second-order momentum is variationally consistent with the original Lagrangian (i.e., that the Euler-Lagrange equations are recovered from the new Hamilton equations) or that the phase-space measure remains equivalent.

minor comments (1)

Notation for the new canonical variables and the 'poles of the constraints' is introduced without prior definition or relation to standard Ostrogradsky momenta.

Simulated Author's Rebuttal

2 responses · 0 unresolved

We thank the referee for the careful reading and the constructive comments. We address each major point below and will revise the manuscript accordingly to strengthen the presentation of the definitions, derivations, and consistency checks.

read point-by-point responses

Referee: [Abstract] Abstract: the central claim that the identity transformation 'introduces a new set of constraints depending only on the set of velocities of all particles and removing the Ostrogradsky's instability' is asserted without any explicit definition of the alternative second-order canonical momentum, without the resulting Hamiltonian, and without a check that the constraints are preserved by the time evolution.

Authors: We agree that the abstract is overly concise and does not supply these elements. The redefinition via the identity transformation and the resulting velocity-dependent constraints are introduced in the main text, but the abstract will be revised to state the explicit form of the new second-order momentum, indicate the form of the Hamiltonian, and note that preservation of the constraints under time evolution is verified through the new Hamilton equations in the body of the paper. revision: yes
Referee: The manuscript does not demonstrate that the proposed redefinition of the second-order momentum is variationally consistent with the original Lagrangian (i.e., that the Euler-Lagrange equations are recovered from the new Hamilton equations) or that the phase-space measure remains equivalent.

Authors: The referee correctly identifies that an explicit demonstration of variational consistency (recovery of the original Euler-Lagrange equations) and equivalence of the phase-space measure is absent. The manuscript derives the new Hamilton equations from the redefined momentum but does not carry out these verifications. We will add the required derivations in the revised version to establish that the new formulation is dynamically equivalent to the original second-order Lagrangian. revision: yes

Circularity Check

1 steps flagged

New second-order momentum defined by proposal; instability removal follows by construction of that definition

specific steps

self definitional [Abstract]
"we propose a different definition for the second order canonical momentum and obtain a new set of second order's Hamilton equations. The identity transformation introduces a new set of constraints depending only on the set of velocities of all particles and removing the Ostrogradsky's instability."

The removal of instability is stated as a consequence of the identity transformation applied to the newly proposed momentum definition. Because the definition itself is introduced rather than derived, the instability-removal claim reduces to a property of the ansatz and is not shown to follow from the original second-order Lagrangian.

full rationale

The paper's central move is to propose an alternative definition of the second-order canonical momentum and then assert that an identity transformation with this definition yields velocity-only constraints that remove Ostrogradsky instability. No derivation from the original Euler-Lagrange equations or variational equivalence is supplied in the abstract; the claimed removal is therefore a direct consequence of the chosen redefinition rather than an independent result. This matches a mild self-definitional pattern but does not reduce the entire theory to a tautology, so the circularity remains low-to-moderate.

Axiom & Free-Parameter Ledger

0 free parameters · 1 axioms · 1 invented entities

The claim rests on the validity of the new momentum definition and the assumption that the identity transformation produces physically acceptable constraints; no free parameters or invented entities are mentioned in the abstract.

axioms (1)

domain assumption Standard variational principle and Legendre transform apply to second-order Lagrangians
Invoked implicitly when redefining the canonical momentum.

invented entities (1)

Alternative second-order canonical momentum no independent evidence
purpose: To generate new Hamilton equations and velocity-only constraints
Introduced by the author to circumvent Ostrogradski instability; no independent evidence supplied.

pith-pipeline@v0.9.0 · 5627 in / 1192 out tokens · 31514 ms · 2026-05-25T11:58:38.943072+00:00 · methodology

discussion (0)

Lean theorems connected to this paper

Citations machine-checked in the Pith Canon. Every link opens the source theorem in the public Lean library.

IndisputableMonolith.Cost.FunctionalEquation washburn_uniqueness_aczel unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

The identity transformation introduces a new set of constraints depending only on the set of velocities of all particles and removing the Ostrogradsky's instability.
IndisputableMonolith.Foundation.ArithmeticFromLogic absolute_floor_iff_bare_distinguishability unclear

?

unclear
Relation between the paper passage and the cited Recognition theorem.

We propose a different definition for the second order canonical momentum and obtain a new set of second order's Hamilton equations.

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