Pith Number

pith:6IQO4YTY

pith:2026:6IQO4YTYQ7HNWNL5PA3WCGNTUX

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Spectral separation of variables from equivalent Lagrangian systems

Mattia Scomparin

Requiring two quadratic Lagrangians to produce the same equations of motion imposes a commutation condition that spectrally decomposes the configuration space and decouples the dynamics.

arxiv:2605.15679 v1 · 2026-05-15 · math-ph · math.MP

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Claims

C1strongest claim

Requiring two quadratic Lagrangians to generate the same Euler-Lagrange equations imposes a compatibility condition between the kinetic matrices and the potential. For constant symmetric kinetic matrices, this condition reduces to a commutation relation with the Hessian of the potential, yielding an orthogonal spectral decomposition of the configuration space.

C2weakest assumption

The kinetic matrices are constant and symmetric, which is required for the compatibility condition to reduce to a commutation relation with the Hessian (as stated in the abstract for the constant symmetric case).

C3one line summary

Dynamical equivalence of quadratic Lagrangians implies a commutation relation with the potential Hessian that yields orthogonal spectral decomposition and decoupled equations of motion.

References

19 extracted · 19 resolved · 0 Pith anchors

[1] Sergio Benenti , Intrinsic characterization of the variable separation in t he Hamilton–Jacobi equation , Journal of Mathematical Physics 38 (1997), no. 12, 6578–6602 1997

[2] Sergio Benenti, Claudia Chanu, and Giovanni Rastelli , Variable sepa- ration for natural hamiltonians with scalar and vector pote ntials on riemannian manifolds, Journal of Mathematical Physics 42 (20 2001

[3] Sergio Benenti, Claudia Chanu, and Giovanni Rastelli , Variable- separation theory for the null Hamilton–Jacobi equation , Journal of Mathemati- cal Physics 46 (2005), 042901 2005

[4] B/suppress laszak and S 1994

[5] C. M. Cosgrove and G. Scoufis , Painlev´ e classiﬁcation of a class of dif- ferential equations of the second order and second degree , Studies in Applied Mathematics 88 (1993), no. 1, 25–87 1993

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Receipt and verification

First computed	2026-05-20T00:01:11.984610Z
Builder	pith-number-builder-2026-05-17-v1
Signature	Pith Ed25519 (`pith-v1-2026-05`) · public key
Schema	pith-number/v1.0

Canonical hash

f220ee627887cedb357d78376119b3a5c550438830ac467ba7a9d82bafd128da

Aliases

arxiv: 2605.15679 · arxiv_version: 2605.15679v1 · doi: 10.48550/arxiv.2605.15679 · pith_short_12: 6IQO4YTYQ7HN · pith_short_16: 6IQO4YTYQ7HNWNL5 · pith_short_8: 6IQO4YTY

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Verify this Pith Number yourself

curl -sH 'Accept: application/ld+json' https://pith.science/pith/6IQO4YTYQ7HNWNL5PA3WCGNTUX \
  | jq -c '.canonical_record' \
  | python3 -c "import sys,json,hashlib; b=json.dumps(json.loads(sys.stdin.read()), sort_keys=True, separators=(',',':'), ensure_ascii=False).encode(); print(hashlib.sha256(b).hexdigest())"
# expect: f220ee627887cedb357d78376119b3a5c550438830ac467ba7a9d82bafd128da

Canonical record JSON

{
  "metadata": {
    "abstract_canon_sha256": "81631051bb8e8cb3ace28d925648991259f3e72c4e78f07f4baa195d87c6afe7",
    "cross_cats_sorted": [
      "math.MP"
    ],
    "license": "http://creativecommons.org/licenses/by/4.0/",
    "primary_cat": "math-ph",
    "submitted_at": "2026-05-15T07:01:35Z",
    "title_canon_sha256": "8edc86f27dcab524da00d6d9e40e454f493ce909bb6c46c4943c99cb45dc540f"
  },
  "schema_version": "1.0",
  "source": {
    "id": "2605.15679",
    "kind": "arxiv",
    "version": 1
  }
}