dorsal/arxiv
View SchemaDouble Complexes and Cohomological Hierarchy in a Space of Weakly Invariant Lagrangians of Mechanics
| Authors | O. M. Khudaverdian, D. A. Sahakyan |
|---|---|
| Categories | |
| ArXiv ID | physics/9712052 |
| URL | https://arxiv.org/abs/physics/9712052 |
| Journal | Acta Appl.Math. 56 (1999) 181-215 |
Abstract
For a given configuration space $M$ and Lie algebra $g$ whose action is defined on $M$, the space $V_{0.0}$ of weakly $g$-invariant Lagrangians (i.e. Lagrangians whose motion equations left hand sides are $g$-invariant) is studied. The problem is reformulated in the terms of the double complex of Lie algebra cochains with values in the complex of Lagrangians. Calculating the cohomology of this complex using the method of spectral sequences, we come to the hierarchy in the space $V_{0.0}$: The double filtration $V_{s.r}$ ($s=0,1,2,3,4;r=0,1$) and the homomorphisms on every space $V_{s.r}$ are constructed. These homomorphisms take values in cohomologies of the Lie algebra $g$ and configuration space $M$. On one hand every space $V_{s.r}$ is the kernel of the corresponding homomorphism, on the other hand this space is defined by its physical properties.
{
"annotation_id": "eef3c641-1988-4d43-94bc-4ffa07f6bc14",
"date_created": "2026-03-02T18:01:21.639000Z",
"date_modified": "2026-03-02T18:01:21.639000Z",
"file_hash": "02e968ff64b99815a2aca37308efa83e5d8f7c90dd95069e46653e26f0a903a6",
"private": false,
"record": {
"abstract": "For a given configuration space $M$ and Lie algebra $g$ whose action is\ndefined on $M$, the space $V_{0.0}$ of weakly $g$-invariant Lagrangians (i.e.\nLagrangians whose motion equations left hand sides are $g$-invariant) is\nstudied.\n The problem is reformulated in the terms of the double complex of Lie algebra\ncochains with values in the complex of Lagrangians. Calculating the cohomology\nof this complex using the method of spectral sequences, we come to the\nhierarchy in the space $V_{0.0}$:\n The double filtration $V_{s.r}$ ($s=0,1,2,3,4;r=0,1$) and the homomorphisms\non every space $V_{s.r}$ are constructed.\n These homomorphisms take values in cohomologies of the Lie algebra $g$ and\nconfiguration space $M$. On one hand every space $V_{s.r}$ is the kernel of the\ncorresponding homomorphism, on the other hand this space is defined by its\nphysical properties.",
"arxiv_id": "physics/9712052",
"authors": [
"O. M. Khudaverdian",
"D. A. Sahakyan"
],
"categories": [
"math-ph",
"hep-th",
"math.DG",
"math.MP"
],
"journal_ref": "Acta Appl.Math. 56 (1999) 181-215",
"title": "Double Complexes and Cohomological Hierarchy in a Space of Weakly Invariant Lagrangians of Mechanics",
"url": "https://arxiv.org/abs/physics/9712052"
},
"schema_id": "dorsal/arxiv",
"source": {
"execution_id": "90459a54-d037-4d75-b8bc-c70f16777127",
"id": "arXiv Dataset IDs",
"type": "Model",
"variant": "snapshot-2026-03-01",
"version": "0.1.0"
},
"user_id": 1000002
}