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Presymplectic form

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A closed degenerate differential 2-form of constant rank

In geometric mechanics a presymplectic form is a closed differential 2-form of constant rank on a manifold. However, some authors use different definitions. Recently, Hajduk and Walczak defined a presymplectic form as a closed differential 2-form of maximal rank on a manifold of odd dimension. A symplectic form is a presymplectic form that is also nondegenerate. Lack of nondegeneracy, leading to presymplectic forms, occurs in dynamical systems with singular Lagrangians, Hamiltonian systems with constraints and control theory.

References

  1. Vaisman, Izu (1983). "Geometric quantization on presymplectic manifolds". Monatshefte für Mathematik. 96 (4): 293–310. doi:10.1007/BF01471212. ISSN 0026-9255. S2CID 123233096.
  2. Boguslaw Hajduk & Rafa Walczak (2009). "Presymplectic manifolds". arXiv:0912.2297 .
  3. Martınez, Eduardo. "Symplectic, Presymplectic, Poisson, Dirac, ..." (PDF). Archived from the original (PDF) on 12 June 2013. Retrieved 26 July 2013.
  4. Alishah, Hassan Najafi. "KAM Theory, Presymplectic Dynamics and Lie algebroids" (PDF). UNIVERSIDADE TÉCNICA DE LISBOA INSTITUTO SUPERIOR TÉCNICO. Retrieved 26 July 2013.


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