In geometric mechanics a presymplectic form is a closed differential 2-form of constant rank on a manifold.[1] 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.[2] A symplectic form is a presymplectic form that is also nondegenerate.[3] Lack of nondegeneracy, leading to presymplectic forms, occurs in dynamical systems with singular Lagrangians, Hamiltonian systems with constraints and control theory.[4]
References
edit- ^ Vaisman, Izu (1983). "Geometric quantization on presymplectic manifolds". Monatshefte für Mathematik. 96 (4): 293–310. doi:10.1007/BF01471212. ISSN 0026-9255. S2CID 123233096.
- ^ Boguslaw Hajduk & Rafa Walczak (2009). "Presymplectic manifolds". arXiv:0912.2297 [math.SG].
- ^ Martınez, Eduardo. "Symplectic, Presymplectic, Poisson, Dirac, ..." (PDF). Archived from the original (PDF) on 12 June 2013. Retrieved 26 July 2013.
- ^ 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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