An invitation to multisymplectic geometry

Abstract

In this article we study multisymplectic geometry, i.e., the geometry of manifolds with a non-degenerate, closed differential form. First we describe the transition from Lagrangian to Hamiltonian classical field theories, and then we reformulate the latter in multisymplectic terms. Furthermore, we investigate basic questions on normal forms of multisymplectic manifolds, notably the questions wether and when Darboux-type theorems hold, and how many diffeomorphisms certain, important classes of multisymplectic manifolds possess. Finally, we survey recent advances in the area of symmetries and conserved quantities on multisymplectic manifolds.

The article 1608.07424 is incorporated and now obsolete

References in corpus (2)

• Linear orbits of alternating forms on real vector spaces
• A multisymplectic manifold not covered by Darboux charts

Cited by in corpus (18)

• Reduction of multisymplectic manifolds
• Covariant canonical formulations of classical field theories
• Quantization of Polysymplectic Manifolds
• On the prequantisation map for 2-plectic manifolds
• Polysymplectic Reduction and the Moduli Space of Flat Connections
• Coisotropic reduction in Multisymplectic Geometry
• Properties of Multisymplectic Manifolds
• Graded Poisson and Graded Dirac structures
• A hydrodynamical homotopy co-momentum map and a multisymplectic interpretation of higher order linking numbers
• Brackets in multicontact geometry and multisymplectization
• Conditions for symmetry reduction of polysymplectic and polycosymplectic structures
• Lagrangian submanifolds of standard multisymplectic manifolds
• Multisymplectic actions of compact Lie groups on spheres
• Reduction of $L_{\infty }$-Algebras of Observables on Multisymplectic Manifolds
• Multisymplectic Formalism for Cubic Horndeski Theories
• Generalizing Hamiltonian Mechanics with Closed Differential Forms
• Multisymplectic observable reduction using constraint triples
• Spin(7)-Manifolds and Multisymplectic Geometry