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We introduce the concepts of a multisymplectic structure and a polysymplecticstructure on a general fiber bundle over a general base manifold, define theconcept of the symbol of a multisymplectic form, which is a polysymplectic formrepresenting its leading order contribution, and prove Darboux theorems for theexistence of canonical local coordinates.

Multisymplectic and polysymplectic structures on fiber bundles