Variational principles and symmetries on fibered multisymplectic manifolds

• Published in: Communications in Mathematics 2016-12, Vol.24 (2), p.137-152
• Main Authors: Gaset, Jordi, Prieto-Martínez, Pedro D., Román-Roy, Narciso
• Format: Article
• Language: English
• Subjects: 49 Calculus of variations and optimal control; optimization; 49S05 Variational principles of physics; 53 Differential geometry; 53D Symplectic geometry, contact geometry; 55 Algebraic topology; 55R Fiber spaces and bundles; 70 Mechanics of particles and systems; 70S Classical field theories; Anàlisi matemàtica; Calculus of variations; Classificació AMS; Conserved quantities; Càlcul de variacions; Espais fibrats (Matemàtica); Fiber bundles; Fiber spaces (Mathematics); Field theory (Physics); Física matemàtica; Geometria; Geometria computacional; Geometria simplèctica; Matemàtiques i estadística; Mathematics; Multisymplectic manifolds; Noether theorem; Symmetries; Symplectic geometry; Variational principles; Àrees temàtiques de la UPC
• ISSN: 2336-1298; 1804-1388; 2336-1298
• DOI: 10.1515/cm-2016-0010

The standard techniques of variational calculus are geometrically stated in the ambient of fiber bundles endowed with a (pre)multi-symplectic structure. Then, for the corresponding variational equations, conserved quantities (or, what is equivalent, conservation laws), symmetries, Cartan (Noether) symmetries, gauge symmetries and different versions of Noether's theorem are studied in this ambient. In this way, this constitutes a general geometric framework for all these topics that includes, as special cases, first and higher order field theories and (non-autonomous) mechanics.