Symmetries, pseudosymmetries and conservation laws in Lagrangian and Hamiltonian \(k\)-symplectic formalisms (Q2802583)

The author presents Lagrangian and Hamiltonian \(k\)-symplectic formalisms, he recalls the notions of symmetry and conservation law and defines the notion of pseudosymmetry as a natural extension of symmetry. The author revisits the study of symmetries, conservation laws and relationship between this in the framework of \(k\)-symplectic geometry and he improves the results obtained in [An. Ştiinţ. Univ. Al. I. Cuza Iaşi, Ser. Nouă, Mat. 53, 249--267 (2007; Zbl 1189.70138); ``Symmetries and conservation laws in \(k\)-symplectic geometry'', BSGP 20, 16--35 (2013)]. More exactly, he intends to extend the study of symmetries and conservation laws from classical mechanics to the first-order classical field theories, both for the Lagrangian and Hamiltonian formalisms, using Günther's \(k\)-symplectic description, and considering only the regular case. The author obtains new kinds of conservation laws for \(k\)-symplectic Hamiltonian systems and \(k\)-symplectic Lagrangian systems, nonclassical, without the help of a Noether's type theorem, using only the relationship between symmetries, pseudosymmetries and conservation laws.