On k -polycosymplectic Marsden–Weinstein reductions

We review and slightly improve the known k -polysymplectic Marsden–Weinstein reduction theory by removing some technical conditions on k -polysymplectic momentum maps by developing a theory of affine Lie group actions for k -polysymplectic momentum maps, removing the necessity of their co-adjoint equivariance. Then, we focus on the analysis of a particular case of k -polysymplectic manifolds, the so-called fibred ones, and we study their k -polysymplectic Marsden–Weinstein reductions. Previous results allow us to devise a k -polycosymplectic Marsden–Weinstein reduction theory, which represents one of our main results. Our findings are applied to study coupled vibrating strings and, more generally, k -polycosymplectic Hamiltonian systems with field symmetries. We show that k -polycosymplectic geometry can be understood as a particular type of k -polysymplectic geometry. Finally, a k -cosymplectic to ℓ -cosymplectic geometric reduction theory is presented, which reduces, geometrically, the space-time variables in a k -cosymplectic framework. An application of this latter result to a vibrating membrane with symmetries is given.