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In this paper we aim at addressing the globalization problem ofHamilton-DeDonder-Weyl equations on a local $k$-symplectic framework and weintroduce the notion of {\it locally conformal $k$-symplectic (l.c.k-s.)manifolds}. This formalism describes the dynamical properties of physicalsystems that locally behave like multi-Hamiltonian systems. Here, we describethe local Hamiltonian properties of such systems, but we also provide a globaloutlook by introducing the global Lee one-form approach. In particular, thedynamics will be depicted with the aid of the Hamilton--Jacobi equation, whichis specifically proposed in a l.c.k-s manifold.

The globalization problem of the Hamilton-DeDonder-Weyl equations on a local $k$-symplectic framework