A k-symplectic Hamiltonian system lifts to a k-contact Hamiltonian system such that any projectable solution of the lifted system projects to a solution of the original system.
Vilariño, S.: Methods of Differential Geometry in Classical Field Theories
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A relation between k-symplectic and k-contact Hamiltonian systems
A k-symplectic Hamiltonian system lifts to a k-contact Hamiltonian system such that any projectable solution of the lifted system projects to a solution of the original system.