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This work presents a comprehensive review of the $k$-polysymplecticMarsden-Weinstein reduction theory, rectifying prior errors and inaccuracies inthe literature while introducing novel findings. It also emphasises the genuinepractical significance of seemingly minor technical details. On this basis, weintroduce a novel $k$-polysymplectic energy-momentum method, new relatedstability analysis techniques, and apply them to Hamiltonian systems ofordinary differential equations relative to a $k$-polysymplectic manifold. Weprovide detailed examples of both physical and mathematical significance,including the study of complex Schwarz equations related to the Schwarzderivative, a series of isotropic oscillators, integrable Hamiltonian systems,quantum oscillators with dissipation, affine systems of differential equations,and polynomial dynamical systems.

An energy-momentum method for ordinary differential equations with an underlying $k$-polysymplectic manifold