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Almost product structures and Poisson reduction of presymplectic systems. (English) Zbl 0892.58032

The authors associate to an almost product structure \(F\) adapted to a presymplectic structure a bracket which satisfies all the properties of a Poisson bracket except the Jacobi identity. It is shown that the Jacobi identity is equivalent with the integrability of \(F\). Then, an integrable almost product structure yields a Poisson structure which is used to obtain a Poisson reduction.

MSC:

37J15 Symmetries, invariants, invariant manifolds, momentum maps, reduction (MSC2010)
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
70H03 Lagrange’s equations
70H33 Symmetries and conservation laws, reverse symmetries, invariant manifolds and their bifurcations, reduction for problems in Hamiltonian and Lagrangian mechanics
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