Abstract. Recently, M. de León el al. ([9]) have developed a geometric Hamilton-Jacobi theory for Classical Field Theories in the setting of multisymplectic geometry. Our purpose in the current paper is to establish the corresponding Hamilton-Jacobi theory in the Cauchy data space, and relate both approaches.

In this paper we develop a Hamilton-Jacobi theory in the setting of almost Poisson manifolds. The theory extends the classical Hamilton-Jacobi theory and can be also applied to very general situations including nonholonomic mechanical systems and time dependent systems with external forces.

In this paper we introduce poly-Poisson structures as a higher-order extension of Poisson structures. It is shown that any poly-Poisson structure is endowed with a polysymplectic foliation. It is also proved that if a Lie group acts polysymplectically on a polysymplectic manifold then, under certain regularity conditions, the reduced space is a poly-Poisson manifold. In addition, some interesting examples of poly-Poisson manifolds are discussed.Under the existence of symmetries, it is possible to perform a reduction procedure in which some of the variables are reduced. The interest of the reduction procedure is twofold: not only we are able to reduce the dynamics but it is a way to generate new examples of symplectic manifolds. One of the reduction methods is the Marsden-Weinstein-Meyer reduction procedure [16]: Given a Hamiltonian action of a Lie group G on a symplectic manifold (M, Ω) with equivariant moment map J : M → g * , it is possible to obtain a symplectic structure on the quotient manifold J −1 (µ)/G µ . An interesting example is the action of a Lie group G on its cotangent bundle T * G by cotangent lifts of left translations and the moment map J : T * G → g * is given by J(α g ) = (T e R g ) * (α g ). Here, the reduced space J −1 (µ)/G µ is just the coadjoint orbit along the element µ ∈ g * endowed with the Kirillov-Kostant-Souriau symplectic structure.On the other hand, the existence of a Lie group of symmetries for a symplectic manifold is one of the justifications for the introduction of Poisson manifolds, which generalize symplectic manifolds. Indeed if (M, Ω) is a symplectic manifold and G is a Lie group acting freely and 2010 Mathematics Subject Classification. 53D05, 53D17, 70645.