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Affine Hamiltonians in higher order geometry

Affine hamiltonians are defined in the paper and their study is based especially on the fact that in the hyperregular case they are dual objects of lagrangians defined on affine bundles, by mean of natural Legendre maps. The variational problems for affine hamiltonians and lagrangians of order $k\geq 2$ are studied, relating them to a Hamilton equation. An Ostrogradski type theorem is proved: the Hamilton equation of an affine familtonian $h$ is equivalent with Euler-Lagrange equation of its dual lagrangian $L$. Zermelo condition is also studied and some non-trivial examples are given.