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Abstract
In this paper, we derive a "Hamiltonian formalism" for a wide
class of mechanical systems, that includes, as particular cases,
classical Hamiltonian systems, nonholonomic systems, some classes of
servomechanisms... This construction strongly relies on the
geometry characterizing the different systems. The main result of
this paper is to show how the general construction of the
Hamiltonian symplectic formalism in classical mechanics remains
essentially unchanged starting from the more general framework of
algebroids. Algebroids are, roughly speaking, vector bundles
equipped with a bilinear bracket of sections and two vector bundle
morphisms (the anchors maps)
satisfying a
Leibniz-type property. The bilinear bracket is not, in general,
skew-symmetric and it does not satisfy, in general, the Jacobi
identity. Since skew-symmetry is related with preservation of the
Hamiltonian, our Hamiltonian framework also covers some examples of
dissipative systems. On the other hand, since the Jacobi identity is
related with the preservation of the associated linear Poisson
structure, then our formalism also admits a Hamiltonian description
for systems which do not preserve this Poisson structure, like
nonholonomic systems.
Some examples of interest
are considered: gradient extension of dynamical systems,
nonholonomic mechanics and generalized nonholonomic mechanics, showing the applicability of our theory and constructing the corresponding Hamiltonian formalism.
Mathematics Subject Classification: Primary: 70H05; Secondary: 70G45; 53D05; 37J60.
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