Incomplete Dirac reduction of constrained Hamiltonian systems

Incomplete Dirac reduction of constrained Hamiltonian systems

Numerical integration of constrained Hamiltonian systems using Dirac brackets

We study the numerical properties of the equations of motion of constrained systems derived with Dirac brackets. This formulation is compared with one based on the extended Hamiltonian. As concrete examples, a pendulum in Cartesian coordinates and a chain molecule are treated.

Model reduction of port-Hamiltonian systems based on reduction of Dirac structures

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Integrability and Reduction of Hamiltonian Actions on Dirac Manifolds

For a Hamiltonian, proper and free action of a Lie group G on a Dirac manifold (M,L), with a regular moment map μ :M → g∗, the manifolds M/G, μ−1(0) and μ−1(0)/G all have natural induced Dirac structures. If (M,L) is an integrable Dirac structure, we show thatM/G is always integrable, but μ−1(0) and μ−1(0)/G may fail to be integrable, and we describe the obstructions to their integrability.

Geometric reduction of Hamiltonian systems

Given a foliation S of a manifold M, a distribution Z in M transveral to S and a Poisson bivector Π on M we present a geometric method of reducing this operator on the foliation S along the distribution Z. It encompasses the classical ideas of Dirac (Dirac reduction) and more modern theory of J. Marsden and T. Ratiu, but our method leads to formulas that allow for an explicit calculation of the...