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The presence of symmetries in a Hamiltonian system usually implies the existence of conservation laws that are represented mathematically in terms of the dynamical preservation of the level sets of a momentum mapping. The symplectic or Marsden–Weinstein reduction procedure takes advantage of this and associates to the original system a new Hamiltonian… (More)

We present reduction and reconstruction procedures for the solutions of symmetric stochastic differential equations, similar to those available for ordinary differential equations. Additionally, we use the local tangent-normal decomposition, available when the symmetry group is proper, to construct local skew-product splittings in a neighborhood of any… (More)

- Jerrold E. Marsden, Gerard Misio, Juan-Pablo Ortega, Matthew Perlmutter, Tudor S. Ratiu
- 2007

- Joan-Andreu Lázaro-Camı, Juan-Pablo Ortega
- 2007

We use the global stochastic analysis tools introduced by P. A. Meyer and L. Schwartz to write down a stochastic generalization of the Hamilton equations on a Poisson manifold that, for exact symplectic manifolds, satisfy a natural critical action principle similar to the one encountered in classical mechanics. Several features and examples in relation with… (More)

An estimate on the number of distinct relative periodic orbits around a stable relative equilibrium in a Hamiltonian system with continuous symmetry is given. This result constitutes a generalization to the Hamiltonian symmetric framework of a classical result by Weinstein and Moser on the existence of periodic orbits in the energy levels surrounding a… (More)

We provide a model for an open invariant neighborhood of any orbit in a symplectic manifold endowed with a canonical proper symmetry. Our results generalize the constructions of Marle [Mar84, Mar85] and Guillemin and Sternberg [GS84] for canonical symmetries that have an associated momentum map. In these papers the momentum map played a crucial role in the… (More)

The authors' recent paper in Reports in Mathematical Physics develops Dirac reduction for cotangent bundles of Lie groups, which is called Lie– Dirac reduction. This procedure simultaneously includes Lagrangian, Hamil-tonian, and a variational view of reduction. The goal of the present paper is to generalize Lie–Dirac reduction to the case of a general… (More)

The estimation of multivariate GARCH time series models is a difficult task mainly due to the significant overparameterization exhibited by the problem and usually referred to as the " curse of dimensionality ". For example, in the case of the VEC family, the number of parameters involved in the model grows as a polynomial of order four on the… (More)

The relative equilibria of a symmetric Hamiltonian dynamical system are the critical points of the so-called augmented Hamiltonian. The underlying geometric structure of the system is used to decompose the critical point equations and construct a collection of implicitly defined functions and reduced equations describing the set of relative equilibria in a… (More)

- Joan-Andreu Lázaro-Camı, Juan-Pablo Ortega
- 2008

We present reduction and reconstruction procedures for the solutions of symmetric stochastic differential equations, similar to those available for ordinary differential equations. The general methods introduced in the first part of the paper are then adapted to the Hamiltonian case, which is studied with special care and illustrated with several examples.