In this paper, we study the integrability of contact Hamiltonian systems, both time-dependent and independent. In order to do so, we construct a Hamilton–Jacobi theory for these systems following two ...approaches, obtaining two different Hamilton–Jacobi equations. Compared to conservative Hamiltonian systems, contact Hamiltonian systems depend of one additional parameter. The fact of obtaining two equations reflects whether we are looking for solutions depending on this additional parameter or not. In order to illustrate the theory developed in this paper, we study three examples: the free particle with a linear external force, the freely falling particle with linear dissipation and the damped and forced harmonic oscillator.
The nonholonomic dynamics can be described by the so-called nonholonomic bracket on the constrained submanifold, which is a non-integrable modification of the Poisson bracket of the ambient space, in ...this case, of the canonical bracket on the cotangent bundle of the configuration manifold. This bracket was defined in 6,21 although there was already some particular and less direct definition. On the other hand, another bracket, also called nonholonomic bracket, was defined using the description of the problem in terms of skew-symmetric algebroids 13,20. Recently, reviewing two older papers by R. J. Eden 17,18, we have defined a new bracket which we call Eden bracket. In the present paper, we prove that these three brackets coincide. Moreover, the description of the nonholonomic bracket à la Eden has allowed us to make important advances in the study of Hamilton–Jacobi theory and the quantization of nonholonomic systems.
We present a unified geometric framework for describing both the Lagrangian and Hamiltonian formalisms of contact autonomous mechanical systems, which is based on the approach of the pioneering work ...of R. Skinner and R. Rusk. This framework permits to skip the second order differential equation problem, which is obtained as a part of the constraint algorithm (for singular or regular Lagrangians), and is especially useful to describe singular Lagrangian systems. Some examples are also discussed to illustrate the method.
A unified geometric framework is presented for describing both the Lagrangian and Hamiltonian formalisms of contact autonomous mechanical systems, which is based on the approach of the pioneering work of R. Skinner and R. Rusk. This framework permits to skip the second order differential equation problem, which is obtained as a part of the constraint algorithm (for singular or regular Lagrangians), and is especially useful to describe singular Lagrangian systems. Some examples are also discussed to illustrate the method.
The specific parasite of Lactuca (Compositae) described in 2005 from the Iberian Peninsula as Phelipanche inexspectata (Orobanchaceae) and known so far in northeastern Spain and southern France, is ...shown to also occur in mountain areas of northern Africa, where it had been previously described under the neglected name Phelipanche cernua. Given the recent proposal to consider the aforementioned Lactuca parasite a mere variant of Phelipanche schultzii, we stress the neat differences between both species. RESUMEN: Phelipanche cernua Pomel (Orobanchaceae), un nombre prioritario para la especie del Mediterráneo Occidental recientemente descrita como Ph. inexspectata. Se dan pruebas de que la parásita específica de Lactuca (Compositae) descrita en 2005 de la Península Ibérica como Phelipanche inexspectata (Orobanchaceae), y conocida hasta ahora del noreste de España y el sur de Francia, alcanza las montañas del norte de África, de las que ya había sido descrita bajo el nombre Phelipanche cernua. Dada la reciente afirmación de que dicha parásita de Lactuca es una mera variante de Phelipanche schultzii, recalcamos las netas diferencias entre ambas especies.
The aim of this paper is to develop a Hamilton--Jacobi theory for contact Hamiltonian systems. We find several forms for a suitable Hamilton-Jacobi equation accordingly to the Hamiltonian and the ...evolution vector fields for a given Hamiltonian function. We also analyze the corresponding formulation on the symplectification of the contact Hamiltonian system, and establish the relations between these two approaches. In the last section, some examples are discussed.
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