Hamiltonian Mechanical Systems Research Articles

The geometry of the solution space of first order Hamiltonian field theories I: From particle dynamics to free electrodynamics

We analyse the problem of defining a Poisson bracket structure on the space of solutions of the equations of motions of first order Hamiltonian field theories. The cases of Hamiltonian mechanical point systems – as a (0+1)-dimensional field – and more general field theories without gauge symmetries are addressed by showing the existence of a symplectic (and, thus, a Poisson) structure on the space of solutions. Also the easiest case of gauge theory, namely free electrodynamics, is considered: within this problem, a pre-symplectic tensor on the space of solutions is introduced, and a Poisson structure is induced in terms of a flat connection on a suitable bundle associated to the theory.

Accurate and efficient simulations of Hamiltonian mechanical systems with discontinuous potentials

This article considers Hamiltonian mechanical systems with potential functions admitting jump discontinuities. The focus is on accurate and efficient numerical approximations of their solutions, which will be defined via the laws of reflection and refraction. Despite of the success of symplectic integrators for smooth mechanical systems, their construction for the discontinuous ones is nontrivial, and numerical convergence order can be impaired too. Several rather-usable numerical methods are proposed, including: a first-order symplectic integrator for general problems, a third-order symplectic integrator for problems with only one linear interface, arbitrarily high-order reversible integrators for general problems (no longer symplectic), and an adaptive time-stepping version of the previous high-order method. Interestingly, whether symplecticity leads to favorable long time performance is no longer clear due to discontinuity, as traditional Hamiltonian backward error analysis does not apply any more. Therefore, at this stage, our recommended default method is the last one. Various numerical evidence, on the order of convergence, long time performance, momentum map conservation, and consistency with the computationally-expensive penalty method, are supplied. A complex problem, namely the Sauteed Mushroom, is also proposed and numerically investigated, for which multiple bifurcations between trapped and ergodic dynamics are observed.

A summary on symmetries and conserved quantities of autonomous Hamiltonian systems

A complete geometric classification of symmetries of autonomous Hamiltonian mechanical systems is established; explaining how to obtain their associated conserved quantities in all cases. In particular, first we review well-known results and properties about the symmetries of the Hamiltonian and of the symplectic form and then some new kinds of non-symplectic symmetries and their conserved quantities are introduced and studied.

On the integrable deformations of a system related to the motion of two vortices in an ideal incompressible fluid

Altering the first integrals of an integrable system integrable deformations of the given system are obtained. These integrable deformations are also integrable systems, and they generalize the initial system. In this paper we give a method to construct integrable deformations of maximally superintegrable Hamiltonian mechanical systems with two degrees of freedom. An integrable deformation of a maximally superintegrable Hamiltonian mechanical system preserves the number of first integrals, but is not a Hamiltonian mechanical system, generally. We construct integrable deformations of the maximally superintegrable Hamiltonian mechanical system that describes the motion of two vortices in an ideal incompressible fluid, and we show that some of these integrable deformations are Hamiltonian mechanical systems too.

Hamiltonian Mechanical System with Split Quaternions

In this article, firstly we study about geometrical applications of split quaternions. Then, we obtain Hamitonian mechanical systems with Split quaternions. Quaternionic and Coquaternionic (split analoque of quaternions) extensions of Hamiltonian mechanics are introduced and are shown as offer a unifying framework for quantum mechanics. This study leads to the possibility of employing algebraic techniques of quaternions and coquaternions to absorbing in quantum mechanics. The founded equations are compared with the Hamiltonian energy equations generally are known and the Hamilton energy equations are obtained in Minkowski space.

Reduction of Hamiltonian Mechanical Systems With Affine Constraints: A Geometric Unification

This paper presents a geometrical approach to the dynamical reduction of a class of constrained mechanical systems. The mechanical systems considered are with affine nonholonomic constraints plus a symmetry group. The dynamical equations are formulated in a Hamiltonian formalism using the Hamilton–d'Alembert equation, and constraint forces determine an affine distribution on the configuration manifold. The proposed reduction approach consists of three main steps: (1) restricting to the constrained submanifold of the phase space, (2) quotienting the constrained submanifold, and (3) identifying the quotient manifold with a cotangent bundle. Finally, as a case study, the dynamical reduction of a two-wheeled rover on a rotating disk is detailed. The symmetry group for this example is the relative configuration manifold of the rover with respect to the inertial space. The proposed approach in this paper unifies the existing reduction procedures for symmetric Hamiltonian systems with conserved momentum, and for Chaplygin systems, which are normally treated separately in the literature. Another characteristic of this approach is that although it tracks the structure of the equations in each reduction step, it does not insist on preserving the properties of the system. For example, the resulting dynamical equations may no longer correspond to a Hamiltonian system. As a result, the invariance condition of the Hamiltonian under a group action that lies at the heart of almost every reduction procedure is relaxed.

Discrete Lagrangian mechanics for nonseparable nonsmooth systems

SummaryWe consider event‐driven schemes for the simulation of nonseparable mechanical systems subject to holonomic unilateral constraints. Systems are modeled in discrete time using variational integrator (VI) theory, by which equations of motion follow from discrete variational principles. For smooth dynamics, VIs are known to exactly conserve a discrete symplectic form and a modified Hamiltonian function. The latter of these conservation laws can play a pivotal role in stabilizing the energy behavior of collision simulations. Previous efforts to leverage modified Hamiltonian conservation have been limited to integrators using the Störmer–Verlet method on separable, nonsmooth Hamiltonian mechanical systems. We generalize the existing approach to the family of all VIs applied to nonseparable, potentially nonconservative Lagrangian mechanical systems. We examine the properties of the resulting integrators relative to other structured collision simulation methods in terms of conserved quantities, trajectory errors as a function of initial condition, and required computation time. Interestingly, we find that the modified collision Verlet algorithm (MCVA) using the Störmer–Verlet integrator defined as a composition method leads to the best accuracy. Although relative to this method, the VI‐based generalized MCVA method offers computational savings when collisions are particularly sparse. Copyright © 2015 John Wiley & Sons, Ltd.

Nonholonomic dynamical reduction of open-chain multi-body systems: A geometric approach

This paper studies the geometry behind nonholonomic Hamilton's equation to present a two-stage reduction procedure for the dynamical equations of nonholonomic open-chain multi-body systems with multi-degree-of-freedom joints. In this process, we use the Chaplygin reduction and an almost symplectic reduction theorem. We first restate the Chaplygin reduction theorem on cotangent bundle for nonholonomic Hamiltonian mechanical systems with symmetry. Then, under some conditions we extend this theorem to include a second reduction stage using an extended version of the symplectic reduction theorem for almost symplectic manifolds. We briefly introduce the displacement subgroups and accordingly open-chain multi-body systems consisting of such joints. For a holonomic open-chain multi-body system, the relative configuration manifold corresponding to the first joint is a symmetry group. Hence, we focus on a class of nonholonomic distributions on the configuration manifold of an open-chain multi-body system that is invariant under the action of this group. As the first stage of reduction procedure, we perform the Chaplygin reduction for such systems. We then introduce a number of sufficient conditions for a reduced system to admit more symmetry due to the action of the relative configuration manifolds of other joints. Under these conditions, we present the second stage of the reduction process for nonholonomic open-chain multi-body systems with multi-degree-of-freedom joints. Finally, we explicitly derive the reduced dynamical equations in the local coordinates for an example of a two degree-of-freedom crane mounted on a four-wheel car to illustrate the results of this paper.

MECHANICAL SYSTEMS ON GENERALIZED-QUATERNIONIC KÄHLER MANIFOLDS

In this paper, we introduce generalized-quaternionic Kähler analogue of Lagrangian and Hamiltonian mechanical systems. Finally, the geometrical-physical results related to generalized-quaternionic Kähler mechanical systems are also given.

From Chemical Langevin Equations to Fokker—Planck Equation: Application of Hodge Decomposition and Klein—Kramers Equation

The stochastic systems without detailed balance are common in various chemical reaction systems, such as metabolic network systems. In studies of these systems, the concept of potential landscape is useful. However, what are the sufficient and necessary conditions of the existence of the potential function is still an open problem. Use Hodge decomposition theorem in differential form theory, we focus on the general chemical Langevin equations, which reflect complex chemical reaction systems. We analysis the conditions for the existence of potential landscape of the systems. By mapping the stochastic differential equations to a Hamiltonian mechanical system, we obtain the Fokker—Planck equation of the chemical reaction systems. The obtained Fokker—Planck equation can be used in further studies of other steady properties of complex chemical reaction systems, such as their steady state entropies.

A Survey on Geometric Dynamics of 4-Walker Manifold

A Walker n-manifold is a semi-Riemannian n-manifold, which admits a field of parallel null r-planes, with r ≤ 2/n . It is well-known that semi-Riemannian geometry has an important tool to describe spacetime events. Therefore, solutions of some structures about 4-Walker manifold can be used to explain spacetime singularities. Then, here we present complex and paracomplex analogues of Lagrangian and Hamiltonian mechanical systems on 4-Walker manifold. Finally, the geometrical-physical results related to complex (paracomplex) mechanical systems are also discussed.

Hamiltonian mechanical systems on Lie algebroids, unimodularity and preservation of volumes

In this paper we discuss the relation between the unimodularity ofa Lie algebroid $\tau_{A}: A \to Q$ and the existence of invariantvolume forms for the dynamics of hamiltonian mechanical systems onthe dual bundle $A$*. The results obtained in this direction areapplied to several hamiltonian systems on different examples ofLie algebroids.

Higher order complex Lagrangian and Hamiltonian mechanics systems

In this Letter, it was present higher order vertical and complete lifts of Euler–Lagrange and Hamiltonian equations introduced on Kählerian manifold to its extensions. Finally, it was discussed differential geometric and physical results for the higher order Lagrangian and Hamiltonian mechanical systems obtained on the extended Kählerian manifolds.

Discrete dynamics of implicit time integration schemes for a dissipative system

AbstractA dissipative one degree of freedom dynamical system is investigated. Three time integration schemes are applied to the time continuous system leading to time discrete systems. Their energy dissipation and phase space contraction behaviour is analysed and differs from the properties of the differential equation. One of the schemes guarantees energy dissipation, whereas another is characterized by phase space contraction; the third scheme does not accurately discretize both properties. It was found that the scheme with unconditional phase space contraction and the standard scheme possess spurious steady states, higher order fixed points, as a result of tangled stable and unstable manifolds of a saddle. However, the consistently energy dissipative scheme exhibits no spurious steady states. It was concluded that the property of correct energy dissipation has a beneficial influence on the quality of the numerical solution. This result is analogous to that obtained for Hamiltonian mechanical systems for which algorithms with energy‐conservation are often considered preferable to symplectic schemes. Copyright © 2003 John Wiley & Sons, Ltd.

Geometric Quantization of Relativistic Hamiltonian Mechanics

A relativistic Hamiltonian mechanical system is seen as a conservative Dirac constraint system on the cotangent bundle of a pseudo-Riemannian manifold. We provide geometric quantization of this cotangent bundle where the quantum constraint serves as a relativistic quantum equation.

Theq-symplectic form on the quantum hyperplane and noncommutative Hamiltonian mechanics

Theq-fields,q-curves, andq-symplectic forms on the quantum hyperplane are given by the use of theq-sequences method. With these structures we discuss a possible noncommutative Hamiltonian mechanical system and give two concrete examples.

QUANTUM-DEFORMED GEOMETRY ON PHASE-SPACE

In this paper we extend the standard Moyal formalism to the tangent and cotangent bundle of the phase-space of any Hamiltonian mechanical system. In this manner we build the quantum analog of the classical Hamiltonian vector-field of time evolution and its associated Lie-derivative. We also use this extended Moyal formalism to develop a quantum analog of the Cartan calculus on symplectic manifolds.

On observable algebras of a class of associative mechanical systems

This paper classifies the associative algebras of functions invariant with respect to the adjoint action of the group of affine transformations of the line, these functions containing an everywhere dense subalgebra of polynomials. The applications of the results to the problem of describing possible Hamiltonian mechanical systems are discussed.

On the Construction and Comparison of Difference Schemes

On the Construction and Comparison of Difference Schemes