Hamiltonian system

Weinstein A 1973 Normal modes for nonlinear Hamiltonian systems Inv. Math. 20 47... 

D. Okunbor and R. D. Skeel. Explicit canonical methods for Hamiltonian systems. Working document. Numerical Computing Group, University of Illinois at UrbanarChampaign, 1991. 

Backward Analysis In this type of analysis, the discrete solution is regarded as an exact solution of a perturbed problem. In particular, backward analysis of symplectic discretizations of Hamiltonian systems (such as the popular Verlet scheme) has recently achieved a considerable amount of attention (see [17, 8, 3]). Such discretizations give rise to the following feature the discrete solution of a Hamiltonian system is exponentially close to the exact solution of a perturbed Hamiltonian system, in which, for consistency order p and stepsize r, the perturbed Hamiltonian has the form [11, 3]... 

Let us introduce a suitably simple example in order to illustrate the notion of almost invariant sets and the performance of our algorithm for Hamiltonian systems. For p = pi,P2),q = (91,92) consider the potential... 

Fig. 5. The left hand side figure shows a contour plot of the potential energy landscape due to V4 with equipotential lines of the energies E = 1.5, 2, 3 (solid lines) and E = 7,8,12 (dashed lines). There are minima at the four points ( 1, 1) (named A to D), a local maximum at (0, 0), and saddle-points in between the minima. The right hand figure illustrates a solution of the corresponding Hamiltonian system with total energy E = 4.5 (positions qi and qs versus time t).

Moreover, our Hamiltonian system possesses an additional symmetry — it is equivariant under the transformation (52,P2) —(92, 2). In other words each of these sets is a candidate for a set B mentioned in the assumptions of Corollary 4. Thus, by this result, both of these sets are almost invariant with... 

J.M. Sanz-Serna and M.P. Calvo. Numerical Hamiltonian Systems. Chapman and Hall, London, Glasgow, New York, Tokyo (1994)... 

B. Leimkuhler and R. D. Skeel. Symplectic numerical integrators in constrained Hamiltonian systems. J. Comp. Phys., 112 117-125, 1994. 

O. Gonzales and J. C. Simo. On the stability of symplectic and energy-momentum conserving algorithms for nonlinear Hamiltonian systems with symmetry. Comp. Meth. App. Mech. Engin., 134 197, 1994. 

On Some Difficulties in Integrating Highly Oscillatory Hamiltonian Systems... 

In this paper, we discuss semi-implicit/implicit integration methods for highly oscillatory Hamiltonian systems. Such systems arise, for example, in molecular dynamics [1] and in the finite dimensional truncation of Hamiltonian partial differential equations. Classical discretization methods, such as the Verlet method [19], require step-sizes k smaller than the period e of the fast oscillations. Then these methods find pointwise accurate approximate solutions. But the time-step restriction implies an enormous computational burden. Furthermore, in many cases the high-frequency responses are of little or no interest. Consequently, various researchers have considered the use of scini-implicit/implicit methods, e.g. [6, 11, 9, 16, 18, 12, 13, 8, 17, 3]. 

W, g potential functions, k 1, has been discussed in various papers (see, for example, [6, 11, 9, 16, 3]). It has been pointed out that, for step-sizes /j > e = 1/ /k, the midpoint method can become unstable due to resonances [9, 16], i.e., for specific values of k. However, generic instabilities arise if the step-size k is chosen such that is not small [3, 6, 18], For systems with a rotational symmetry this has been shown rigorously in [6j. This effect is generic for highly oscillatory Hamiltonian systems, as argued for in [3] in terms of decoupling transformations and proved for a linear time varying system without symmetry. 

Given a general autonomous, separable Hamiltonian system (1), the Hamiltonian... 

This latter modified midpoint method does work well, however, for the long time integration of Hamiltonian systems which are not highly oscillatory. Note that conservation of any other first integral can be enforced in a similar manner. To our knowledge, this method has not been considered in the literature before in the context of Hamiltonian systems, although it is standard among methods for incompressible Navier-Stokes (where its time-reversibility is not an issue, however). 

For highly oscillatory Hamiltonian systems, the best energy conserving midpoint variant that we are aware of is (6). In the sequel we therefore examine only its performance. 

U. Ascher and S. Reich. The midpoint scheme and variants for Hamiltonian systems advantages and pitfalls. SIAM J. Sci. Comput., to appear. 

F.A. Bornemann and Ch. Schiitte. Homogenization of Hamiltonian systems with a strong constraining potential. Physica D, 102 57-77, 1997. 

The exact propagator for a Hamiltonian system for any given time increment At is symplectic. As a consequence it possesses the Liouville property of preserving volume in phase space. 

Symplecticness is a characterization of Hamiltonian systems in terms of their solution. The solution operator t, to) defined by... 

A mapping is said to be symplectic or canonical if it preserves the differential form dp A dq which defines the symplectic structure in the phase space. Differential forms provide a geometric interpretation of symplectic-ness in terms of conservation of areas which follows from Liouville s theorem [14]. In one-degree-of-freedom example symplecticness is the preservation of oriented area. An example is the harmonic oscillator where the t-flow is just a rigid rotation and the area is preserved. The area-preserving character of the solution operator holds only for Hamiltonian systems. In more then one-degree-of-freedom examples the preservation of area is symplecticness rather than preservation of volume [5]. 

Recent mathematical work suggests that—especially for nonlinear phenomena—certain geometric properties can be as important as accuracy and (linear) stability. It has long been known that the flows of Hamiltonian systems posess invariants and symmetries which describe the behavior of groups of nearby trajectories. Consider, for example, a two-dimensional Hamiltonian system such as the planar pendulum H = — cos(g)) or the... 

An example of a symplectic/time-reversible method is the Verlet (leap-frog) scheme. This method is applicable to separataP Hamiltonian systems of the... 

We will introduce the following notation to describe the flow map of a Hamiltonian system with Hamiltonian H ... 

A key feature required of a Hamiltonian system that leads to an efficient method based on splitting is the ability to separate the Hamiltonian into p-dependent and g-dependent terms. 

The form of the Hamiltonian impedes efficient symplectic discretization. While symplectic discretization of the general constrained Hamiltonian system is possible using, e.g., the methods of Jay [19], these methods will require the solution of a nontrivial nonlinear system of equations at each step which can be quite costly. An alternative approach is described in [10] ( impetus-striction ) which essentially converts the Lagrange multiplier for the constraint to a differential equation before solving the entire system with implicit midpoint this method also appears to be quite costly on a per-step basis. 

Since we have discovered the underlying Hamiltonian structure of the QCMD model we are able to apply methods commonly used to construct suitable numerical integrators for Hamiltonian systems. Therefore we transform the QCMD equations (1) into the Liouville formalism. To this end, we introduce a new state z in the phase space, z = and define the nonlinear... 

D. Okunbor, Canonical methods for Hamiltonian systems Numerical experiments , Physica D, 60, 314-322, 1992. 

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