Hamiltonian flow

The cissertion that Hamiltonian flows preserve phase-space volumes is known as Louiville s Theorem, and is easily verified from equation 4.3 by using the Hamiltonian equations 4.5 ... 

In his interesting paper Professor Nicolis raises the question whether models can be envisioned which lead to a spontaneous spatial symmetry breaking in a chemical system, leading, for example, to the production of a polymer of definite chirality. It would be even more interesting if such a model would arise as a result of a measure preserving process that could mimic a Hamiltonian flow. Although we do not have such an example of a chiral process, which imbeds an axial vector into the polymer chain, several years ago we came across a stochastic process that appears to imbed a polar vector into a growing infinite chain. 

However, using a method proposed [60,62,95,112] for experimental analysis of the Hamiltonian flow in an extended phase space of the fluctuating system, we can exploit the analogy between the Wentzel-Freidlin and Pontryagin Hamiltonians arising in the analysis of fluctuations, and the energy-optimal control problem in a nonlinear oscillator. To see how this can be done, let us consider the fluctuational dynamics of the nonlinear oscillator (35). 

Maxwell theory, soliton flows are Hamiltonian flows. Such Hamiltonian functions define symplectic structures6 for which there is an absence of local invariants but an infinite-dimensional group of diffeomorphisms which preserve global properties. In the case of solitons, the global properties are those permitting the matching of the nonlinear and dispersive characteristics of the medium through which the wave moves. 

With the connection of PDEs, and especially soliton forms, to group symmetries established, one can conclude that if the Maxwell equation of motion that includes electric and magnetic conductivity is in soliton (SGE) form, the group symmetry of the Maxwell field is SU(2). Furthermore, because solitons define Hamiltonian flows, their energy conservation is due to their symplectic structure. 

An alternative procedure to generate a canonical transformation is to use the Hamiltonian flow itself. Consider an arbitrary Hamiltonian system of the same dimension as the original system. The associated functional dependence of the final state ait = tf on the initial state at t = t, can be represented by... 

The classical dynamics of a system can also be analyzed on the so-caUed Poincare surface of section (PSS). Hamiltonian flow in the entire phase space then reduces to a Poincare map on a surface of section. One important property of the Poincare map is that it is area-preserving for time-independent systems with two DOFs. In such systems Poincare showed that all dynamical information can be inferred from the properties of trajectories when they cross a PSS. For example, if a classical trajectory is restricted to a simple two-dimensional toms, then the associated Poincare map will generate closed KAM curves, an evident result considering the intersection between the toms and the surface of section. If a Poincare map generates highly erratic points on a surface of section, the trajectory under study should be chaotic. The Poincare map has been a powerful tool for understanding chemical reaction dynamics in few-dimensional systems. 

The Poincare map is a method to transform the continuous flow in n-dimensional phase space to an equivalent discrete flow (map) in a phase space of (n — l)-dimensions (or (n — 2)-dimensions for Hamiltonian flows). 

The intersections of the continuous Hamiltonian flow in the 2n-dimensional phase space, defined by Equation (71), with the surface of section defined by Equation (72), transforms the continuous flow to an equivalent discrete flow (map), on a (2n — 2)-dimensional surface of section (see Figure 12). 

The same investigation in the case of the Hamiltonian flow (Figure 17) shows exactly the same qualitative situation. Not only we are confident that we are not facing anomalous diffusion but also that the orbits correspond really to chaotic motion and not to regular tori which could have FLI larger than log T because of the proximity to a separatrix. 

Indeed, one can prove that the observable G computed on the Hamiltonian flow of the truncated normal form has the representation ... 

The equations of motion (14) also show that pi - cji < 0 on the forward dividing surface B q (E) and pi - Cji > 0 on the backward dividing surface Bj jjf(E). Hence, except for the NHIM, which is an invariant manifold, the dividing surface is everywhere transverse to the Hamiltonian flow. This means that a trajectory, after having crossed the forward or backward dividing surface, Bj q (E) or respectively, must leave the neighborhood of... 

It would be valuable to have a general method for deriving volume conserving methods. It turns out that volume conservation is, itself, most readily obtained as a consequence of a more fundamental property of Hamiltonian flows, the conservation of the symplectic form. 

In this chapter, we show that a symplectic integrator can be viewed as being effectively equivalent to the flow map of a certain Hamiltonian system. The starting point is that symplectic integrators are symplectic maps that are near to the identity since they depend on a parameter (the stepsize h) which can be chosen as small as needed, and, if consistent, in the limit -> 0, such a map must tend to the identity map. We can express the fundamental consequence as follows not only are Hamiltonian flow maps symplectic, but also near-identity symplectic maps are (in an approximate sense) Hamiltonian flow maps [31], The fact leads to the existence of a modifled (perturbed) Hamiltonian from which the discrete trajectory may be derived (as snapshots of continuous trajectories). In some cases we may derive this perturbed Hamiltonian as an expansion in powers of the stepsize. 

For a Hamiltonian flow, we typicaUy simplify notation by writing Ch in place of... 

This bound tends to zero extremely rapidly (more rapidly than any power of h) as h 0. The assumption is that if this error in the approximation of the numerical map by the flow of a truncated perturbed Hamiltonian flow map is much smaller than the other errors present in our model, any drift due to lack of convergence would be minor, perhaps even invisible, on the timescale of simulation. 

As both volume and energy are invariant under the Hamiltonian flow, /r(M) must be invariant under the flow, thus... 

Theorem 4.2.4 is applicable to the case of left-invariant Hamiltonian flows on Lie groups. In particular, Theorem 4.2.4 implies the following assertion. Let F T M — G be momentum mapping. 

Theorem 5.2.1 can be reformulated as follows. Let be a real-analytic two-dimensional compact closed connected manifold endowed with an arbitrary real-analytic Riemannian metric. If the genus of a manifold is higher than unity then the geodesic flow of this metric (as the Hamiltonian flow on a four-dimensional manifold T M ) is not completely Liouville integrable, that is, does not have an additional (second) integral which is independent of the energy integral and is in involution with it. 

Troflmov, V. V., and Fomenko, A. T. Methods of constructing Hamiltonian flows on symmetric spaces and integrability of seveal hydrodynamic systems. DoU. Akad. Nauk SSSR, 254 (1980), No. 6, 1349-1353. 

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