Symplectic map

G. Benettin and A. Giorgilli. On the Hamiltonian interpolation of near to the identity symplectic mappings with applications to symplectic integration algorithms. J. Stat. Phys. 74 (1994)... 

Wisdom, J. Holman, M. Symplectic Maps for the JV-body Problem. Astron. J. 102 (1991) 1528-1538... 

A convenient and constructive approach to attain symplectic maps is given by the composition of symplectic maps, which yields again a symplectic map. For appropriate Hk, the splittings (6) and (7) are exactly of this form If the Hk are Hamiltonians with respect to the whole system, then the exp rLnk) define the phase flow generated by these Hk- Thus, the exp TL-Hk) are symplectic maps on the whole phase space and the compositions in (6) and (7) are symplectic maps, too. Moreover, in order to allow for a direct numerical realization, we have to find some Hk for which either exp(rL-Kfc) has an analytic solution or a given symplectic integrator. 

The resulting numerical method is obviously symplectic since exp L-Hi) and exp AtL-H ) are symplectic maps and the composition of symplectic maps yields a symplectic map. 

A. J. Dragt and J. M. Finn, Lie series and invariant functions for analytic symplectic maps,... 

In two-dimensional Hamiltonian systems, the trajectories can be visualized by means of the Poincare surface of section plot. It is also possible to study two-dimensional Hamiltonian systems using the two-dimensional symplectic mapping. A typical phase space portrait of generic nonhyperbolic phase space is... 

N > 1 (for symplectic maps), N > 2 (for systems with energy conservation)... 

It is very reasonable to compute the diffusion coefficient in order to study transport in symplectic maps. Diffusion coefficient is defined by... 

In previous papers (Froeschle et al. 2000, Guzzo et al. 2002, Lega et al. 2002) we used the FLI to describe the geometry of the resonances, integrating orbits of the Hamiltonian system of equation 6 and of the following 4-dimensional symplectic map ... 

Both for the Hamiltonian system (equation 6) and for the 4 dimensional symplectic mapping (equation 7) we have computed the FLI charts for different values of the perturbing parameter and we have selected a low order resonance. In the case of the Hamiltonian system we have considered the unperturbed resonance l = 2/2, while for the mapping we have chosen 2 = —x/2. 

Froeschle, C. and Lega, E. (2000). On the structure of symplectic mappings. The fast Lyapunov indicator a very sensitive tool. Celest. Mech. and Dynamical Astronomy, 78 167-195. 

K. R. Meyer, Normal forms for Hamiltonian systems. Celestial Mech. 9 (1974) 517-522. A.J. Dragt, J.M. Finn, Lie series and invariant functions for analytic symplectic maps, J. Math. Phys. 17 (1976) 2215-2227. 

A map that conserves the symplectic 2-form, or, in coordinates, satisfies (2.17), is termed a symplectic map. 

This proves that the flow map of a Hamiltonian system is a symplectic map. 

Thus the composition of any pair of symplectic maps is a symplectic map. The determinant of a symplectic map is 1, hence these maps are always invertible, and the inverse of a symplectic map is symplectic since = J implies J =... 

Thus the symplectic maps form a group under composition. 

This is just the leapfrogA erlet method in its Hamiltonian form. Since we have obtained the method as the composition of two symplectic maps, and the symplectic maps form a group, we know that this method will also be symplectic. 

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. 

Thus A is an eigenvalue of which, in turn, implies that 1 /A is an eigenvalue of T. The matrix being real implies that the conjugates of A and 1 /A are also eigenvalues, thus we have the same eigenvalue quadruplets as for a linear symplectic map. 

In such a case the iterates of the two maps will also be conjugate and, if they are numerical methods, they will have similar stability properties and performance (e.g. the same effective order). It is difficult to separate the relevance of the two properties in cases where symplectic and reversible maps are conjugate. This is however rarely the case and certainly does not hold generically for discrete maps in many dimensions [209]. There is no direct correspondence between reversible and symplectic maps, however each class of maps admits certain theorems of dynamical systems which are in many ways analogous (for example, the Kolmogorov-Arnold-Moser, or KAM, theory for symplectic maps near elliptic fixed points [386] has an analogue for reversible maps [97]). 

Of particular importance for molecular dynamics are the following properties a symplectic map will preserve volume, whereas a time-reversible map need not do so, and a symplectic integrator will approximately conserve energy due to the existence of the perturbed Hamiltonian, whereas a time-reversible integrator may... 

Thus we see that the projected version of the symplectic form in the ambient space is conserved by the differential equation system (4.7)-(4.9). This is what is meant by saying that the constrained system is analogous to a Hamiltonian system. We may also think of its flow map as being a symplectic map of the co-tangent bundle. 

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