Physics
Symplectic Methods
Symplectic methods are numerical techniques used to solve differential equations that arise in classical mechanics. These methods preserve the symplectic structure of the equations, which ensures that the solutions remain physically meaningful and accurate over long periods of time. They are particularly useful for simulating the motion of particles in complex systems.
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5 Key excerpts on "Symplectic Methods"
eBook - PDF- Fanzhang Li, Li Zhang, Zhao Zhang(Authors)
- 2018(Publication Date)
- De Gruyter(Publisher)
The author proved the effectiveness of the symplectic algorithm. (7) Zhang Kai applied a type of multi-symplectic algorithm to the PML (perfectly matched layer) method in an electromagnetic field [7]. The symplectic method is used to reveal the equivalence of two Hamiltonian forms of a Hamiltonian microsystem. For the nonlinear wave equation, the relationship between the algorithm and the general difference scheme is established using this multi-sym-plectic algorithm. Then, according to the different forms of the nonlinear wave equation and the effect of different methods in the symplectic space and time direction, an effective symplectic algorithm is obtained. In addition, the author considered the stability of the hyperbolic equation numerical format, gave a crite-rion for evaluating the form of the symplectic form in many aspects, and discussed the Euler box format of a variety of symplectic algorithms as well as some results of the stability. (8) Xu et al. gave the numerical solution of a differential strategy based on a Hamilto-nian system and the symplectic algorithm [8]. It is useful to employ a differential 86 | 3 Symplectic group learning solution to solve the two-point problem. The differential game problem can be at-tributed to the Hamiltonian system. An important geometric feature of the Hamil-tonian system is that its phase flow maintains the symplectic structure; hence, the symplectic geometry algorithm has a good effect on the linear optimal prob-lem. The symplectic geometry algorithm can maintain the symmetry; hence, the algorithm can maintain the Hamiltonian symplectic structure and all the features, such that the energy of the system does not change for a long time, and it yields good results in solving the differential game calculation problem. (9) Sun et al. presented an application of the symplectic semi-analytical method based on a Hamiltonian system in heterogeneous electromagnetic waveguides [9].
- Sergio Blanes, Fernando Casas(Authors)
- 2017(Publication Date)
- CRC Press(Publisher)
In this case, the solution of both systems can be written in closed form and any splitting method can be used as a method. Alternatively, one can solve each system using, for example, implicit Symplectic Methods to build high order methods using appropriate compositions. □Other techniques exist to generate volume-preserving integrators. Among them, we can mention in particular the approach in [215], and the volume-preserving schemes obtained from generating functions (analogously to symplectic integrators) [99]. Moreover, the conditions to be satisfied for the existence of volume-preserving Runge–Kutta schemes have also been analyzed [99 , 121 ] (see also [10] for a more recent study).4.4 Lie group methods
Lie groups play a fundamental role in many branches of science, since they constitute the natural framework to express symmetries originating in the very formulation of the models. Thus, in Hamiltonian mechanics the set of symplectic maps defined on the phase space form the symplectic group, a particular instance of a Lie group. Very often, the solution of the initial value problem, evolves in a differentiable manifold acted upon transitively by a Lie group, i.e., in ahomogeneous space [205]. Examples of homogeneous spaces include spheres, tori, isospectral orbits and also Lie groups themselves. Under such circumstances, it is certainly advantageous to design integration methods such that the corresponding numerical flow respects this structure, i.e., is defined in the same Lie group as the continuous flow [85]. These are called Lie group methodsx ˙= f( t , x ) , x( 0 ) =,x 0
eBook - PDF- Ronald Devore, Arieh Iserles, Endre Süli(Authors)
- 2001(Publication Date)
- Cambridge University Press(Publisher)
2.12. A symplectic multistep method: the torus has dimension 2 instead of 1 as in Figs. 2.8-2.10. A single orbit is shown, with the first 50 time steps marked by x. lie on an invariant torus, the preservation of some invariant tori nearby helps a great deal. In Section 2.3, we talked about the importance of staying in the right phase space. The multistep method Xn+1 = Xn-l + 2r f(xn} is a map on the product phase space ]R2 x]R2. It can be shown to be symplectic in this larger space, but its KAM tori have dimension 2, instead of 1 as in the real system. When projected to the original phase space, they fill out a solid region, instead of a curve-a disaster for long-time simulations. This effect is illustrated in Fig. 2.12. 2.8 Summary Systems may have many geometric or structural features. Integrators must balance costs, local, global, and long-time errors, stability, and structural preservation. You can't expect to do well at all of these simul- taneously! Also, numerical studies can have different goals. Demanding very small local errors for a large class of ODEs tilts the balance in favour of highly-developed standard methods; seeking reliability over long times with simple, fast methods tilts in favour of geometric integrators. 180 R.I. McLachlan & G.R. W. Quispel The remaining lectures look at preserving different properties. Here we sum up what is known about preserving several properties at once. (i) Symplecticity and energy: If, by "integrator", we mean that the method is defined for all Hamiltonian ODEs, then by a theorem of Ge, this is impossible. For exceptional ("completely integrable") problems, such as the free rigid body, this can be done. (ii) Symplecticity and integrals apart from energy: Not known, al- though doable in principle. (iii) Symplecticity and linear symmetries: Achieved by, e.g., the im- plicit midpoint rule. (iv) Poisson and linear symmetries: Not known. (v) Volume preservation and linear symmetries: Not known.
eBook - PDF- Ahmed Lesfari(Author)
- 2022(Publication Date)
- Wiley-ISTE(Publisher)
1 Symplectic Manifolds 1.1. Introduction This chapter is devoted to the study of symplectic manifolds and their connection with Hamiltonian systems. It is well known that symplectic manifolds play a crucial role in classical mechanics, geometrical optics and thermodynamics, and currently have conquered a rich territory, asserting themselves as a central branch of differential geometry and topology. In addition to their activity as an independent subject, symplectic manifolds are strongly stimulated by important interactions with many mathematical and physical specialties, among others. The aim of this chapter is to study some properties of symplectic manifolds and Hamiltonian dynamical systems, and to review some operations on these manifolds. This chapter is organized as follows. In the second section, we begin by briefly recalling some notions about symplectic vector spaces. The third section defines and develops explicit calculation of symplectic structures on a differentiable manifold and studies some important properties. The forth section is devoted to the study of some properties of one-parameter groups of diffeomorphisms or flow, Lie derivative, interior product and Cartan’s formula. We review some interesting properties and operations on differential forms. The fifth section deals with the study of a central theorem of symplectic geometry, namely Darboux’s theorem: the symplectic manifolds (M, ω) of dimension 2m are locally isomorphic to (R 2m , ω). The sixth section contains some technical statements concerning Hamiltonian vector fields. The latter form a Lie subalgebra of the space vector field and we show that the matrix associated with a Hamiltonian system forms a symplectic structure. Several properties concerning Hamiltonian vector fields, their connection with symplectic manifolds, Poisson manifolds or Hamiltonian manifolds as well as some interesting examples are studied in the seventh section.
- Luigi Brugnano, Felice Iavernaro(Authors)
- 2016(Publication Date)
- Chapman and Hall/CRC(Publisher)
Consequently, the unperturbed torus T m ˆ t I 0 u is analytically deformed into a torus which is invariant for the perturbed system, whose solution remains quasi-periodic. The framework stemming from the studies of Kolmogorov, Arnold, and Moser is referred to as KAM theory. 13 1.4 Symplectic Methods The above properties, and the fact that symplecticity is a characterizing property of Hamiltonian systems, reinforce the search of Symplectic Methods for their numerical integration. A one-step method y 1 “ Φ h p y 0 q is per se a transformation of the phase space. Therefore the method is sym-plectic if Φ h is a symplectic map, i.e., if B Φ h p y 0 q B y 0 J J B Φ h p y 0 q B y 0 “ J. 13 A closely related problem in complex dynamics concerns the existence of centers. Given a complex analytic function f p z q “ e 2 πiα z ` ř 8 k “ 2 a k z k , with α P p 0 , 1 q , a formidable question in complex dynamics was whether it was possible for f to be conjugate to its linear part. In such an event, the origin would be a center for the discrete dynamical system z n “ f p z n ´ 1 q , where z 0 is an initial point close to the origin. This problem was addressed by Pfeiffer in 1915 and challenged several prominent figures such as Julia, Fatou, and Cremer. It was finally solved by Siegel in 1942 (prior to Kolmogorov), who proved the existence of centers assuming a Diophantine condition analogous to (1.34). The center problem is a small divisors problem, and Siegel’s solution played an inspirational role in the development of Moser’s approach to solve the small divisors problem in the context of celestial mechanics [3]. A primer on line integral methods 19 An important consequence of symplecticity in Runge-Kutta methods is the conservation of all quadratic first integrals of a Hamiltonian system.
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