Symplectic Condition Research Articles

Explicit Symplectic Runge–Kutta–Nyström Methods Based on Roots of Shifted Legendre Polynomial

To date, all explicit symplectic Runge–Kutta–Nyström methods of order five or above are derived by numerical solutions of order condition equations and symplectic condition. In this paper, we derive 124 sets of seven-stage fifth-order explicit symplectic Runge–Kutta–Nyström methods with closed-form coefficients in the Butcher tableau using the roots of a degree-3 shifted Legendre polynomial. One method is analyzed and its P-stable interval is derived. Numerical tests on the two newly discovered methods are performed, showing their long-time stability and large step size stability over some existing methods.

A new symplectic integrator for stochastic Hamiltonian systems on manifolds

This paper presents a significant development in the field of geometric numerical stochastic integration schemes. Specifically, the Geometric Symplectic Itô-Taylor 1.0 strong numerical integration scheme, tailored for Hamiltonian systems evolving on 2-sphere or S2 manifold is proposed. The core of this proposed algorithm centers around a new weak symplectic condition. This condition ensures the numerical stability for long duration time integration that can be readily achieved through numerical steps rather than relying solely on analytical satisfaction of the condition. This proposed advancement over the state-of-the-art caters to improved accuracy and stability of numerical simulations in manifold-based dynamical systems for long duration simulations. Through extensive analysis and numerical experiments, the effectiveness and reliability of the proposed scheme and symplectic criterion are validated. The findings of this study promises to offer valuable insights for researchers and practitioners working in the field of symplectic numerical integration methods.

Wigner distribution associated with the symplectic coordinates transformation

The celebrated τ-Wigner distribution (τ-WD), a generalization of the ordinary Wigner distribution (WD), has attracted much attention in the literature for its wide use in time-frequency analysis. However, it fails to break through the time-frequency resolution limit of the WD. Due to more degrees of freedom, the matrix Wigner distribution (MWD), a generalization of the τ-WD and a special case of the A-Wigner distribution, seems to be an effective tool to tackle this challenge. In this study, we focus on a specific MWD, i.e., the so-called symplectic Wigner distribution (SWD), through the change of coordinates with the general symplectic matrix. We disclose an equivalence relation between the uncertainty product in SWD domains and that in the classical settings, from which we obtain the optimal symplectic matrix constraint condition under which the SWD achieves higher time-frequency resolution than the WD. We also provide the simulation example associated with linear frequency-modulated signals for verifying the established superresolution theory.

Symplectic Flatness and Twisted Primitive Cohomology

We introduce the notion of symplectic flatness for connections and fibre bundles over symplectic manifolds. Given an A_infty -algebra, we present a flatness condition that enables the twisting of the differential complex associated with the A_infty -algebra. The symplectic flatness condition arises from twisting the A_infty -algebra of differential forms constructed by Tsai, Tseng and Yau. When the symplectic manifold is equipped with a compatible metric, the symplectic flat connections represent a special subclass of Yang–Mills connections. We further study the cohomologies of the twisted differential complex and give a simple vanishing theorem for them.

Structure-preserving techniques in accelerator physics

To very good approximation, particularly for hadron machines, charged-particle trajectories in accelerators obey Hamiltonian mechanics. During routine storage times of eight hours or more, such particles execute some revolutions about the machine, oscillations about the design orbit, and passages through various bending and focusing elements. Prior to building, or modifying, such a machine, we seek to identify accurately the long-term behaviour and stability of particle orbits over such large numbers of interactions. This demanding computational effort does not yield easily to traditional methods of symplectic numerical integration, including both explicit Yoshida-type and implicit Runge–Kutta or Gaussian methods. As an alternative, one may compute an approximate one-turn map and then iterate that map. We describe some of the essential considerations and techniques for constructing such maps to high order and for realistic magnetic field models. Particular attention is given to preserving the symplectic condition characteristic of Hamiltonian mechanics.

Diffusion and drift in volume-preserving maps

A nearly-integrable dynamical system has a natural formulation in terms of actions, y (nearly constant), and angles, x (nearly rigidly rotating with frequency Ω(y)).We study angleaction maps that are close to symplectic and have a twist, the derivative of the frequency map, DΩ(y), that is positive-definite. When the map is symplectic, Nekhoroshev’s theorem implies that the actions are confined for exponentially long times: the drift is exponentially small and numerically appears to be diffusive. We show that when the symplectic condition is relaxed, but the map is still volume-preserving, the actions can have a strong drift along resonance channels. Averaging theory is used to compute the drift for the case of rank-r resonances. A comparison with computations for a generalized Froeschl´e map in four-dimensions shows that this theory gives accurate results for the rank-one case.

A modified symplectic scheme for seismic wave modeling

Symplectic integrators are well known for their excellent performance in solving partial differential equation of dynamical systems because they are capable of preserving some conservative properties of dynamic equations. However, there are not enough high-order, for example third-order symplectic schemes, which are suitable for seismic wave equations. Here, we propose a strategy to construct a symplectic scheme that is based on a so-called high-order operator modification method. We first employ a conventional two-stage Runge–Kutta–Nyström (RKN) method to solve the ordinary differential equations, which are derived from the spatial discretization of the seismic wave equations. We then add a high-order term to the RKN method. Finally, we obtain a new third-order symplectic scheme with all positive symplectic coefficients, and it is defined based on the order condition, the symplectic condition, the stability condition and the dispersion relation. It is worth noting that the new scheme is independent of the spatial discretization type used, and we simply apply some finite difference operators to approximate the spatial derivatives of the isotropic elastic equations for a straightforward discussion. For the theoretical analysis, we obtain the semi-analytic stability conditions of our scheme with various orders of spatial approximation. The stability and dispersion properties of our scheme are also compared with conventional schemes to illustrate the favorable numerical behaviors of our scheme in terms of precision, stability and dispersion characteristics. Finally, three numerical experiments are employed to further demonstrate the validity of our method. The modified strategy that is proposed in this paper can be used to construct other explicit symplectic schemes.

The Continuous Finite Element Methods for a Simple Case of Separable Hamiltonian Systems

Combined with the characteristics of separable Hamiltonian systems and the finite element methods of ordinary differential equations, we prove that the composition of linear, quadratic, cubic finite element methods are symplectic integrator to separable Hamiltonian systems, i.e. the symplectic condition is preserved exactly, but the energy is only approximately conservative after compound. These conclusions are confirmed by our numerical experiments.

Transfer matrix of linear solenoid fringe superimposed magnet

The general transfer matrix of the superimposed solenoid magnet linear fringe field is given in this paper. It is an exact analytic matrix. The symplectic condition for the arbitrary shape solenoid will be discussed also.

BFT Hamiltonian Embedding of Massive Theory with One- and Two-Form Gauge Fields

Without resorting to the symplectic condition, we rigorously study the constraint structure of the topologically massive theory with one- and two-form fields in the framework of Batalin–Fradkin–Tyutin Hamiltonian embedding procedure. Through this full analysis we obtain a new type of Wess–Jumino action with novel symmetry, which is originated from the topological coupling term, as well as the Stückelberg action related to the explicit gauge breaking mass terms from the original theory.

Symplectic integration of nonlinear Hamiltonian systems

There exist several standard numerical methods for integrating ordinary differential equations. However, if one is interested in integration of Hamiltonian systems, these methods can lead to wrong results. This is due to the fact that these methods do not explicitly preserve the so-called 'symplectic condition' (that needs to be satisfied for Hamiltonian systems) at every integration step. In this paper, we look at various methods for integration that preserve the symplectic condition.

Global Symplectic Polynomial Approximation of Area-Preserving Maps

Different methods are proposed and tested for transforming a nonlinear differential system, and more particularly a Hamiltonian one, into a map without having to integrate the whole orbit as in the well known Poincaré map technique . We construct a synthetic map, based on a global fitting, which satisfies the symplectic condition. Taking the Standard Map as a model problem we point our attention on methods suitable for comparing the model map and its synthetic counterpart. We test the agreement of the fitting on finer scales through the visual representation, the computation of the rotation number and the measure of the local distribution of the Lyapunov characteristic exponents. Comparing these results with those obtained by Froeschlé and Petit using a method based on Taylor interpolation, we show that the symplectic character is a crucial condition for the recovering of the finest details of a dynamical system. On the other hand the global character of our method makes the generalization to any system of differential equations difficult.

COSY 5.0 — The fifth order code for corpuscular optical systems

COSY 5.0 is a new computer code for the design of corpuscular optical systems based on the principle of transfer matrices. The particle optical calculations include all image aberrations through fifth order. COSY 5.0 uses canonical coordinates and exploits the symplectic condition to increase the speed of computation. COSY 5.0 contains a library for the computation of matrix elements of all commonly used corpuscular optical elements such as electric and magnetic multipoles and sector fields. The corresponding formulas were generated algebraically by the computer code HAMILTON. Care was taken that the optimization of optical elements is achieved with minimal numerical effort. Finally COSY 5.0 has a very general mnemonic input code resembling a higher programming language.

Hamiltonian magnetic optics

Starting from Hamilton's principle, the imaging properties of charged particle beams in arbitrary static magnetic fields are outlined. The magnetic scalar potential is expressed as a series of multipoles about an arbitrary space curve which need not necessarily coincide with a trajectory of the beam. Appropriate power series expansions are given for the components of the magnetic vector potential where the expansion coefficients are related to the complex curvature of the axis and the strengths of the multipole components of the magnetic scalar potential. The general laws which govern the propagation of the charged particles are discussed by means of Hamilton's characteristic functions. It is shown that the Poisson and the Lagrange brackets are equivalent representations of the symplectic condition. The Poincaré invariant is used for elucidating some characteristic imaging properties of magnetic fields. A perturbation eikonal is introduced which allows a systematic and simultaneous calculation of the geometrical and chromatic aberrations to any order. The method has the advantage that it reveals at the very beginning all interrelations between the aberration coefficients. The iteration procedure starts from the paraxial rays which are supposed to be known. The algorithm is especially suited for numerical calculation of the higher-order aberrations.