Noncanonical Hamiltonian Systems Research Articles

Semiexplicit K‐symplectic‐like methods with energy conservation for noncanonical Hamiltonian systems

AbstractFor the nonseparable noncanonical Hamiltonian systems, we propose efficient K‐symplectic‐like methods which are semiexplicit and energy‐preserving. By introducing two copies of the phase space and constructing an augmented Hamiltonian, we can separate the noncanonical Hamiltonian system into two explicitly integrable parts. Subsequently, explicit K‐symplectic methods can be constructed by using the splitting and composing method. To enforce constraints on the two copies of the phase space, we provide two transformations with energy conservation property. This enables us to obtain semiexplicit K‐symplectic‐like methods that preserve energy. Two algorithms are provided to implement the semiexplicit K‐symplectic‐like methods with energy conservation and their convergence has been proved. Numerical results on two noncanonical Hamiltonian systems demonstrate that the energy errors of our proposed methods remain bounded within machine precision over long time without exhibiting energy drift. Furthermore, the proposed methods exhibit superior computational efficiency compared to the canonicalized symplectic methods of the same order.

A collision operator for describing dissipation in noncanonical phase space

The phase space of a noncanonical Hamiltonian system is partially inaccessible due to dynamical constraints (Casimir invariants) arising from the kernel of the Poisson tensor. When an ensemble of noncanonical Hamiltonian systems is allowed to interact, dissipative processes eventually break the phase space constraints, resulting in a thermodynamic equilibrium described by a Maxwell–Boltzmann distribution. However, the time scale required to reach Maxwell–Boltzmann statistics is often much longer than the time scale over which a given system achieves a state of thermal equilibrium. Examples include diffusion in rigid mechanical systems, as well as collisionless relaxation in magnetized plasmas and stellar systems, where the interval between binary Coulomb or gravitational collisions can be longer than the time scale over which stable structures are self-organized. Here, we focus on self-organizing phenomena over spacetime scales such that particle interactions respect the noncanonical Hamiltonian structure, but yet act to create a state of thermodynamic equilibrium. We derive a collision operator for general noncanonical Hamiltonian systems, applicable to fast, localized interactions. This collision operator depends on the interaction exchanged by colliding particles and on the Poisson tensor encoding the noncanonical phase space structure, is consistent with entropy growth and conservation of particle number and energy, preserves the interior Casimir invariants, reduces to the Landau collision operator in the limit of grazing binary Coulomb collisions in canonical phase space, and exhibits a metriplectic structure. We further show how thermodynamic equilibria depart from Maxwell–Boltzmann statistics due to the noncanonical phase space structure, and how self-organization and collisionless relaxation in magnetized plasmas and stellar systems can be described through the derived collision operator.

Canonical and noncanonical Hamiltonian operator inference

A method for the nonintrusive and structure-preserving model reduction of canonical and noncanonical Hamiltonian systems is presented. Based on the idea of operator inference, this technique is provably convergent and reduces to a straightforward linear solve given snapshot data and gray-box knowledge of the system Hamiltonian. Examples involving several hyperbolic partial differential equations show that the proposed method yields reduced models which, in addition to being accurate and stable with respect to the addition of basis modes, preserve conserved quantities well outside the range of their training data.

Nearly Periodic Maps and Geometric Integration of Noncanonical Hamiltonian Systems

M. Kruskal showed that each continuous-time nearly periodic dynamical system admits a formal U(1)-symmetry, generated by the so-called roto-rate. When the nearly periodic system is also Hamiltonian, Noether’s theorem implies the existence of a corresponding adiabatic invariant. We develop a discrete-time analog of Kruskal’s theory. Nearly periodic maps are defined as parameter-dependent diffeomorphisms that limit to rotations along a U(1)-action. When the limiting rotation is non-resonant, these maps admit formal U(1)-symmetries to all orders in perturbation theory. For Hamiltonian nearly periodic maps on exact presymplectic manifolds, we prove that the formal U(1)-symmetry gives rise to a discrete-time adiabatic invariant using a discrete-time extension of Noether’s theorem. When the unperturbed U(1)-orbits are contractible, we also find a discrete-time adiabatic invariant for mappings that are merely presymplectic, rather than Hamiltonian. As an application of the theory, we use it to develop a novel technique for geometric integration of non-canonical Hamiltonian systems on exact symplectic manifolds.

Explicit K-symplectic methods for nonseparable non-canonical Hamiltonian systems

We propose efficient numerical methods for nonseparable non-canonical Hamiltonian systems which are explicit, K-symplectic in the extended phase space with long time energy conservation properties. They are based on extending the original phase space to several copies of the phase space and imposing a mechanical restraint on the copies of the phase space. Explicit K-symplectic methods are constructed for two non-canonical Hamiltonian systems. Numerical tests show that the proposed methods exhibit good numerical performance in preserving the phase orbit and the energy of the system over long time, whereas higher order Runge–Kutta methods do not preserve these properties. Numerical tests also show that the K-symplectic methods exhibit better efficiency than that of the same order implicit symplectic, explicit and implicit symplectic methods for the original nonseparable non-canonical systems. On the other hand, the fourth order K-symplectic method is more efficient than the fourth order Yoshida’s method, the optimized partitioned Runge–Kutta and Runge–Kutta–Nyström explicit K-symplectic methods for the extended phase space Hamiltonians, but less efficient than the the optimized partitioned Runge–Kutta and Runge–Kutta–Nyström extended phase space symplectic-like methods with the midpoint permutation.

Linear stability analysis via simulated annealing and accelerated relaxation

Simulated annealing (SA) is a kind of relaxation method for finding equilibria of Hamiltonian systems. A set of evolution equations is solved with SA, which is derived from the original Hamiltonian system so that the energy of the system changes monotonically while preserving Casimir invariants inherent to noncanonical Hamiltonian systems. The energy extremum reached by SA is an equilibrium. Since SA searches for an energy extremum, it can also be used for stability analysis when initiated from a state where a perturbation is added to an equilibrium. The procedure of the stability analysis is explained, and some examples are shown. Because the time evolution is computationally time consuming, efficient relaxation is necessary for SA to be practically useful. An acceleration method is developed by introducing time dependence in the symmetric kernel used in the double bracket, which is part of the SA formulation described here. An explicit formulation for low-beta reduced magnetohydrodynamics (MHD) in cylindrical geometry is presented. Since SA for low-beta reduced MHD has two advection fields that relax, it is important to balance the orders of magnitude of these advection fields.

Energy-Preserving Continuous-Stage Exponential Runge--Kutta Integrators for Efficiently Solving Hamiltonian Systems

As one of the most important properties, energy preservation is a natural requirement for numerical integrators of Hamiltonian systems. Considering the limited second-order accuracy of the existing exponential average vector filed (or discrete gradient) method, this paper is devoted to developing high-order energy-preserving exponential integrators for general semilinear Hamiltonian systems. To this end, we first formulate and analyze the continuous-stage exponential Runge--Kutta (ERK) method. After deriving the order conditions, energy-preserving conditions, and symmetric conditions for the continuous-stage ERK method, we construct a novel class of fourth-order symmetric energy-preserving continuous-stage ERK methods with a free parameter based on the established conditions and present a convergence theorem for the fixed-point iteration associated with the continuous-stage ERK method. Furthermore, we extend the proposed continuous-stage ERK method to noncanonical Hamiltonian systems and discuss implementation issues in detail. Finally, numerical results from the FPU problem, the charged-particle dynamics, and the Klein--Gordon equation demonstrate the high efficiency, good energy-preserving behavior, and applicability of large time stepsizes of the proposed fourth-order energy-preserving continuous-stage ERK method.

A Poisson map from kinetic theory to hydrodynamics with non-constant entropy

Kinetic theory describes a dilute monatomic gas using a distribution function f(q,p,t), the expected phase-space density of particles at a given position q with a given momentum p. The distribution function evolves according to the collisionless Boltzmann equation in the high Knudsen number limit. Fluid dynamics provides an alternative description of the gas using macroscopic hydrodynamic variables that are functions of position and time only. The mass, momentum and entropy densities of the gas evolve according to the compressible Euler equations in the limit of vanishing viscosity and thermal diffusivity. Both systems can be formulated as noncanonical Hamiltonian systems. Each configuration space is an infinite-dimensional Poisson manifold, and the dynamics is the flow generated by a Hamiltonian functional via a Poisson bracket. We construct a map J1 from the space of distribution functions to the space of hydrodynamic variables that respects the Poisson brackets on the two spaces. This map is therefore a Poisson map. It maps the p-integral of the Boltzmann entropy flogf to the hydrodynamic entropy density. This map belongs to a family of Poisson maps to spaces that include generalised entropy densities as additional hydrodynamic variables. The whole family can be generated from the Taylor expansion of a further Poisson map that depends on a formal parameter. If the kinetic-theory Hamiltonian factors through the Poisson map J1, an exact reduction of kinetic theory to fluid dynamics is possible. However, this is not the case. Nonetheless, by ignoring the contribution to the Hamiltonian from the entropy of the distribution function relative to its local Maxwellian, a distribution function defined by the p-moments ∫dnp(1,p,|p|2)f, we construct an approximate Hamiltonian that factors through the map. The resulting reduced Hamiltonian, which depends on the hydrodynamic variables only, generates the compressible Euler equations. We can thus derive the compressible Euler equations as a Hamiltonian approximation to kinetic theory. We also give an analogous Hamiltonian derivation of the compressible Euler–Poisson equations with non-constant entropy, starting from the Vlasov–Poisson equation.

Hamiltonian Perturbation Theory on a Lie Algebra. Application to a non-autonomous Symmetric Top

We propose a perturbation algorithm for Hamiltonian systems on a Lie algebra $\mathbb{V}$, so that it can be applied to non-canonical Hamiltonian systems. Given a Hamiltonian system that preserves a subalgebra $\mathbb{B}$ of $\mathbb{V}$, when we add a perturbation the subalgebra $\mathbb{B}$ will no longer be preserved. We show how to transform the perturbed dynamical system to preserve $\mathbb{B}$ up to terms quadratic in the perturbation. We apply this method to study the dynamics of a non-autonomous symmetric Rigid Body. In this example our algebraic transform plays the role of Iterative Lemma in the proof of a KAM-like statement.

Exponential Integrators Based on Discrete Gradients for Linearly Damped/Driven Poisson Systems

Exponential integrators based on discrete gradient methods are applied to non-canonical Hamiltonian systems with added linear forcing/damping terms, which may be time-dependent. Changes in the dynamics, such as conservation of energy or Casimirs, which result from inclusion of the linear forcing/damping terms, are not exactly preserved by standard discrete gradient methods. However, those changes are shown to be exactly preserved by the exponential integrators in special circumstances. The methods are also symmetric, second order, and linearly stable. To demonstrate advantages in both accuracy and efficiency over other standard methods, the exponential integrators are applied to a three dimensional Lotka-Volterra system and a damped/driven Ablowitz-Ladik system.

Deformation of Lie–Poisson algebras and chirality

Linearization of a Hamiltonian system around an equilibrium point yields a set of Hamiltonian symmetric spectra: If λ is an eigenvalue of the linearized generator, −λ and λ¯ (hence, −λ¯) are also eigenvalues—the former implies a time-reversal symmetry, while the latter guarantees the reality of the solution. However, linearization around a singular equilibrium point (which commonly exists in noncanonical Hamiltonian systems) works out differently, resulting in breaking of the Hamiltonian symmetry of spectra; time-reversal asymmetry causes chirality. This interesting phenomenon was first found in analyzing the chiral motion of the rattleback, a boat-shaped top having misaligned axes of inertia and geometry [Z. Yoshida et al., Phys. Lett. A 381, 2772–2777 (2017)]. To elucidate how chiral spectra are generated, we study the three-dimensional Lie–Poisson systems and classify the prototypes of singularities that cause symmetry breaking. The central idea is the deformation of the underlying Lie algebra; invoking Bianchi’s list of all three-dimensional Lie algebras, we show that the so-called class-B algebras, which are produced by asymmetric deformations of the simple algebra so(3), yield chiral spectra when linearized around their singularities. The theory of deformation is generalized to higher dimensions, including the infinite-dimensional Poisson manifolds relevant to fluid mechanics.

Dissipative brackets for the Fokker–Planck equation in Hamiltonian systems and characterization of metriplectic manifolds

It is shown that the Fokker–Planck equation describing diffusion processes in noncanonical Hamiltonian systems exhibits a metriplectic structure, i.e. an algebraic bracket formalism that generates the equation in consistency with the thermodynamic principles of energy conservation and entropy growth. First, a microscopic metriplectic bracket is derived for the stochastic equations of motion that characterize the random walk of the elements constituting the statistical ensemble. Such bracket is fully determined by the Poisson operator generating the Hamiltonian dynamics of an isolated (unperturbed) particle. Then, the macroscopic metriplectic bracket associated with the evolution of the distribution function of the ensemble is induced from the microscopic metriplectic bracket. Similarly, the macroscopic Casimir invariants are inherited from microscopic dynamics. The theory is applied to construct the Fokker–Planck equation of an infinite dimensional Hamiltonian system, the Charney–Hasegawa–Mima equation. Finally, the canonical form of the symmetric (dissipative) part of the metriplectic bracket is identified in terms of a ‘canonical metric tensor’ corresponding to an Euclidean metric tensor on the symplectic leaves foliated by the Casimir invariants.

Symplectic simulation of dark solitons motion for nonlinear Schrödinger equation

In this paper, we study symplectic simulation of dark solitons motion of nonlinear Schrodinger equation (NLSE). The Ablowitz-Ladik model (A-L model) of NLSE can be expressed as a non-canonical Hamiltonian system. By using splitting technique, we construct explicit splitting K-symplectic methods for the A-L model. On the other hand, the A-L model can be transformed into a canonical system and standard symplectic methods can be employed to perform numerical simulation. A second order K-symplectic method and a second order symplectic method are employed to simulate one dark soliton and two dark solitons motion for the A-L model and its canonicalized system respectively. By comparing with a third-order non-symplectic Runge-Kutta method, we show the superiorities of the two symplectic methods in long-term tracking the motion of dark solitons and preserving the invariants. We also compare the CPU times of K-symplectic methods and standard symplectic methods and show that the former ones are more efficient. The energy-preserving scheme is also applied for non-canonical Hamiltonian systems. The numerical results demonstrate that the K-symplectic methods can nearly preserve the energy, the discrete invariants of A-L model and conserved quantities of NLSE, but the energy-preserving scheme can only exactly preserve the energy.

Diagnosing diabatic effects on the available energy of stratified flows in inertial and non-inertial frames

The concept of available energy in a stratified fluid is revisited from the point of view of non-canonical Hamiltonian systems. We show that the concept of available energy arises when we minimize the energy subject to the constraints associated with the existence of Lagrangian invariants. The non-canonical structure implies that there exists a class of dynamically equivalent Hamiltonians, related by a local (in phase space) gauge symmetry. A local diagnostic energy can be defined via the Hamiltonian density chosen imposing a specific gauge-fixing condition on the class of dynamically similar Hamiltonians. The gauge-fixing condition that we introduce selects a specific local diagnostic energy which is well suited to study the effect of diabatic processes on the evolution of the available energy. Non-inertial effects, which are notoriously elusive to capture within an energetic framework, are naturally included via conservation of potential vorticity. We apply the framework to stratified flows in inertial and non-inertial frames. For stratified Boussinesq flows, when the initial distribution of potential vorticity is even around the origin, our framework recovers the available potential energy introduced by Holliday & McIntyre (J. Fluid Mech., vol. 107, 1981, pp. 221–225), and as such, depends only on the mass distribution of the flow. In rotating flows, the isopycnals of the ground state are generally not flat, and the ground state may have kinetic energy. We finally demonstrate that flows in non-inertial frames characterized by a low Rossby number ($Ro$), the local diagnostic energy has, to lowest order in $Ro$, a universal character.

Construction of Darboux coordinates and Poincaré–Birkhoff normal forms in noncanonical Hamiltonian systems

We demonstrate a general method to construct Darboux coordinates via normal form expansions in noncanonical Hamiltonian system obtained from e.g. a variational approach to quantum systems. The procedure serves as a tool to naturally extract canonical coordinates out of the variational parameters and at the same time to transform the energy functional into its Poincaré–Birkhoff normal form. The method is general in the sense that it is applicable for arbitrary degrees of freedom, in arbitrary orders of the local expansion, and it is independent of the precise form of the Hamilton operator. The method presented allows for the general and systematic investigation of quantum systems in the vicinity of fixed points, which e.g. correspond to ground, excited or transition states. Moreover, it directly allows to calculate classical and quantum reaction rates by applying transition state theory.

Splitting K-symplectic methods for non-canonical separable Hamiltonian problems

Non-canonical Hamiltonian systems have K-symplectic structures which are preserved by K-symplectic numerical integrators. There is no universal method to construct K-symplectic integrators for arbitrary non-canonical Hamiltonian systems. However, in many cases of interest, by using splitting, we can construct explicit K-symplectic methods for separable non-canonical systems. In this paper, we identify situations where splitting K-symplectic methods can be constructed. Comparative numerical experiments in three non-canonical Hamiltonian problems show that symmetric/non-symmetric splitting K-symplectic methods applied to the non-canonical systems are more efficient than the same-order Gauss' methods/non-symmetric symplectic methods applied to the corresponding canonicalized systems; for the non-canonical Lotka–Volterra model, the splitting algorithms behave better in efficiency and energy conservation than the K-symplectic method we construct via generating function technique. In our numerical experiments, the favorable energy conservation property of the splitting K-symplectic methods is apparent.

Weakly nonlinear dynamics in noncanonical Hamiltonian systems with applications to fluids and plasmas

A method, called beatification, is presented for rapidly extracting weakly nonlinear Hamiltonian systems that describe the dynamics near equilibria of systems possessing Hamiltonian form in terms of noncanonical Poisson brackets. The procedure applies to systems like fluids and plasmas in terms of Eulerian variables that have such noncanonical Poisson brackets, i.e., brackets with nonstandard and possibly degenerate form. A collection of examples of both finite and infinite dimensions is presented.

Stabilizing the coupled orbit–attitude dynamics of a rigid body in a J2 gravity field using Hamiltonian structure

The gravitationally coupled orbit–attitude dynamics of a rigid body in a J2 gravity field is a high-precision model for the spacecraft in the close proximity of a small spheroid celestial body, since the gravitational orbit–attitude coupling of the spacecraft is naturally taken into account in this model. A Hamiltonian structure-based feedback control law is proposed to stabilize relative equilibria of the rigid body in the J2 gravity field. The proposed stabilization control law is consisted of three parts: potential shaping, momentum control, and energy control. The potential shaping is used to modify the gravitational potential artificially so that the relative equilibrium is a minimum of the modified Hamiltonian. It is shown that the unstable relative equilibrium can always be stabilized in the Lyapunov sense by the potential shaping with sufficiently large feedback gains. Then the momentum control leads the motion to the same invariant manifold with the relative equilibrium, which is actually a momentum level set. The energy control introduces energy dissipation into the system and the motion will asymptotically converge to the minimum of the modified Hamiltonian on the invariant manifold, i.e., the relative equilibrium. The feasibility of the proposed stabilization control law is validated through numerical simulations. The control law can be implemented by the attitude control system and low-thrust engines onboard and the fuel consumption is reasonable. This Hamiltonian structure-based stabilization approach is applicable to non-canonical Hamiltonian systems commonly existing in the astrodynamics. Since the natural dynamical behaviors of the system is fully utilized, the proposed control law is very simple and is easy to implement autonomously with little computation in the space applications.

Multiple periodic solutions for non-canonical Hamiltonian systems with application to differential delay equations

We first establish Maslov index for non-canonical Hamiltonian system by using symplectic transformation for Hamiltonian system. Then the existence of multiple periodic solutions for the non-canonical Hamiltonian system is obtained by applying the Maslov index and Morse theory. As an application of the results, we study a class of non-autonomous differential delay equation which can be changed to non-canonical Hamiltonian system and obtain the existence of multiple periodic solutions for the equation by employing variational method.

Dynamically accessible variations for two-fluid plasma model

Dynamically accessible perturbation is a type of Lie perturbation for noncanonical Hamiltonian systems. Firstly, a set of first-order constraint variations that preserve all the Casimir functions is presented based on the two-fluid Poisson bracket. Then the equilibrium equations are given by minimizing the two-fluid Hamiltonian with these variations.