Hamiltonian System Of Differential Equations Research Articles

Discrete symmetries of equations of dynamics with polynomial integrals of higher degrees

We consider systems with toric configuration space and kinetic energy in the form of a "flat" Riemannian metric on the torus. The potential energy $V$ is a smooth function on the configuration torus. The dynamics of such systems is described by "natural" Hamiltonian systems of differential equations. If $V$ is replaced by $\varepsilon V$, where $\varepsilon$ is a small parameter, then the study of such Hamiltonian systems for small $\varepsilon$ is a part of the "main problem of dynamics" according to Poincaré. We discuss the well-known conjecture on the existence of single-valued momentum-polynomial integrals of motion equations: if there is a momentum-polynomial integral of degree $m$, then there exist a momentum-linear or momentum-quadratic integral. This conjecture was verified in full generality for $m=3$ and $m=4$. We study the cases of "higher" degrees $m=5$ and $m=6$. Similarly to the theory of perturbations of Hamiltonian systems, we introduce resonance lines on the momentum plane. If a system admits a polynomial integral, then the number of these lines is finite. The symmetries of the set of resonance lines are found, from which, in particular, necessary conditions for integrability are derived. Some new criteria for the existence of single-valued polynomial integrals are obtained.

On stability of motion of satellite

In this paper we consider the translational and rotational motion of satellites around the Earth. First, we study the motion of an arrow-type satellite. We consider the system of equations of motion in the first approximation. We consider the linear Hamiltonian systems of differential equations. The Hamiltonian systems arise in transportation problems. We consider the normalization of Hamiltonian matrix. We solve the system of matrix equations to find the generating function of the canonical transformation. We obtain stability criterion in several cases. Further, we study the motion of a spoke-type satellite. We consider eigenvalues of characteristic equation corresponding to the motion equations. We get the normal form of the Hamiltonian matrix. We obtain the stability criterion.

On stability of motion of unbalanced gyroscope

In this paper we consider the stability of motion of an unbalanced gyroscope with a flexible shaft in gimbal suspension.Gyroscopes are used to control the stability of boom tower cranes. First, we consider the linear Hamiltonian systems of differential equations. The Hamiltonian systems arise in transportation problems. We prove several properties of the Hamiltonian matrix. Next, we consider the normalization of Hamiltonian matrix. We solve the system of matrix equations to find the generating function of the canonical transformation. Further, we find the symmetric solution of nonlinear algebraic matrix Riccati equation in one case. We propose the analytical method for solving this equation. Further, we investigate the motion of gyroscope. We consider the system of equations of motion of the gyroscope rotor and frames in the first approximation. We consider eigenvalues of characteristic equation corresponding to the motion equations. We obtain stability criterion in several cases.

A new class of G(ϵ)-symplectic general linear methods

A new class of G(ϵ)-symplectic general linear methods for numerical integration of Hamiltonian systems of differential equations is described. Order conditions for these methods are derived using Albrecht approach and the construction of G(ϵ)-symplectic method is described based on solving minimization problems with nonlinear inequality constrains. Examples of methods up to the order four are presented. Numerical experiments confirm that these methods achieve the expected order of accuracy and that they approximately preserve Hamiltonians of differential systems.

Construction of [formula omitted]- or [formula omitted]-symplectic general linear methods

We describe the construction of G- or G(ϵ)-symplectic, and parasitism free or ϵ-parasitism free general linear methods for numerical integration of Hamiltonian systems of differential equations. Examples of such methods are presented up to the order p=4 and stage order q=p−1. Numerical experiments confirm that all methods achieve the expected order of accuracy, and that these methods approximately preserve Hamiltonians as well as quadratic invariants of differential systems.

Invariant Nijenhuis Tensors and Integrable Geodesic Flows

We study invariant Nijenhuis $(1,1)$-tensors on a homogeneous space $G/K$ of a reductive Lie group $G$ from the point of view of integrability of a Hamiltonian system of differential equations with the $G$-invariant Hamiltonian function on the cotangent bundle $T^*(G/K)$. Such a tensor induces an invariant Poisson tensor $\Pi_1$ on $T^*(G/K)$, which is Poisson compatible with the canonical Poisson tensor $\Pi_{T^*(G/K)}$. This Poisson pair can be reduced to the space of $G$-invariant functions on $T^*(G/K)$ and produces a family of Poisson commuting $G$-invariant functions. We give, in Lie algebraic terms, necessary and sufficient conditions of the completeness of this family. As an application we prove Liouville integrability in the class of analytic integrals polynomial in momenta of the geodesic flow on two series of homogeneous spaces $G/K$ of compact Lie groups $G$ for two kinds of metrics: the normal metric and new classes of metrics related to decomposition of $G$ to two subgroups $G=G_1\cdot G_2$, where $G/G_i$ are symmetric spaces, $K=G_1\cap G_2$.

Lax Pairs for Linear Hamiltonian Systems

In the paper Lax pairs for linear Hamiltonian systems of differential equations are constructed. In particular, Gr\"obner bases are used for the computations. It is proved that the maps which appear in the construction of Lax pairs are Poisson. Various properties of first integrals of the system which are obtained from the Lax pairs are investigated.

Climate Vibrations and Chaotic Behavior of Earth’s Inclination in the Laskar’s Model of a Moonless Earth

The model of the moonless Earth, introduced by J. Laskar, has the form of a non-autonomous Hamiltonian system of differential equations for two variables: the cosine of the angle of inclination and the longitude of the axis of rotation of the Earth. The system describes the rotational dynamics of the Earth under the influence of the sun and planets. Earth perturbations from other planets of the solar system are considered periodic and are taken into account using the first four terms of the Fourier expansion of the corresponding part of the Hamilton function with known amplitudes and frequencies. The initial inclination of the Earth is considered as a parameter of the problem. The system was numerically integrated over a time period of 18 million years for various values of the initial inclination from 0 to 180 degrees. Three chaotic gaps of the initial inclination were found. During the bifurcation study, singular points were found and special segments of the non-autonomous system were obtained. A bifurcation diagram of the system is constructed by the initial inclination parameter. Poincare cartographic maps are constructed. The system is written in variations on the initial conditions for the Laskar system, and with its help the dependences of the problem parameter of the senior Lyapunov exponent and the averaged MEGNO indicator are calculated. The results confirm the presence of three chaotic and one regular region of variation of the bifurcation parameter of the problem.

Global versus local superintegrability of nonlinear oscillators

Liouville (super)integrability of a Hamiltonian system of differential equations is based on the existence of globally well-defined constants of the motion, while Lie point symmetries provide a local approach to conserved integrals. Therefore, it seems natural to investigate in which sense Lie point symmetries can be used to provide information concerning the superintegrability of a given Hamiltonian system. The two-dimensional oscillator and the central force problem are used as benchmark examples to show that the relationship between standard Lie point symmetries and superintegrability is neither straightforward nor universal. In general, it turns out that superintegrability is not related to either the size or the structure of the algebra of variational dynamical symmetries. Nevertheless, all of the first integrals for a given Hamiltonian system can be obtained through an extension of the standard point symmetry method, which is applied to a superintegrable nonlinear oscillator describing the motion of a particle on a space with non-constant curvature and spherical symmetry.

On First Integrals of Linear Hamiltonian Systems

In the paper, Lax pairs for linear Hamiltonian systems of differential equations are found. Besides, first integrals of the system which are obtained from the Lax pairs are investigated.

$\mathcal C$-symmetric Hamiltonian systems with almost constant coefficients

We consider a \mathcal C -Symmetric Hamiltonian System of differential equations on a half interval or the real line. We determine the spectrum and construct the resolvent for the system. The essential spectrum is found to be a subset of an algebraic curve \Sigma defined by a characteristic polynomial for the system. The results are first proved for a constant coefficient system and then for an almost constant coefficient system. The results are applied to a number of examples including the complex hydrogen atomand the complex relativistic electron.

Multi-stage splitting integrators for sampling with modified Hamiltonian Monte Carlo methods

Modified Hamiltonian Monte Carlo (MHMC) methods combine the ideas behind two popular sampling approaches: Hamiltonian Monte Carlo (HMC) and importance sampling. As in the HMC case, the bulk of the computational cost of MHMC algorithms lies in the numerical integration of a Hamiltonian system of differential equations. We suggest novel integrators designed to enhance accuracy and sampling performance of MHMC methods. The novel integrators belong to families of splitting algorithms and are therefore easily implemented. We identify optimal integrators within the families by minimizing the energy error or the average energy error. We derive and discuss in detail the modified Hamiltonians of the new integrators, as the evaluation of those Hamiltonians is key to the efficiency of the overall algorithms. Numerical experiments show that the use of the new integrators may improve very significantly the sampling performance of MHMC methods, in both statistical and molecular dynamics problems.

Properties of canonical transformations of linear Hamiltonian systems

In this paper we consider the linear Hamiltonian systems of differential equations. The Hamiltonian systems have an important role in fluid mechanics and in statistical mechanics. Firstly, we prove some properties of canonical transformations of linear Hamiltonian systems. Further, we get the new method to find the generating function of a canonical transformation. We get a system of matrix equations for finding the generating function of a canonical transformation. In various cases we obtain the solution of the system. Then we use the system for normalization the Hamiltonian matrix. We apply the system of matrix equations to transform the Hamiltonian matrix from one normal form to another. Further, we get the solution of the matrix Riccati equation. This nonlinear equation has an important role in optimal control problems, multivariable and large-scale systems, scattering theory, estimation, detection and transportation. Finally, we transform the Hamiltonian from the complex form to the real form. An illustrative example for the proposed method is given. Therefore, we obtain the new method of normalization of the quadratic Hamiltonian. With this method we can investigate the stability of the solution of the Hamiltonian systems.

A nonlinear singular eigenvalue problem for a Hamiltonian system of differential equations with redundant condition

A nonlinear eigenvalue problem for a self-adjoint Hamiltonian system of differential equations is examined on an infinite half-line. It is assumed that the original data (that is, the system matrix and the matrix of boundary conditions) satisfy certain monotonicity conditions for the spectral parameter. In addition to the initial condition and the requirement that the solution be bounded on infinity, a redundant nonlocal condition specified by a Stieltjes integral is imposed. In order to make the resulting problem nontrivially solvable, it is replaced by an auxiliary problem, which is consistent subject to all the above conditions. This auxiliary problem is examined, and a numerical method that solves it is given.

Erratum to: “Nonlinear Spectral Problem for a Hamiltonian System of Differential Equations with Redundant Conditions”

We consider a nonlinear spectral problem for a self-adjoint Hamiltonian system of differential equations. The boundary conditions correspond to a self-adjoint problem. It is assumed that the input data (the matrix of the system and the matrices of the boundary conditions) satisfy certain conditions of monotonicity with respect to the spectral parameter. In addition to the main boundary conditions, a redundant nonlocal condition given by a Stieltjes integral is imposed on the solution. For the nontrivial solvability of the over-determined problem thus obtained, the original problem is replaced by an auxiliary problem that is consistent with the entire set of conditions. This auxiliary problem is obtained from the original one by allowing a discrepancy of a specific form. We study the resulting problem and give a numerical method for its solution.

Some notes on oscillation of two-dimensional system of difference equations

Oscillatory properties of solutions to the system of first-order linear difference equations $$ \begin {aligned} \Delta u_k & = q_k v_k \\ \Delta v_k & = -p_k u_{k+1}, \end {aligned} $$ are studied. It can be regarded as a discrete analogy of the linear Hamiltonian system of differential equations. \endgraf We establish some new conditions, which provide oscillation of the considered system. Obtained results extend and improve, in certain sense, results presented in Opluštil (2011).

On the index of the boundary value problem for a homogeneous Hamiltonian system of differential equations

The index of the homogeneous self-adjoint boundary value problem for the Hamiltonian systems of ordinary differential equations is introduced. It is assumed that the system has a nontrivial solution. The relationship between the index of an eigenvalue of the nonlinear eigenvalue problem and the index of the corresponding homogeneous problem is established. Properties of the index of the problem and those of the eigenvalue are examined.

Preserving poisson structure and orthogonality in numerical integration of differential equations

We consider the numerical integration of two types of systems of differential equations. We first consider Hamiltonian systems of differential equations with a Poisson structure. We show that symplectic Runge-Kutta methods preserve this structure when the Poisson tensor is constant. Using nonlinear changes of coordinates this structure can also be preserved for nonconstant Poisson tensors, as exemplified on the Euler equations for the free rigid body. We also consider orthogonal flows and the closely related class of isospectral flows. To numerically preserve the orthogonality property we take the approach of formulating an equivalent system of differential-algebraic equations (DAEs) and of integrating the system with a special combination of a particular class of Runge-Kutta methods. This approach requires only matrix-matrix products and can preserve geometric properties of the flow such as reversibility.

Multisymplectic geometry, covariant Hamiltonians, and water waves

This paper concerns the development and application of the multisymplectic Lagrangian and Hamiltonian formalism for nonlinear partial differential equations. This theory generalizes and unifies the classical Hamiltonian formalism of particle mechanics as well as the many pre-symplectic 2-forms used by Bridges. In this theory, solutions of a partial differential equation are sections of a fibre bundle Y over a base manifold X of dimension n+1, typically taken to be spacetime. Given a connection on Y, a covariant Hamiltonian density [Hscr ] is then intrinsically defined on the primary constraint manifold P[Lscr ], the image of the multisymplectic version of the Legendre transformation. One views P[Lscr ] as a subbundle of J1(Y)*, the affine dual of J1(Y)*, the first jet bundle of Y. A canonical multisymplectic (n+2)-form Ω[Hscr ] is then defined, from which we obtain a multisymplectic Hamiltonian system of differential equations that is equivalent to both the original partial differential equation as well as the Euler–Lagrange equations of the corresponding Lagrangian. Furthermore, we show that the n+1 2-forms ω(μ) defined by Bridges are a particular coordinate representation for a single multisymplectic (n+2)-form and, in the presence of symmetries, can be assembled into Ω[Hscr ]. A generalized Hamiltonian Noether theory is then constructed which relates the action of the symmetry groups lifted to P[Lscr ] with the conservation laws of the system. These conservation laws are defined by our generalized Noether's theorem which recovers the vanishing of the divergence of the vector of n+1 distinct momentum mappings defined by Bridges and, when applied to water waves, recovers Whitham's conservation of wave action. In our view, the multisymplectic structure provides the natural setting for studying dispersive wave propagation problems, particularly the instability of water waves, as discovered by Bridges. After developing the theory, we show its utility in the study of periodic pattern formation and wave instability.

Sympletic Runge--Kutta Shemes I: Order Conditions

Much recent work has indicated that considerable benefit arises from the use of symplectic algorithms when numerically integrating Hamiltonian systems of differential equations. Runge--Kutta schemes are symplectic subject to a simple algebraic condition. Starting with Butcher's formalism it is shown that there exists a more natural basis for the set of necessary and sufficient order conditions for these methods, involving only s(s + 1)/2 free parameters for a symplectic s-stage scheme. A graph theoretical process for determining the new order conditions is outlined. Furthermore, it is shown that any rooted tree arising from the same free tree enforces the same algebraic constraint on the parametrized coefficients. When coupled with the standard simplifying assumptions for implicit schemes the number of order conditions may be further reduced. In the new framework a simple symmetry of the parameter matrix yields (not necessarily symplectic) self-adjoint methods. In this case the order conditions associated with even trees become redundant.