Hamiltonian Differential Equations Research Articles

Center conditions: rigidity of logarithmic differential equations

In this paper, we prove that any degree d deformation of a generic logarithmic polynomial differential equation with a persistent center must be logarithmic again. This is a generalization of Ilyashenko's result on Hamiltonian differential equations. The main tools are Picard–Lefschetz theory of a polynomial with complex coefficients in two variables, specially the Gusein-Zade/A'Campo's theorem on calculating the Dynkin diagram of the polynomial, and the action of Gauss–Manin connection on the so-called Brieskorn lattice/Petrov module of the polynomial. We will also generalize J.P. Francoise recursion formula and (∗) condition for a polynomial which is a product of lines in a general position. Some applications on the cyclicity of cycles and the Bautin ideals will be given.

Splitting Methods for Non-autonomous Hamiltonian Equations

We present an algorithm for numerically integrating non-autonomous Hamiltonian differential equations. Special attention is paid to the separable case and, in particular, a new fourth-order splitting method is presented which in a certain measure is optimal. In combination with a new way of handling non-autonomous problems, the schemes we present are based on Magnus expansions and they show very promising results when applied to Hamiltonian ODEs and PDEs.

Field lines of gravity, their curvature and torsion, the Lagrange and the Hamilton equations of the plumbline

The length of the gravitational field lines/of the orthogonal trajectories of a family of gravity equipotential surfaces/of the plumbline between a terrestrial topographic point and a point on a reference equipotential surface like the geoid í also known as the orthometric height í plays a central role in Satellite Geodesy as well as in Physical Geodesy. As soon as we determine the geometry of the Earth pointwise by means of a satellite GPS (Global Positioning System: «global problem solver») we are left with the problem of converting ellipsoidal heights (geometric heights) into orthometric heights (physical heights). For the computation of the plumbline we derive its three differential equations of first order as well as the three geodesic equations of second order. The three differential equations of second order take the form of a Newton differential equation when we introduce the parameter time via the Marussi gauge on a conformally flat three-dimensional Riemann manifold and the generalized force field, the gradient of the superpotential, namely the modulus of gravity squared and taken half. In particular, we compute curvature and torsion of the plumbline and prove their functional relationship to the second and third derivatives of the gravity potential. For a spherically symmetric gravity field, curvature and torsion of the plumbline are zero, the plumbline is straight. Finally we derive the three Lagrangean as well as the six Hamiltonian differential equations of the plumbline, in particular in their star form with respect to Marussi gauge.

HOMOCLINIC ORBITS IN PERTURBED GENERALIZED HAMILTONIAN SYSTEMS

In this paper, we obtain existence and bifurcation theorems for homoclinic orbits in three-dimensional, time dependent and independent, perturbations of generalized Hamiltonian differential equations defined on three-dimensional Poisson manifolds. Then we apply them to a truncated spectral model of the quasi-geostrophic flow on a cyclic β-plane.

On the stability boundaries of linear periodic Hamiltonian or reversible differential equations

Two-parameter families of time periodic reversible or Hamiltonian differential equations are considered. The goal is to describe conditions such that the stability boundary is a smooth curve, at least locally. This is done by transforming the monodromy matrix of the system to a suitable normal form.

PERIODIC ORBITS IN PERTURBED GENERALIZED HAMILTONIAN SYSTEMS

In this paper, we develop a global perturbation technique for the study of periodic orbits in three-dimensional, time dependent and independent, perturbations of generalized Hamiltonian differential equations defined on three-dimensional Poisson manifolds. We give existence, stability and bifurcation theorems and illustrate our results with a truncated spectral model of the forced, dissipative quasi-geostrophic flow on a cyclic β-plane.

Momentum conserving symplectic integrators

In this paper, we show that symplectic partitioned Runge-Kutta methods conserve momentum maps corresponding to linear symmetry groups acting on the phase space of Hamiltonian differential equations by extended point transformation. We also generalize this result to constrained systems and show how this conservation property relates to the symplectic integration of Lie-Poisson systems on certain submanifolds of the general matrix group GL( n).

Sympletic capacities in two dimensions

Some years ago a class of new symplectic invariants was discovered, the so-called symplectic capacities. In this note we compute the Hofer-Zehnder-capacitycHZ, which is defined via periodic solutions of Hamiltonian differential equations, for two-dimensional, connected manifoldsM with an area element ω. It turns out thatcHZ(M,ω) is just the area |∫Mω|. Moreover, some examples illustrate the dynamics standing behind the definition ofcHZ. In the last part we treat the special case of the real plane where also another type of capacities exists, an example of which is Hofer's displacement energy.

Morse theory for periodic solutions of hamiltonian systems and the maslov index

AbstractIn this paper we prove Morse type inequalities for the contractible 1‐periodic solutions of time dependent Hamiltonian differential equations on those compact symplectic manifolds M for which the symplectic form and the first Chern class of the tangent bundle vanish over π2 (M). The proof is based on a version of infinite dimensional Morse theory which is due to Floer. The key point is an index theorem for the Fredholm operator which plays a central role in Floer homology. The index formula involves the Maslov index of nondegenerate contractible periodic solutions. This Maslov index plays the same role as the Morse index of a nondegenerate critical point does in finite dimensional Morse theory. We shall use this connection between Floer homology and Maslov index to establish the existence of infinitely many periodic solutions having integer periods provided that every 1‐periodic solution has at least one Floquet multiplier which is not equal to 1.

Global Approximation of Perturbed Hamiltonian Differential Equations with Several Turning Points

Given a perturbed Hamiltonian system (E) $i\varepsilon V' = [H_0 (x) + \varepsilon H_1 (x,\varepsilon )]V$, where $H_0 (x)$ is a Hermitian matrix analytic on an interval I. Any two eigenvalues of $H_0 (x)$ are allowed to coalesce finitely many times in the interval I (however, no two of them are identical on I). Assume that $H_1 (x,\varepsilon ) \in C^1 (I \times \bar S_c $ with $S_c = (0,c]$. Let $U(x)$ be the unitary matrix such that $D(x) = U^{ - 1} (x)H_0 (x)U(x)$ is diagonal. Assume that the entries of $U^{ - 1} H_1 U - iU^{ - 1} U'$ are bounded on $I \times S_c $, absolutely integrable over I. Then, (E) is shown to have a fundamental matrix of the form $V = U(x)Y(I_n + P(x,\varepsilon ))$ where Y is the exponential of a diagonal matrix, $I_n $ is the n-dimensional unit matrix and $P(x,\varepsilon ) \to 0$ as $\varepsilon \to 0^ + $. This result is applied to prove an adiabatic approximation theorem in quantum mechanics and provide a criterion measuring the phenomenon of degeneracy by the orders of coalescing of eigenvalues. Two examples are given.

Periodic Orbits in Slowly Varying Oscillators

We develop a global perturbation technique for the study of periodic orbits in three-dimensional, time dependent and independent, perturbations of planar Hamiltonian differential equations. We give...

Reduction of Dimension and Turnpike Theory

We give here the description of asymptotical properties of solutions to a class of scale-invariant dynamic optimization models arising in mathematical economics. These results are based on the geometric theory of Hamiltonian differential equations possessing the group of symmetries.

Linear Hamiltonian systems are integrable with quadratics

A new proof of a theorem of Williamson on the complete integrability of time-independent, real, linear Hamiltonian differential equations with quadratic integrals is given. The sets where these integrals are functionally dependent are explicitly found.

Nonlinear Oscillations

Part 1 The mathematical pendulum as an illustration of linear and non-linear oscillations - systems which are similar to a simple linear oscillator: Undamped free oscillations of the pendulum damped free oscillations forced oscillations. Part 2 Liapounov stability theory and bifurcations: The concept of Liapounov stability the direct method of Liapounov stability by the first approximation the Poincare map the critical case of a conjugate pair of eigenvalues simple bifurcation of equilibria and the Hopf bifurcation. Part 3: Self-excited oscillations in mechanical and electrical systems analytical approximation methods for the computation of self-excited oscillations analytical criteria for the existence of limit cycles forced oscillations in self-excited systems self-excited oscillations in systems with several degrees of freedom Part 4 Hamiltonian systems: Hamiltonian differential equations in mechanics canonical transformations the Hamilton-Jacobi differential equation canonical transformations and the motion perturbation theory Part 5 Introduction to the theory of optimal control: Control problems, controllability the Pontryagin maximum principle transversality conditions and problems with target sets canonical perturbation theory in optimal control.

Spectral theory for a singular linear system of Hamiltonian differential equations

In 1910 Weyl [9] inaugurated the modern theory of singular self-adjoint differential operators, considering, in particular, second-order differential operators with real coefficients. Since then, his results have been generalized by various authors to the case of arbitrary even order self-adjoint differential equations with real coefficients. Also, in 1954 Coddington [2] treated the case of an arbitrary n-th order self-adjoint differential equation with complex coefficients, He obtained the Parseval equality and spectral expansion associated with the singular case directly, through the consideration of such a problem as a limiting case of corresponding self-adjoint two-point boundary-value problems on compact subintervals of the reals. Using the same general concept, Coddington and Levinson [3] derived a Green’s function, which in turn enabled them to obtain the spectral matrix for a singular problem involving a linear homogeneous n-th order differential operator, and then proceeded to derive the Parseval equality and spectral expansions. Recently Brauer [I] treated similar problems which involve a definitely self-adjoint vector differential operator, of the sort that has been treated by Reid [6] and others. He obtained the Green’s matrix and spectral matrix through the use of the spectral theorem and the theory of direct integrals. The main purpose of the present paper is to employ the general method

New Results in Linear Filtering and Prediction Theory

A nonlinear differential equation of the Riccati type is derived for the covariance matrix of the optimal filtering error. The solution of this “variance equation” completely specifies the optimal filter for either finite or infinite smoothing intervals and stationary or nonstationary statistics. The variance equation is closely related to the Hamiltonian (canonical) differential equations of the calculus of variations. Analytic solutions are available in some cases. The significance of the variance equation is illustrated by examples which duplicate, simplify, or extend earlier results in this field. The Duality Principle relating stochastic estimation and deterministic control problems plays an important role in the proof of theoretical results. In several examples, the estimation problem and its dual are discussed side-by-side. Properties of the variance equation are of great interest in the theory of adaptive systems. Some aspects of this are considered briefly.

Kleine Schwingungen dynamischer Systeme

The Hamiltonian differential equations for small displacements of a dynamical system with a finite number of degrees of freedom are solved by use of the methods of linear algebra. In this way the stability of the equilibrium configuration or uniform motion in the case of positive definite total energy is demonstrated. Especially gyroscopic terms may appear in the hamiltonian, making impossible the usual proof of stability by means of transformation to normal coordinates. This method is then applied to the Weierstrass procedure of solution.