Stability Of Hamiltonian Systems Research Articles

On types of stability in Hamiltonian systems

On types of stability in Hamiltonian systems

О типах устойчивости в системах Гамильтона

О типах устойчивости в системах Гамильтона

On stability in Hamiltonian systems with two degrees of freedom

We consider the stability of the equilibrium position at the origin of coordinates of a Hamiltonian system with two degrees of freedom whose unperturbed part describes oscillators with restoring force of odd order greater than 1. It is proved that if the exponents of the restoring force of the oscillators are not equal, then the equilibrium position is Lyapunov stable. If the exponents are equal, then the equilibrium position is conditionally stable for trajectories not belonging to some level surface of the Hamiltonian. The reduction of the system to this surface shows that the equilibrium position is stable in the case of general position.

JACOBI STABILITY OF HAMILTONIAN SYSTEMS

The stability of Hamiltonian systems with one degree of freedom is investigated from Jacobi stability point of view (the KCC theory) and it is shown that the Jacobi stability conditions of non-degenerate equilibrium points are the same as known conditions for Liapunov stability.

On the stability of Hamiltonian systems with weakly time dependent potentials

We show here that a recently developed criterion for the stability of conservative Hamiltonian systems can be extended to Hamiltonians with weak time dependence. In this method, the geodesic equations contain the Hamilton equations of the original potential model through an inverse map in the tangent space in terms of a geometric embedding. The second covariant derivative of the geodesic deviation generates a dynamical curvature, resulting in a (energy dependent) local criterion for unstable behavior different from the usual Lyapunov criterion. We show by direct simulation that our geometrical criterion predicts correctly the stability/instability of motions, sometimes contrary to indications of the local Lyapunov method.

SUBTLE IS THE LORD: ON THE DIFFERENCE BETWEEN NEWTONIAN (LYAPUNOV) STABILITY ANALYSIS AND GEOMETRICAL STABILITY ANALYSIS OF GRAVITATIONAL ORBITS

In this essay we discuss the geometrical embedding method (GEM) for the analysis of the stability of Hamiltonian systems using geometrical techniques familiar from general relativity. This method has proven to be very effective. In particular, we show that although the application of standard Lyapunov analysis predicts that completely integrable Kepler motion is unstable, this geometrical analysis predicts the observed stability. Moreover, we apply this approach to the three body problem in which the third body is restricted to move on a circle of large radius which induces an adiabatic time dependent potential on the second body. This causes the second body to move in a very intricate but periodic trajectory. The geometric approach predicts the correct stable motion in this case as well.

Periodic motions of a nearly dynamically symmetric satellite in the neighborhood of hyperboloidal precession

The motion of a satellite close to a dynamically symmetric solid body in a Newtonian gravitational field over a circular orbit is studied. The system of differential equations describing the body’s motion is close to a system with cyclic coordinate. New classes of periodic motions are constructed in the neighborhood of a known partial solution to an unperturbed problem, hyperboloidal precession of a dynamically symmetric satellite. In the resonance case, when the ratio of one frequency of small oscillations of a reduced system with two degrees of freedom in the neighborhood of a stable equilibrium position to the frequency of cyclic coordinate variation is close to an integer number, there exist one or three families of periodic motions that are analytic in terms of fractional powers of a small parameter. A study of stability of these motions was performed with the help of KAM (Kolmogorov-Arnold-Moser) theoty. Faling the described resonance there exists a unique family of periodic motions that is analytic in terms of integer powers of a small parameter. The check-up of stability of these motrons was carried out. We distinguished the cases of parametric resonance, resonances of the third and fourth orders, and a non-resonant case. In the resonance cases our study relies on well-known results on stability of Hamiltonian systems during resonances [1]. In the non-resonant case we use the KAM theory [2].

KAM stability and celestial mechanics

KAM theory is a powerful tool apt to prove perpetual stability in Hamiltonian systems, which are a perturbation of integrable ones. The smallness requirements for its applicability are well known to be extremely stringent. A long standing problem, in this context, is the application of KAM theory to systems for observable values of the perturbation parameters. We consider the Restricted, Circular, Planar, Three-Body Problem (RCP3BP), i.e., the problem of studying the planar motions of a small body subject to the gravitational attraction of two primary bodies revolving on circular Keplerian orbits (which are assumed not to be influenced by the small body). When the mass ratio of the two primary bodies is small, the RCP3BP is described by a nearly-integrable Hamiltonian system with two degrees of freedom; in a region of phase space corresponding to nearly elliptical motions with non small eccentricities, the system is well described by Delaunay variables. The Sun-Jupiter observed motion is nearly circular and an asteroid of the Asteroidal belt may be assumed not to influence the Sun-Jupiter motion. The Jupiter-Sun mass ratio is slightly less than 1/1000. We consider the motion of the asteroid 12 Victoria taking into account only the Sun-Jupiter gravitational attraction regarding such a system as a prototype of a RCP3BP. For values of mass ratios up to 1/1000, we prove the existence of two-dimensional KAM tori on a fixed three-dimensional energy level corresponding to the observed energy of the Sun-Jupiter-Victoria system. Such tori trap the evolution of phase points to the observed physical data of the Sun-Jupiter-Victoria system. As a consequence, in the RCP3BP description, the motion of Victoria is proven to be forever close to an elliptical motion. The proof is based on: 1) a new iso-energetic KAM theory; 2) an algorithm for computing iso-energetic, approximate Lindstedt series; 3) a computer-aided application of 1)+2) to the Sun-Jupiter-Victoria system. The paper is self-contained but does not include the (similar to 12000 lines) computer programs, which may be obtained by sending an e-mail to one of the authors.

Krein's formula for indefinite multipliers in linear periodic Hamiltonian systems

This paper is concerned with Hamiltonian systems of linear differential equations with periodic coefficients under a small perturbation. It is well known that Krein's formula determines the behavior of definite multipliers on the unit circle and is quite useful in studying the (strong) stability of Hamiltonian system. Our aim is to give a simple formula that determines the behavior of indefinite multipliers with two multiplicity, which is generic case. The result does not require analyticity and is proved directly. Applying this formula, we obtain instability criteria for solutions with periodic structure in nonlinear dissipative systems such as the Swift–Hohenberg equation and reaction–diffusion systems of activator–inhibitor type.

A cotangent bundle slice theorem

This article concerns cotangent-lifted Lie group actions; our goal is to find local and “semi-global” normal forms for these and associated structures. Our main result is a constructive cotangent bundle slice theorem that extends the Hamiltonian slice theorem of Marle [C.-M. Marle, Modèle d'action hamiltonienne d'un groupe de Lie sur une variété symplectique, Rendiconti del Seminario Matematico, Università e Politecnico, Torino 43 (2) (1985) 227–251] and Guillemin and Sternberg [V. Guillemin, S. Sternberg, A normal form for the moment map, in: S. Sternberg (Ed.), Differential Geometric Methods in Mathematical Physics, in: Mathematical Physics Studies, vol. 6, D. Reidel, 1984]. The result applies to all proper cotangent-lifted actions, around points with fully-isotropic momentum values. We also present a “tangent-level” commuting reduction result and use it to characterise the symplectic normal space of any cotangent-lifted action. In two special cases, we arrive at splittings of the symplectic normal space. One of these cases is when the configuration isotropy group is contained in the momentum isotropy group; in this case, our splitting generalises that given for free actions by Montgomery et al. [R. Montgomery, J.E. Marsden, T.S. Ratiu, Gauged Lie–Poisson structures, Cont. Math. AMS 128 (1984) 101–114]. The other case includes all relative equilibria of simple mechanical systems. In both of these special cases, the new splitting leads to a refinement of the so-called reconstruction equations or bundle equations [J.-P. Ortega, Symmetry, reduction, and stability in Hamiltonian systems, PhD thesis, University of California, Santa Cruz, 1998; J.-P. Ortega, T.S. Ratiu, A symplectic slice theorem, Lett. Math. Phys. 59 (1) (2002) 81–93; M. Roberts, C. Wulff, J.S.W. Lamb, Hamiltonian systems near relative equilibria, J. Differential Equations 179 (2) (2002) 562–604]. We also note cotangent-bundle-specific local normal forms for symplectic reduced spaces.

Stability of Hamiltonian Systems with Three Degrees of Freedom and the Three Body-Problem

Results are obtained about formal stability and instability of Hamiltonian systems with three degrees of freedom, two equal frequencies and the matrix of the linear part is not diagonalizable, in terms of the coefficients of the development in Taylor series of the Hamiltonian of the system. The results are applied to the study of stability of the Lagrangian solutions of the Three Body-Problem in the case in which the center of mass is over the curve ρ*, on the border of the region of linear stability of Routh. The curve ρ* is divided symmetrically in three arcs in such a way that if the center of mass of the three particles lies on the central arc, the Lagrangian solution is unstable in the sense of Liapunov (in finite order), while if the center of mass determines one point that lies on one of the other two arcs of ρ*, then the Lagrangian solution is formally stable.

A note on discrete-time stabilization of hamiltonian systems

In this paper we present some preliminary results on the stabilization problem for Hamiltonian systems using approximate discrete-time models. The issues of constructing a discrete-time model for Hamiltonian system are in general different from those for dissipative systems. We propose an algorithm for constructing an approximate discrete-time model, which guarantees the Hamiltonian conservation, and apply the algorithm to a class of port-controlled Hamiltonian systems. We illustrate the usefulness of the algorithm in designing a discrete-time controller to stabilize the angular velocity of a rigid body.

Critical manifolds and stability in Hamiltonian systems with non-holonomic constraints

We explore a particular approach to the analysis of dynamical and geometrical properties of autonomous, Pfaffian non-holonomic systems in classical mechanics. The method is based on the construction of a certain auxiliary constrained Hamiltonian system, which comprises the non-holonomic mechanical system as a dynamical subsystem on an invariant manifold. The embedding system possesses a completely natural structure in the context of symplectic geometry, and using it in order to understand properties of the subsystem has compelling advantages. We discuss generic geometric and topological properties of the critical sets of both embedding and physical system, using Conley–Zehnder theory, and by relating the Morse–Witten complexes of the ‘free’ and constrained system to one another. Furthermore, we give a qualitative discussion of the stability of motion in the vicinity of the critical set. We point out key relations to sub-Riemannian geometry, and a potential computational application.

Critical manifolds and stability in Hamiltonian systems with non-holonomic constraints

We explore a particular approach to the analysis of dynamical and geometrical properties of autonomous, Pfaffian non-holonomic systems in classical mechanics. The method is based on the construction of a certain auxiliary constrained Hamiltonian system, which comprises the non-holonomic mechanical system as a dynamical subsystem on an invariant manifold. The embedding system possesses a completely natural structure in the context of symplectic geometry, and using it in order to understand properties of the subsystem has compelling advantages. We discuss generic geometric and topological properties of the critical sets of both embedding and physical system, using Conley–Zehnder theory, and by relating the Morse–Witten complexes of the ‘free’ and constrained system to one another. Furthermore, we give a qualitative discussion of the stability of motion in the vicinity of the critical set. We point out key relations to sub-Riemannian geometry, and a potential computational application.

On the stability boundary of hamiltonian systems

The criterion for the points in the parameter space being on the stability boundary of linear Hamiltonian system depending on arbitrary numbers of parameters was given, through the sensitivity analysis of eigenvalues and eigenvectors. The results show that multiple eigenvalues with Jordan chain take a very important role in the stability of Hamiltonian systems.

Stabilization of Hamiltonian systems with nonholonomic constraints based on time-varying generalized canonical transformations

This paper is concerned with the stabilization of nonholonomic systems in port-controlled Hamiltonian formulae based on time-varying generalized canonical transformations. A special class of time-varying generalized canonical transformations are introduced which modify the kinetic energy of the original system without changing the generalized Hamiltonian structure with passivity. Utilizing these transformations, time-varying asymptotically stabilizing controllers for the nonholonomic Hamiltonian systems are derived. Since the proposed method is a natural generalization of passivity based control for conventional holonomic systems, it is expected that the tools developed for conventional systems will be applicable to nonholonomic systems based on the proposed method.

Stabilization of Hamiltonian systems with dissipation

Stabilization of generalized Hamiltonian systems with dissipation is investigated. First, we generalize the Casimir submanifold approach with constant controls to the static state feedback control case. Secondly, a direct Hamiltonian function method is proposed and it is shown that this method is equivalent to the Casimir sub-manifold approach. Furthermore, a dynamic state feedback control is proposed. Some sufficient conditions and a constructive process for determining controllers are provided. It is shown that the dynamic state feedback control is more powerful than the static one. Finally, the problem of a dissipative type realization of the general controlled Hamiltonian system is discussed.

Time-varying Stabilization of Hamiltonian Systems Via Generalized Canonical Transformations

Abstract This paper focuses on the stabilization of port-controlled Hamiltonian systems employing possibly time-varying controllers. At first we refer to the generalized canonical transformation which preserves the structure of Hamiltonian systems and the passivity property that physical systems innately possess. Next we show a general stabilization strategy for port-controlled Hamiltonian systems based on it which is a natural generalization of well-known passivity based control. Finally we utilize this method to mechanical Hamiltonian systems with nonholonomic constraints by modifying the kinetic energy of the system. Furthermore some examples are given to show how this technique works for physical systems.

Noninteracting control with stability for Hamiltonian systems

The problem of noninteraction with stability via dynamic state feedback is addressed and solved for a class of nonlinear Hamiltonian systems. A simple way to check necessary and sufficient conditions is proposed. It is well known that to decide if the problem is solvable, and which class of state feedback has to be used, the stability properties of some special dynamics are to be investigated. For this reason, on the way to the main result, it is shown that such dynamics are not necessarily Hamiltonian. Several examples, clarifying the role of different classes of state-feedback control laws (either static or dynamic) in the solution of the problem, are proposed.

On the passive stabilization of the equilibrium state of Lagrangian systems

The idea of passive stabilization of a dynamical system whose motion is not asymptotically stable was advanced for the first time in the monograph [1] by introducing supplementary degrees of freedom. Based on this idea, Savchenko [2] discussed the stabilization of Hamiltonian systems by a nonlinear method (namely the method of passive stabilization by defreezing parameters). The authors of this paper investigate the effectiveness of its application to a Lagrangian system by a mechanical model which has independent scientific meaning. A comparison is also made between this model and another similar mechanical model. The problem of optimal passive stabilization is solved at the end of the paper. It is shown that this problem is closely connected with the resonance situations.