Jan-Cees van der Meer

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• Dr.
• UD (retired) at Eindhoven University of Technology

We present a geometrical description of the symmetries and reduction of the full gravitational 2-body problem after complete averaging over fast angles. Our variables allow for a well-suited formulation in action-angle type coordinates associated with the averaged angles, which provide geometric insight into the problem. After introducing extra fic...

In this note we will consider reduction techniques for Hamiltonian systems that are invariant under the action of a compact Lie group $G$ acting by symplectic diffeomorphisms and the related work on stability of relative equilibria. We will focus on reduction by invariants in which case it is possible to describe a reduced phase space within the or...

The KS regularization connects the dynamics of the harmonic oscillator to the dynamics of bounded Kepler orbits. Using orbit space reduction, it can be shown that reduced harmonic oscillator orbits can be identified with re-parametrized Kepler orbits by factorizing the KS map as reduction mapping followed by a chart on the reduced phase space. In t...

A geometrical approach to a radial intermediary model for an axisymmetric rigid body in roto-orbital motion is presented. The presence of symmetries enables a well-suited formulation by choosing action–angle type variables. Singularities associated with the angles are avoided by introducing extra fictitious variables and performing a symplectic tra...

It is shown that the generalized Hopf map H × H → H × ℝ × ℝ quaternion formulation can be interpreted as an SO(3) orbit map for a symplectic SO(3) action. As a consequence the generalized Hopf fibration S7 → S4 appears in the SO(3) geometric symplectic reduction of the 4DOF isotropic harmonic oscillator. Furthermore it is shown how the Hopf fibrati...

In this paper we review the connection between the Kepler problem and the harmonic oscillator. More specifically we consider how the Kepler system can be obtained through geometric reduction of the harmonic oscillator. We use the method of constructive geometric reduction and explicitly construct the reduction map in terms of invariants. The Kepler...

Relative equilibria of an intermediary in attitude dynamics of a generic triaxial spacecraft in a circular orbit under gravity-gradient perturbation are discussed. Intermediary defines a Poisson flow over a large parameter space: three physical parameters (moments of intertia) and three distinguished parameters, the integrals M,G3 and n. In the cas...

The 2-D sextic oscillator is studied as a family of axial symmetric parametric integrable Hamiltonian systems, presenting a bifurcation analysis of the different flows. It includes the "elliptic core" model in 1-D nonlinear oscillators, recently proposed in the literature. We make use of the energy-momentum mapping, which will give us the fundament...

This paper deals with the analysis of Hamiltonian Hopf as well as saddle-center bifurcations in 4-DOF systems defined by perturbed isotropic oscillators (1:1:1:1 resonance), in the presence of two quadratic symmetries Ξ and L 1. When we normalize the system with respect to the quadratic part of the energy and carry out a reduction with respect to a...

A uniparametric 4-DOF family of perturbed Hamiltonian oscillators in 1:1:1:1 resonance is studied as a generalization for several models for perturbed Keplerian systems. Normalization by Lie transforms (only first order is considered here) as well as geometric reduction related to the invariants associated to the symmetries is used based on the pre...

The autocatalytic equation derived in this study describes and even predicts the evolution of the number average molecular weight of aliphatic polyesters upon hydrolytic degradation. The main reaction in the degradation of aliphatic polyesters is autocatalytic hydrolysis of ester bonds, which causes the molecular weight to decrease. During hydrolys...

Consider a Poisson structure on C∞(R^3, R) with bracket {, } and suppose that C is a Casimir function. Then {f, g} =< ∇C,(∇g × ∇f) > is a possible Poisson structure. This confirms earlier observations concerning the Poisson structure for Hamiltonian systems that are reduced to a one degree of freedom system and generalizes the Lie-Poisson structure...

The objective of this study was to develop a mathematical model that describes and even predicts the hydrolytic degradation of aliphatic polyesters. From literature, it is known that the main process of degradation of aliphatic polyesters is the autocatalytic hydrolysis of ester bonds. Because of this hydrolysis, polymer chains are cleaved and the...

We consider Hamiltonian systems in 1:1 resonance in the presence of symmetry. We give some new proofs for known results concerning the classifi-cation of generic one-parameter deformations of equivariant linear systems and the passing and splitting of eigenvalues. We show that for nonlinear systems in two degrees of freedom the bifurcation of perio...

In this communication we deal with the analysis of Hamiltonian Hopf bifurcations in 4-DOF systems defined by perturbed isotropic oscillators (1-1-1-1 resonance), in the presence of two quadratic symmetries I 1 and I 2. As a perturbation we consider a polynomial function with a parameter. After normalization, the truncated normal form gives rise to...

A method is sketched to determine the presence of non-degenerate Hamiltonian Hopf bifurcations in three-degree-of-freedom systems by putting the bifurcation into standard form. Detailed computations are performed for the non-trivial example of the 3D Hnon–Heiles family. After a careful formulation of the local once reduced system in terms of proper...

We give a short review of available methods to determine the nondegeneracy of Hamiltonian-Hopf bifurcations in three-degree-of-freedom systems. We illustrate the geometric method to more detail, using the example of the Lagrange top.

In this note we study the local behavior of singularities occurring in scale space under Gaussian blurring. Based on ideas from singularity theory for vector fields this is done by considering deformations or unfoldings. To deal with the special nature of the problem the concept of Gaussian deformation is introduced. Using singularity theory the st...

In this note we study the local behavior of singularities occurring in scale space tinder Gaussian blurring. Based on ideas from singularity theory for vector fields this is done by considering deformations or unfoldings. To deal with the special nature of the problem the concept of Gaussian deformation is introduced. Using singularity theory the s...

We consider Hamiltonian systems in 1:1 resonance in the presence of symmetry. We give some new proofs for known results concerning the classification of generic one-parameter deformations of equivariant linear systems and the passing and split- ting of eigenvalues. We show that for nonlinear systems in two degrees of freedom the bifurcation of peri...

An axially symmetric perturbed isotropic harmonic oscillator undergoes several bifurcations as the parameter adjusting the relative strength of the two terms in the cubic potential is varied. We show that three of these bifurcations are Hamiltonian Hopf bifurcations. To this end we analyse an appropriately chosen normal form. It turns out that the...

Studies Hamiltonian Hopf bifurcation in the presence of a compact symmetry group G. The author classifies the expected actions of G and show that near four-dimensional fixed point subspaces of subgroups of G*Si the bifurcation of periodic solutions is diffeomorphic to the standard Hamiltonian Hopf bifurcation in two degrees of freedom. Examples are...

We show that a reversible non-Hamiltonian vector field at nonsemisimple 1:1 resonance can be split into a Hamiltonian and a non-Hamiltonian part in such a way that after reduction to the orbit space for the S 1 -action coming from the semisimple part of the linearized vector field the non-Hamiltonian part vanishes. As a consequence the reduced reve...

We show that a reversible non-Hamiltonian vector field at nonsemisimple 1:1 resonance can be split into a Hamiltonian and a non-Hamiltonian part in such a way that after reduction to the orbit space for the S 1-action coming from the semisimple part of the linearized vector field the non-Hamiltonian part vanishes. As a consequence the reduced rever...

We show that the Lagrange top undergoes a Hamiltonian Hopf bifurcation when the angular momentum corresponding to rotation about the symmetry axis of the body passes through a value where the sleeping top changes stability.

Consider a Hamiltonian system (H, 2n ,). LetM be a symplectic submanifold of (2n ,). The system (H, 2n ,) constrained toM is (HM, M, M). In this paper we give an algorithm which normalizes the system on 2n in such a way that restricted toM we have normalized the constrained system. This procedure is then applied to perturbed Kepler systems such as...

In this paper a description is given of the bifurcation of periodic solutions occurring when a Hamiltonian system of two degrees of freedom passes through nonsemisimple 1–1 resonance at an equilibrium. A bifurcation like this is found in the planar circular restricted problem of three bodies at the Lagrange equilibriumL 4 when the mass parameter pa...

Periodic solutions of a family of Hamiltonian systems of two degrees of freedom are studied. The general normal form theory and the normalization of the Hamiltonian function are examined. The application of the equivalent theory of stability of maps to the derivation of normal forms for energy-momentum maps of Hamiltonian systems is discussed. The...

Consider a Hamiltonian system of two degrees of freedom at an equilibrium. Suppose that the linearized vectorfield has eigenvaluesi,i,–i,–i ( , >0) and is not semisimple. In this paper we discuss the real normalization of the Hamiltonian function of such a system. We normalize the Hamiltonian up to 4th order and show how to compute its coefficients...