Hamiltonian Normal Form Research Articles

Hamiltonian normal forms in the neighbourhoods of collinear libration points in the restricted circular three body problem

The fourth-degree Hamiltonian normal forms in the neighbourhoods of the collinear libration points in the restricted plane circular three-body problem are found using an invariant normalization algorithm. Exact relations between the coefficients of the normal forms and the ratio of the masses of the heavy bodies are obtained. The subspaces of the phase variables, the initial values of which determine the periodic trajectories in the neighbourhoods of the libration points, are found.

Hamiltonian normal form in the vicinity of libration point L 1 in the restricted circular three-body problem

Hamiltonian normal form in the vicinity of libration point L 1 in the restricted circular three-body problem

Relevance of the 1:1 resonance in galactic dynamics

This paper aims to illustrate the applications of resonant Hamiltonian normal forms to some problems of galactic dynamics. We detail the construction of the 1:1 resonant normal form corresponding to a wide class of potentials with self-similar elliptical equipotentials and apply it to investigate relevant features of the orbit structure of the system. We show how to compute the bifurcation of the main periodic orbits in the symmetry planes of a triaxial ellipsoid and in the meridional plane of an axisymmetric spheroid and briefly discuss how to refine these results with higher-order approaches.

Lagrangian approach to weakly nonlinear stability of elliptical flow

Rotating flows with elliptically strained streamlines suffer from parametric resonance instability between a pair of Kelvin waves whose azimuthal wavenumbers are separated by two. We address the weakly nonlinear amplitude evolution of three-dimensional (3D) Kelvin waves, in resonance, on a flow confined in a cylinder of elliptic cross-section. In a traditional Eulerian approach, derivation of the mean flow induced by nonlinear interaction of Kelvin waves stands as an obstacle. We show how a topological idea, or the Lagrangian approach, facilitates calculation of the wave-induced mean flow. A steady incompressible Euler flow is characterized as a state of the maximum of the total kinetic energy with respect to perturbations constrained to an isovortical sheet, and the isovortical perturbation is handled only in terms of the Lagrangian variables. The criticality in energy of a steady flow allows us to calculate the wave-induced mean flow only from the linear Lagrangian displacement. With the mean flow at hand, the Lagrangian approach provides us with a shortcut to enter into a weakly nonlinear amplitude evolution regime of 3D disturbances. Unlike the Eulerian approach, the amplitude equations are available directly in the Hamiltonian normal form.

On nonlinear vibrations of a heavy mass point on a spring

We present a complete study of small nonlinear vibrations of a swinging spring with a nonlinear dependence of the spring tension on its elongation. We use the Hamiltonian normal form method. The Hamiltonian normal form profitably differs from the general normal form of differential equations, because it has an additional integral. To reduce the Hamiltonian to normal form, we use the invariant normalization method, which significantly reduces the computations. The normal form asymptotics are obtained by successively calculating the quadratures in the same way for both resonance and nonresonance cases. The solutions of Hamiltonian equations in normal form showed that the periodic change of vibrations from vertical to horizontal modes and vice versa occurs only in the case of 1:1 and 2:1 resonances. In the case of 2:1 resonance, this effect manifests itself in the quadratic terms of the equation, and in the case of 1:1 resonance, it manifests itself if the cubic terms are taken into account. In all other cases, both in the case of resonance and without any resonance, the vibrations occur at two constant frequencies, which slightly differ from the linear approximation frequencies. In the case of 2:1 resonance, we found the maximum frequency detuning at which the effect of the energy pumping from one vibration mode to another disappears. 1:1 resonance is physically possible only for a spring with a negative cubic additional term in the strain law.

Nonlinear 3D vibrations of a heavy material point on a spring at resonance

We consider the problem of nonlinear 3D vibrations of a heavy material point suspended on a weightless spring at a 1:1:2 frequency resonance. To construct an asymptotic solution, we use the Hamiltonian normal form method. Just as in the plane problem, this asymptotic solution describes the periodic process in which the vertical vibration energy passes into the horizontal vibration energy. For an arbitrarily small nonzero angular momentum with respect to the vertical axis, an effect typical of 3D systems manifests itself. The projection of the trajectory of the point onto the horizontal plane (xy) is an ellipse of constant area with axes varying in time. For certain initial conditions, the ellipse almost degenerates into straight-line segments. The direction of the straight line does not vary on the time interval where the vibration energy is in the horizontal mode and then varies almost by a jump on the interval where the vibration energy is transferred into the vertical mode. The analytic results are in good agreement with numerical solutions of equations of motion of the system.

Periodic orbits in the logarithmic potential

Analytic methods to investigate periodic orbits in galactic potentials. To evaluate the quality of the approximation of periodic orbits in the logarithmic potential constructed using perturbation theory based on Hamiltonian normal forms. The solutions of the equations of motion corresponding to periodic orbits are obtained as series expansions computed by inverting the normalizing canonical transformation. To improve the convergence of the series a resummation based on a continued fraction may be performed. This method is analogous to that looking for approximate rational solutions (Prendergast method). It is shown that with a normal form truncated at the lowest order incorporating the relevant resonance it is possible to construct quite accurate solutions both for normal modes and periodic orbits in general position.

Hamiltonian normal forms

Hamiltonian normal forms

Calculation of the Hamiltonian normal form

Calculation of the Hamiltonian normal form

Hamiltonian maps and normal forms for intense beams

The dynamics of a beam in a ring with a localized multipolar nonlinearity is described by a polynomial one turn map. The space charge forces act continuously along the ring, but their effect can be included by replacing the linear tune with the depressed tune which depends on the Courant Snyder invariant. This approximation allows to use the normal forms to compute the nonlinear invariants, the nonlinear tune and the islands geometric parameters when a low order resonance is approached.

Regular deformations of completely integrable systems

We study several aspects of the regular deformations of completely integrable systems. Namely, we prove the existence of a Hamiltonian normal form for these deformations and we show the necessary and sufficient conditions a perturbation has to satisfy in order for the perturbed Hamiltonian to be a first order deformation.

ON THE CONSTRUCTION OF TRANSFORMATIONS OF LINEAR HAMILTONIAN SYSTEMS TO REAL NORMAL FORMS

A technique for constructing the transformations to real Hamiltonian normal forms of linear Hamiltonian systems via permutation matrices is presented. In particular, a method is shown for obtaining the symplectomorphism between the symplectic basis of the real Jordan form to the standard symplectic basis in which the real Hamiltonian normal form resides. All possible degeneracies are accounted for since the algebraic and geometric multiplicities of nonsemisimple eigenvalues are not restricted, including the "difficult" cases of zero and imaginary eigenvalues. Since the normal forms are not unique, several possible arrangements of the suggested transformations are given which result in the various normal forms derived previously as well as in a few new ones for degenerate cases which have not appeared before.

Non-autonomous normal forms for Hamiltonian systems

A method for calculating normal forms for non-autonomous periodically perturbed Hamiltonian systems is developed. The solution for an autonomous Hamiltonian normal form is well known, and involves the solution of a homological equation on the vector space of homogeneous scalar polynomials. An algorithm is presented for generating an analogous non-autonomous homological equation using Lie transforms. Solution of this equation will generate a normal form for the non-autonomous Hamiltonian. Although this equation is defined on an infinite-dimensional space, it is shown that the problem can be reduced to an equivalent one on a finite-dimensional space. A solution can then be found in an analogous way to the solution for the autonomous problem. It is also shown that the normal form satisfies invariance properties. Finally, an example problem is presented to illustrate the solution technique.

Resonances, radiation damping and instabilitym in Hamiltonian nonlinear wave equations

We consider a class of nonlinear Klein-Gordon equations which are Hamiltonian and are perturbations of linear dispersive equations. The unperturbed dynamical system has a bound state, a spatially localized and time periodic solution. We show that, for generic nonlinear Hamiltonian perturbations, all small amplitude solutions decay to zero as time tends to infinity at an anomalously slow rate. In particular, spatially localized and time-periodic solutions of the linear problem are destroyed by generic nonlinear Hamiltonian perturbations via slow radiation of energy to infinity. These solutions can therefore be thought of as metastable states. The main mechanism is a nonlinear resonant interaction of bound states (eigenfunctions) and radiation (continuous spectral modes), leading to energy transfer from the discrete to continuum modes. This is in contrast to the KAM theory in which appropriate nonresonance conditions imply the persistence of invariant tori. A hypothesis ensuring that such a resonance takes place is a nonlinear analogue of the Fermi golden rule, arising in the theory of resonances in quantum mechanics. The techniques used involve: (i) a time-dependent method developed by the authors for the treatment of the quantum resonance problem and perturbations of embedded eigenvalues, (ii) a generalization of the Hamiltonian normal form appropriate for infinite dimensional dispersive systems and (iii) ideas from scattering theory. The arguments are quite general and we expect them to apply to a large class of systems which can be viewed as the interaction of finite dimensional and infinite dimensional dispersive dynamical systems, or as a system of particles coupled to a field.

Time-dependent normal form Hamiltonian for dynamical equilibria

A new time-dependent normal-form procedure for dynamical equilibria (undergoing parametric excitation) of one-dimensional (1D) Hamiltonian systems is developed with the method of Lie transforms. The expansion is based on the Lewis invariant for the linearized motion. The time-dependent Hamiltonian normal form reduces smoothly to the usual representation in the autonomous limit. Illustrative examples of the formalism are focused on time-periodic systems and the dynamics of Hamiltonian switching.

Normal forms for three-dimensional parametric instabilities in ideal hydrodynamics

We derive and analyze several low dimensional Hamiltonian normal forms describing system symmetry breaking in ideal hydrodynamics. The equations depend on two parameters (ϵ, λ), where ϵ is the strength of a system symmetry breaking perturbation and λ is a detuning parameter. In many cases the resulting equations are completely integrable and have an interesting Hamiltonian structure. Our work is motivated by three-dimensional instabilities of rotating columnar fluid flows with circular streamlines (such as the Burger vortex) subjected to precession, elliptical distortion or off-center displacement.

Instability induced by symmetry reduction.

We demonstrate that instabilities in a Hamiltonian system can occur via deformations that reduce the symmetry of the system. The movement of eigenvalues at an equilibrium point of a family of Hamiltonian systems is constrained by the symmetry type of the system. If deformations of a family change the symmetry type, then instabilities can appear at multiple eigenvalues that produce large amplitude changes in the system dynamics. We illustrate this phenomenon in the context of a low-dimensional Hamiltonian normal form, and then analyze the instability of a vortex filament in a strain field

Quadratic Hamiltonians in phase-space quantum mechanics

The dynamical evolution is described within the phase-space formalism by means of the Moyal propagator, which is the symbol of the evolution operator. Quadratic Hamiltonians on the phase space are distinguished in that their Moyal bracket with any function equals their Poisson bracket. It is shown that, for general time-independent quadratic Hamiltonians, the Moyal propagators transform covariantly under linear canonical transformations; they are then derived and classified in a fully explicit manner using the theory of Hamiltonian normal forms. The authors present several tables of propagators. It is proved that these propagators belong to the Moyal algebra of distributions, and that the spectrum of the Hamiltonian may be obtained directly as the support of the Fourier transform of the Moyal propagator with respect to time. From that, the quantum-mechanical problem for these Hamiltonians is, in principle, completely solved. The appropriate path-integral formalism for phase-space quantum mechanics, leading back to the same results, is outlined in an appendix.

Global aspects of hamiltonian normal forms

We derive simple global characterizations for the normal forms of hamiltonian systems with constant and periodic coefficients.