Hamiltonian Sense Research Articles

Hamiltonian/Stroh formalism for reversible poroelasticity (and thermoelasticity)

Stroh’s sextic formalism represents the equilibrium equations of anisotropic elasticity in a particularly attractive form, that is most suitable for studying interface-dominated multilayered solids, composite materials and time-harmonic problems. Taking advantage of the fact that the Stroh formalism really amounts to the canonical form of the equations in the Hamiltonian sense, the case of Biot’s reversible (i.e. no fluid dissipation) poroelasticity is here addressed, in the absence of a fluid pressure gradient. This framework is the same as thermoelasticity of perfect conductors. Two Hamiltonian formulations are developed: the first describes both the solid and the fluid phases and it exhibits, besides energy conservation, momentum conservation, as a result of pressure uniformity (perfectly drained conditions). The second is restricted to the solid skeleton and perfectly parallels anisotropic elasticity, where the Stroh matrices refer to the effective stress tensor. The case of weak fluid–solid coupling is also considered and it produces a perturbation from anisotropic elasticity with the same structure as incompressibility, although in an “opposing” manner. This comparison suggests that the incompressibility limit introduced by Biot should be revised. The energy conservation integral and the edge impedance matrix are also illustrated.

Classification of phases for mixed states via fast dissipative evolution

We propose the following definition of topological quantum phases valid for mixed states: two states are in the same phase if there exists a time independent, fast and local Lindbladian evolution driving one state into the other. The underlying idea, motivated by \cite{Konig2014}, is that it takes time to create new topological correlations, even with the use of dissipation.We show that it is a good definition in the following sense: (1) It divides the set of states into equivalent classes and it establishes a partial order between those according to their level of ``topological complexity''. (2) It provides a path between any two states belonging to the same phase where observables behave smoothly.We then focus on pure states to relate the new definition in this particular case with the usual definition for quantum phases of closed systems in terms of the existence of a gapped path of Hamiltonians connecting both states in the corresponding ground state path. We show first that if two pure states are in the same phase in the Hamiltonian sense, they are also in the same phase in the Lindbladian sense considered here.We then turn to analyse the reverse implication, where we point out a very different behaviour in the case of symmetry protected topological (SPT) phases in 1D. Whereas at the Hamiltonian level, phases are known to be classified with the second cohomology group of the symmetry group, we show that symmetry cannot give any protection in 1D in the Lindbladian sense: there is only one SPT phase in 1D independently of the symmetry group.We finish analysing the case of 2D topological quantum systems. There we expect that different topological phases in the Hamiltonian sense remain different in the Lindbladian sense. We show this formally only for theZnquantum double modelsD(Zn). Concretely, we prove that, ifmis a divisor ofn, there cannot exist any fast local Lindbladian connecting a ground state ofD(Zm)with one ofD(Zn), making rigorous the initial intuition that it takes long time to create those correlations present in theZncase that do not exist in theZmcase and that, hence, theZnphase is strictly more complex in the Lindbladian case than theZmphase. We conjecture that such Lindbladian does exist in the opposite direction since Lindbladians can destroy correlations.

Bending reorientational solitons with modulated alignment

While curved waveguides are fundamental elements in photonics, those induced all optically in nonlinear uniform dielectrics tend to be straight. In uniaxial soft matter with a reorientational response, such as nematic liquid crystals (NLCs), light beams in the extraordinary polarization undergo self-focusing via an increase in refractive index and eventually form spatial solitons, i.e., self-induced waveguides. Hereby we investigate the bending of such waveguides by analyzing the trajectory of solitons in NLCs—nematicons—in the presence of a linearly varying transverse orientation of the optic axis. To this extent, we use and compare two approaches: i) a slowly varying (adiabatic) approximation based on momentum conservation of the nematicon in a Hamiltonian sense; and ii) the Frank–Oseen elastic theory coupled with a fully vectorial and nonlinear beam propagation method. The models provide comparable results in such a non-homogeneously oriented uniaxial medium and predict bent soliton paths with either monotonic or non-monotonic curvatures, enabling the design of curved channel waveguides induced by light beams.

Complex Hamiltonian systems on ℂ2 with Hamiltonian function of low laurent degree

We study complex Hamiltonian systems on ℂ × (ℂ \ {0}) with the standard symplectic structure ωℂ = dz^dw and Hamiltonian function f = az2+b/w+Pn(w), where Pn(w) is a polynomial of degree n, the numbers a, b ∈ ℂ, and ab ≠ 0. For some natural classes of these ℂ-Hamiltonian systems, an equivalence relation in the Hamiltonian sense is studies and the topology of the corresponding quotient space is determined. It is also proved that for ℂ-Hamiltonian systems with the Hamiltonian function f = az2 + b/w + Pn(w), where ab ≠ 0, n ≥ 0, the bifurcation complex is homeomorphic to a two-dimensional plane.

The effect of asymmetrical body-mass distribution on the stability and dynamics of quadruped bounding

The effect of asymmetrical body-mass distribution on the stability and dynamics of two-degree-of-freedom quadruped bounding in place is investigated in this study. An analytical stability criterion for bounding of quadrupeds with asymmetrical mass distribution is developed. Bounding is found to be passively stable in the Hamiltonian sense when the dimensionless pitch moment of inertia of the body is less than 1-/spl beta//sup 2/, where /spl beta/ is a dimensionless measure of the asymmetry. The criterion is derived under the assumptions of infinite leg stiffness and no energy loss. With energy dissipation modeled as linear damping in the legs, simulation results show that the criterion is independent of the value of leg stiffness and a conservative estimate of the critical inertia value. Body symmetry appears to be more favorable to stable bounding than asymmetry, but only slightly so in practicality. The results show that asymmetry offers some advantages when the dynamic characteristics of bounding are compared for symmetrical and asymmetrical models. Lower stride frequency, larger vertical displacement, and smaller duty factor are obtained with this more animal-like model.

Control of integrable Hamiltonian systems and degenerate bifurcations.

We discuss control of low-dimensional systems which, when uncontrolled, are integrable in the Hamiltonian sense. The controller targets an exact solution of the system in a region where the uncontrolled dynamics has invariant tori. Both dissipative and conservative controllers are considered. We show that the shear flow structure of the undriven system causes a Takens-Bogdanov bifurcation to occur when control is applied. This implies extreme noise sensitivity. We then consider an example of these results using the driven nonlinear Schrödinger equation.

Dispersionless limit of integrable models

Nonlinear dispersionless equations arise as the dispersionless limit of well know integrable hierarchies of equations or by construction, such as the system of hydrodynamic type. Some of these equations are integrable in the Hamiltonian sense and appear in the study of topological minimal models. In the first part of the review we will give a brief introduction to integrable models, mainly its Lax representation. Then, we will introduce the dispersionless limit and show some of our results concerning the two-component hyperbolic system of equations such as the polytropic gas and Born-Infeld equations.

Waves in the Davey-Stewartson equation

Generalized Backlund gauge transformations depending on arbitrary functions are introduced. They are used to generate wave solutions of the Davey-Stewartson equation. In particular a version of the Davey-Stewartson equation which is completely integrable in the Hamiltonian sense is considered. In this case different forms of extended string-type wave solutions are given.

Possible chaotic pulsations in ZZ Ceti and rapidly oscillating Ap stars due to non-linear harmonic mode coupling

Summary Some ZZ Ceti and rapidly oscillating Ap stars show some multi-periodic behaviour, including prominent peaks in their power spectra at harmonically spaced frequencies with ratios 1 :2:3. The non-linear interaction between three such modes is a combination of the usual three-wave interaction coupled to second-harmonic generation. The amplitude equations for such a combined interaction are fundamentally different, and the Painleve analysis carried out here shows them to be non-integrable in the Hamiltonian sense, so that for long-term behaviour one expects irregular or chaotic phenomena instead of simple periodic or multi-periodic pulsations, especially when other modes and dissipation are included. Some examples of stars to which this might apply are given. These include VY Hor = BPM 3 1594 and BG CVn = GD 154 among the ZZ Ceti stars, PG 1351 + 489 among the DB variable white dwarfs, and HD 60435 and HD 22374 among the rapidly oscillating Ap stars.

Null co-ordinate dynamics of linearized gravitation

The null dynamics of the first-order linearized theory has different qualitative properties with respect to the well-known dynamical analysis made by Arnowitt and Deser when it is chosen a preferredx 0 = const hypersurface. If we focus the study on the hypersurfacesx + = const, the whole set of new constraints is found and after having been introduced into the action it is seen that the dynamics depend upon 2 dynamical independent variables. These 2 independent variables, which correspond to the fact that the linearized gravitational field is a massless helicity-2 field, are self-conjugate in a Hamiltonian sense,i.e. these variables essentially coincide with their momenta. Thus the Hamiltonian formalism is given without doubling the number of independent field variables, as in any integer-spinx 0-formulation happens. The null Hamiltonian is found and it is seen to correspond to the momentumP + of the field. Thereafter, the Heisenberg commutation relations for the dynamical variables are computed. Null electrodynamics is also studied starting from the first-order action and the so-called null gauge is discussed. In particular it is shown that its choice is not compulsory.