Hamiltonian dynamics Research Papers - Academia.edu (original) (raw)
The goal of the present account is to review our efforts to obtain and apply a “collective” Hamiltonian for a few, approximately decoupled, adiabatic degrees of freedom, starting from a Hamiltonian system with more or many more degrees of... more
The goal of the present account is to review our efforts to obtain and apply a “collective” Hamiltonian for a few, approximately decoupled, adiabatic degrees of freedom, starting from a Hamiltonian system with more or many more degrees of freedom. The approach is based on an analysis of the classical limit of quantum-mechanical problems. Initially, we study the classical problem within the framework of Hamiltonian dynamics and derive a fully self-consistent theory of large-amplitude collective motion with small velocities. We derive a measure for the quality of decoupling of the collective degree of freedom. We show for several simple examples, where the classical limit is obvious, that when decoupling is good, a quantization of the collective Hamiltonian leads to accurate descriptions of the low energy properties of the systems studied. In nuclear physics problems we construct the classical Hamiltonian by means of time-dependent mean-field theory, and we transcribe our formalism to this case. We report studies of a model for monopole vibrations, of 28Si with a realistic interaction, several qualitative models of heavier nuclei, and preliminary results for a more realistic approach to heavy nuclei. Other topics included are a nuclear Born–Oppenheimer approximation for an ab initio quantum theory and a theory of the transfer of energy between collective and noncollective degrees of freedom when the decoupling is not exact. The explicit account is based on the work of the authors, but a thorough survey of other work is included.
A time-varying electron density bubble created by the radiation pressure of a tightly focused petawatt laser pulse traps electrons of ambient rarefied plasma and accelerates them to a GeV energy over a few-cm distance. Expansion of the... more
A time-varying electron density bubble created by the radiation pressure of a tightly focused petawatt laser pulse traps electrons of ambient rarefied plasma and accelerates them to a GeV energy over a few-cm distance. Expansion of the bubble caused by the shape variation of the self-guided pulse is the primary cause of electron self-injection in strongly rarefied plasmas (n_e ∼ 10^17 cm^−3). Stabilization and contraction of the bubble extinguishes the injection. After the bubble stabilization, longitudinal non-uniformity of the accelerating gradient results in a rapid phase space rotation that produces a quasi-monoenergetic bunch well before the de-phasing limit. Combination of reduced and fully self-consistent (first-principle) 3-D PIC simulations complemented with the Hamiltonian diagnostics of electron phase space shows that the bubble dynamics and the self-injection process are governed primarily by the driver evolution; collective transverse fields of the trapped electron bunch reduce the accelerating gradient, slow down phase space rotation, and result in a formation of monoenergetic electron beam with higher energy than test-particle modeling predicts.
In the domain of nondissipative unitary Hamiltonian dynamics, the well-known Mandelstam–Tamm–Messiah time–energy uncertainty relation τ F Δ H ≥ ℏ / 2 provides a general lower bound to the characteristic time τ F = Δ F / | d ⟨ F ⟩ / d t |... more
In the domain of nondissipative unitary Hamiltonian dynamics, the well-known Mandelstam–Tamm–Messiah time–energy uncertainty relation τ F Δ H ≥ ℏ / 2 provides a general lower bound to the characteristic time τ F = Δ F / | d ⟨ F ⟩ / d t | with which the mean value of a generic quantum observable F can change with respect to the width Δ F of its uncertainty distribution (square root of F fluctuations). A useful practical consequence is that in unitary dynamics the states with longer lifetimes are those with smaller energy uncertainty Δ H (square root of energy fluctuations). Here we show that when unitary evolution is complemented with a steepest-entropy-ascent model of dissipation, the resulting nonlinear master equation entails that these lower bounds get modified and depend also on the entropy uncertainty Δ S (square root of entropy fluctuations). For example, we obtain the time–energy-and–time–entropy uncertainty relation ( 2 τ F Δ H / ℏ ) 2 + ( τ F Δ S / k B τ ) 2 ≥ 1 where τ is ...
We study the dynamics in the neighborhood of an invariant torus of a nearly integrable system. We provide an upper bound to the diffusion speed, which turns out to be of superexponentially small size exp[-exp(1/σ)], σ being the distance... more
We study the dynamics in the neighborhood of an invariant torus of a nearly integrable system. We provide an upper bound to the diffusion speed, which turns out to be of superexponentially small size exp[-exp(1/σ)], σ being the distance from the invariant torus. We also discuss the connection of this result with the existence of many invariant tori close to the considered one.
The Hamiltonian dynamics of classical planar Heisenberg model is numerically investigated in two and three dimensions. By considering the dynamics as a geodesic flow on a suitable Riemannian manifold, it is possible to analytically... more
The Hamiltonian dynamics of classical planar Heisenberg model is numerically investigated in two and three dimensions. By considering the dynamics as a geodesic flow on a suitable Riemannian manifold, it is possible to analytically estimate the largest Lyapunov exponent in terms of some curvature fluctuations. The agreement between numerical and analytical values for Lyapunov exponents is very good in a wide range of temperatures. Moreover, in the three dimensional case, in correspondence with the second order phase transition, the curvature fluctuations exibit a singular behaviour which is reproduced in an abstract geometric model suggesting that the phase transition might correspond to a change in the topology of the manifold whose geodesics are the motions of the system.
Point-vortex models are presented for the generalized Euler equations, which are characterized by a fractional Laplacian relation between the active scalar and the stream function. Special focus is given to the case of the surface... more
Point-vortex models are presented for the generalized Euler equations, which are characterized by a fractional Laplacian relation between the active scalar and the stream function. Special focus is given to the case of the surface quasigeostrophic (SQG) equations, for which the existence of finite-time singularities is still a matter of debate. Point-vortex trajectories are expressed using Nambu dynamics. The formulation is based on a noncanonical bracket and allows for a geometrical interpretation of trajectories as intersections of level sets of the Hamiltonian and Casimir. Within this setting, we focus on the collapse of solutions for the three-point-vortex model. In particular, we show that for SQG the collapse can be either self-similar or non-self-similar. Self-similarity occurs only when the Hamiltonian is zero, while non-self-similarity appears for nonzero values of the same. For both cases, collapse is allowed for any choice of circulations within a permitted interval. These results differ strikingly from the classical point-vortex model, where collapse is self-similar for any value of the Hamiltonian, but the vortex circulations must satisfy a strict relationship. Results may also shed a light on the formation of singularities in the SQG partial differential equations, where the singularity is thought to be reached only in a self-similar way.
In some previous papers, a geometric description of Lagrangian Mechanics on Lie algebroids has been developed. In the present paper, we give a Hamiltonian description of Mechanics on Lie algebroids. In addition, we introduce the notion of... more
In some previous papers, a geometric description of Lagrangian Mechanics on Lie algebroids has been developed. In the present paper, we give a Hamiltonian description of Mechanics on Lie algebroids. In addition, we introduce the notion of a Lagrangian submanifold of a symplectic Lie algebroid and we prove that the Lagrangian (Hamiltonian) dynamics on Lie algebroids may be described in terms of Lagrangian submanifolds of symplectic Lie algebroids. The Lagrangian (Hamiltonian) formalism on Lie algebroids permits to deal with Lagrangian (Hamiltonian) functions not defined necessarily on tangent (cotangent) bundles. Thus, we may apply our results to the projection of Lagrangian (Hamiltonian) functions which are invariant under the action of a symmetry Lie group. As a consequence, we obtain that Lagrange-Poincare (Hamilton-Poincare) equations are the Euler-Lagrange (Hamilton) equations associated with the corresponding Atiyah algebroid. Moreover, we prove that Lagrange-Poincare (Hamilton-Poincare) equations are the local equations defining certain Lagrangian submanifolds of symplectic Atiyah algebroids.
Clustered motion of particles is found in Hamiltonian dynamics of symplectic coupled map systems. Particles assemble and move with strong correlation. The motion is chaotic but is distinguishable from random chaotic motion. Lyapunov... more
Clustered motion of particles is found in Hamiltonian dynamics of symplectic coupled map systems. Particles assemble and move with strong correlation. The motion is chaotic but is distinguishable from random chaotic motion. Lyapunov analysis distinguishes global instability from local fluctuations. Clustered motions have finite lifetime. They have fractal geometric structure in the phase space, as the orbits are trapped to ruins of KAM tori and islands.
This paper presents an adaptive dynamic analysis of discontinuous smart beam energy harvester systems using a shunt vibration control. The smart structural systems, connected with the shunt and harvesting circuit interfaces, consist of... more
This paper presents an adaptive dynamic analysis of discontinuous smart beam energy harvester systems using a shunt vibration control. The smart structural systems, connected with the shunt and harvesting circuit interfaces, consist of the three types of non-homogeneous structural combinations with different piezoelectric materials. The constitutive coupled dynamic equations with full variational parameters are reduced using the charge type-based Hamiltonian mechanics and the Ritz method-based weak-form analytical approach. Unlike the conventional techniques, this study elaborates the appearance of the two resonances with a wider shift on a specific range of the optimal power output frequencies, using only the first mode of the smart structural systems. Moreover, the two-equal peak of the optimal response may potentially occur to appear not only at the first resonance, but also at the second resonance. This intrinsically represents strong electromechanical effect, depending on the properties and thicknesses of piezoelectric materials and the circuit parameters. The accuracy of the theoretical method is tested using the iterative computational process of the optimal frequency response with full coupled electromechanical system parameters. Further details of the parametric studies are discussed to show the prediction of the energy harvesting with the ability of tuning an adaptive frequency response.
We use the global stochastic analysis tools introduced by P. A. Meyer and L. Schwartz to write down a stochastic generalization of the Hamilton equations on a Poisson manifold that, for exact symplectic manifolds, satisfy a natural... more
We use the global stochastic analysis tools introduced by P. A. Meyer and L. Schwartz to write down a stochastic generalization of the Hamilton equations on a Poisson manifold that, for exact symplectic manifolds, satisfy a natural critical action principle similar to the one encountered in classical mechanics. Several features and examples in relation with the solution semimartingales of these equations are presented.
We show that the strong form of Heisenberg’s inequalities due to Robertson and Schrödinger can be formally derived using only classical considerations. This is achieved using a statistical tool known as the “minimum volume ellipsoid”... more
We show that the strong form of Heisenberg’s inequalities due to Robertson and Schrödinger can be formally derived using only classical considerations. This is achieved using a statistical tool known as the “minimum volume ellipsoid” together with the notion of symplectic capacity, which we view as a topological measure of uncertainty invariant under Hamiltonian dynamics. This invariant provides a right measurement tool to define what “quantum scale” is. We take the opportunity to discuss the principle of the symplectic camel, which is at the origin of the definition of symplectic capacities, and which provides an interesting link between classical and quantum physics.
In the 1830s, W. R. Hamilton established a formal analogy between optics and mechanics by constructing a mathematical equivalence between the extremum principles of ray optics (Fermat's principle) and corpuscular mechanics (Maupertuis's... more
In the 1830s, W. R. Hamilton established a formal analogy between optics and mechanics by constructing a mathematical equivalence between the extremum principles of ray optics (Fermat's principle) and corpuscular mechanics (Maupertuis's principle). Almost a century later, this optical-mechanical analogy played a central role in the development of wave mechanics. Schrödinger was well acquainted with Hamilton's analogy through earlier studies. From Schrödinger's research notebooks, we show how he used the analogy as a heuristic tool to develop de Broglie's ideas about matter waves and how the role of the analogy in his thinking changed from a heuristic tool into a formal constraint on possible wave equations. We argue that Schrödinger only understood the full impact of the optical-mechanical analogy during the preparation of his second communication on wave mechanics: Classical mechanics is an approximation to the new undulatory mechanics, just as ray optics is an approximation to wave optics. This completion of the analogy convinced Schrödinger to stick to a realist interpretation of the wave function, in opposition to the emerging mainstream. The transformations in Schrödinger's use of the optical-mechanical analogy can be traced in his research notebooks, which offer a much more complete picture of the development of wave mechanics than has been previously thought possible.
Contrary to many recent models of growing networks, we present a model with fixed number of nodes and links, where a dynamics favoring the formation of links between nodes with degree of connectivity as different as possible is... more
Contrary to many recent models of growing networks, we present a model with fixed number of nodes and links, where a dynamics favoring the formation of links between nodes with degree of connectivity as different as possible is introduced. By applying a local rewiring move, the network reaches equilibrium states assuming broad degree distributions, which have a power-law form in an intermediate range of the parameters used. Interestingly, in the same range we find nontrivial hierarchical clustering.