Hamiltonian Flows Research Articles

Orbital dynamics on invariant sets of contact Hamiltonian systems

<p style='text-indent:20px;'>In this paper, we shall give new insights on dynamics of contact Hamiltonian flows, which are gaining importance in several branches of physics as they model a dissipative behaviour. We divide the contact phase space into three parts, which are corresponding to three differential invariant sets <inline-formula><tex-math id="M1">\begin{document}$ \Omega_\pm, \Omega_0 $\end{document}</tex-math></inline-formula>. On the invariant sets <inline-formula><tex-math id="M2">\begin{document}$ \Omega_\pm $\end{document}</tex-math></inline-formula>, under some geometric conditions, the contact Hamiltonian system is equivalent to a Hamiltonian system via the Hölder transformation. The invariant set <inline-formula><tex-math id="M3">\begin{document}$ \Omega_0 $\end{document}</tex-math></inline-formula> may be composed of several equilibrium points and heteroclinic orbits connecting them, on which contact Hamiltonian system is conservative. Moreover, we have shown that, under general conditions, the zero energy level domain is a domain of attraction. In some cases, such a domain of attraction does not have nontrivial periodic orbits. Some interesting examples are presented.</p>

Ergodicity, mixing, Ratner's properties and disjointness for classical flows: On the research of Corinna Ulcigrai

<p style='text-indent:20px;'>We present and discuss C. Ulcigrai's results concerning mixing properties of locally Hamiltonian flows, spectral properties of smooth time changes of horocycle flows together with their Möbius orthogonality and the ergodicity problems of directional flows in the wind tree model of Ehrenfest.</p>

Measures on Hilbert space invariant with respect to Hamiltonian flows

Measures on Hilbert space invariant with respect to Hamiltonian flows

English

We construct a class of quantum field theories depending on the data of a holomorphic Poisson structure on a piece of the underlying spacetime. The main technical tool relies on a characterization of deformations and anomalies of such theories in terms of the Gelfand-Fuchs cohomology of formal Hamiltonian vector fields. In the case that the Poisson structure is non-degenerate such theories are topological in a certain weak sense, which we refer to as de Rham topological. While the Lie algebra of translations acts in a homotopically trivial way, we will show that the space of observables of such a theory does not define an E_n-algebra. Additionally, we will highlight a conjectural relationship to theories of supergravity in four and five dimensions.

Vanishing cycles, plane curve singularities and framed mapping class groups

Let f be an isolated plane curve singularity with Milnor fiber of genus at least 5. For all such f, we give (a) an intrinsic description of the geometric monodromy group that does not invoke the notion of the versal deformation space, and (b) an easy criterion to decide if a given simple closed curve in the Milnor fiber is a vanishing cycle or not. With the lone exception of singularities of type $A_n$ and $D_n$, we find that both are determined completely by a canonical framing of the Milnor fiber induced by the Hamiltonian vector field associated to f. As a corollary we answer a question of Sullivan concerning the injectivity of monodromy groups for all singularities having Milnor fiber of genus at least 7.

Hamiltonian nilpotent saddles of linear plus cubic homogeneous polynomial vector fields

We completely characterize the global phase portraits in the Poincaré disk for all planar Hamiltonian vector fields with linear plus cubic homogeneous terms having a nilpotent saddle at the origin.

A weight-homogenous condition to the real Jacobian conjecture in

AbstractIt is known that a polynomial local diffeomorphism $(f,\, g): {\mathbb {R}}^{2} \to {\mathbb {R}}^{2}$ is a global diffeomorphism provided the higher homogeneous terms of $f f_x+g g_x$ and $f f_y+g g_y$ do not have real linear factors in common. Here, we give a weight-homogeneous framework of this result. Our approach uses qualitative theory of differential equations. In our reasoning, we obtain a result on polynomial Hamiltonian vector fields in the plane, generalization of a known fact.

Non-uniform ergodic properties of Hamiltonian flows with impacts

AbstractThe ergodic properties of two uncoupled oscillators, one horizontal and one vertical, residing in a class of non-rectangular star-shaped polygons with only vertical and horizontal boundaries and impacting elastically from its boundaries are studied. We prove that the iso-energy level sets topology changes non-trivially; the flow on level sets is always conjugated to a translation flow on a translation surface, yet, for some segments of partial energies the genus of the surface is strictly greater than $1$ . When at least one of the oscillators is unharmonic, or when both are harmonic and non-resonant, we prove that for almost all partial energies, including the impacting ones, the flow on level sets is uniquely ergodic. When both oscillators are harmonic and resonant, we prove that there exist intervals of partial energies on which periodic ribbons and additional ergodic components coexist. We prove that for almost all partial energies in such segments the motion is uniquely ergodic on the part of the level set that is not occupied by the periodic ribbons. This implies that ergodic averages project to piecewise smooth weighted averages in the configuration space.

The L∞-algebra of a symplectic manifold

We construct an $L_\infty$-algebra on the truncated canonical homology complex of a symplectic manifold, which naturally projects to the universal central extension of the Lie algebra of Hamiltonian vector fields.

THE DYNAMICAL EVOLUTION OF GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM

Geometric Quantum Mechanics is a formulation that demonstrates how quantum theory may be casted in the language of Hamiltonian phase-space dynamics. In this framework, the states are referring to points in complex projective Hilbert space, the observables are real valued functions on the space and the Hamiltonian flow is defined by Schr{\"o}dinger equation. Recently, the effort to cast uncertainty principle in terms of geometrical language appeared to become the subject of intense study in geometric quantum mechanics. One has shown that the stronger version of uncertainty relation i.e. the Robertson-Schr{\"o}dinger uncertainty relation can be expressed in terms of the symplectic form and Riemannian metric. In this paper, we investigate the dynamical behavior of the uncertainty relation for spin $\frac{1}{2}$ system based on this formulation. We show that the Robertson-Schr{\"o}dinger uncertainty principle is not invariant under Hamiltonian flow. This is due to the fact that during evolution process, unlike symplectic area, the Riemannian metric is not invariant under the flow.

Classification of Simple Lie Superalgebras in Characteristic 2

Abstract All results concern characteristic 2. We describe two procedures; each of which to every simple Lie algebra assigns a simple Lie superalgebra. We prove that every simple finite-dimensional Lie superalgebra is obtained as the result of one of these procedures. For Lie algebras, in addition to the known “classical” restrictedness, we introduce a (2,4)-structure on the two non-alternating series: orthogonal and Hamiltonian vector fields. For Lie superalgebras, the classical restrictedness of Lie algebras has two analogs: a $2|4$-structure, which is a direct analog of the classical restrictedness, and a novel $2|2$-structure—one more analog, a $(2,4)|4$-structure on Lie superalgebras is the analog of (2,4)-structure on Lie algebras known only for non-alternating orthogonal and Hamiltonian series.

THE GENERALIZED GEOMETRIC UNCERTAINTY PRINCIPLE FOR SPIN 1/2 SYSTEM`

Geometric Quantum Mechanics is a version of quantum theory that has been formulated in terms of Hamiltonian phase-space dynamics. The states in this framework belong to points in complex projective Hilbert space, the observables are real valued functions on the space, and the Hamiltonian flow is described by the Schr{\"o}dinger equation. Besides, one has demonstrated that the stronger version of the uncertainty relation, namely the Robertson-Schr{\"o}dinger uncertainty relation, may be stated using symplectic form and Riemannian metric. In this research, the generalized Robertson-Schr{\"o}dinger uncertainty principle for spin $\frac{1}{2}$ system has been constructed by considering the operators corresponding to arbitrary direction.

The Limit of Vanishing Diffusivity for Passive Scalars in Hamiltonian Flows

The Limit of Vanishing Diffusivity for Passive Scalars in Hamiltonian Flows

Sasakian Structure Associated with a Second-Order ODE and Hamiltonian Dynamical Systems

We define a contact metric structure on the manifold corresponding to a second order ordinary differential equation $d^2y/dx^2=f(x,y,y')$ and show that the contact metric structure is Sasakian if and only if the 1-form $\frac{1}{2}(dp-fdx)$ defines a Poisson structure. We consider a Hamiltonian dynamical system defined by this Poisson structure and show that the Hamiltonian vector field, which is a multiple of the Reeb vector field, admits a compatible bi-Hamiltonian structure for which $f$ can be regarded as a Hamiltonian function. As a particular case, we give a compatible bi-Hamiltonian structure of the Reeb vector field such that the structure equations correspond to the Maurer-Cartan equations of an invariant coframe on the Heisenberg group and the independent variable plays the role of a Hamiltonian function. We also show that the first Chern class of the normal bundle of an integral curve of a multiple of the Reeb vector field vanishes iff $f_x+ff_p = \Psi (x)$ for some $\Psi$.

Entanglement wedge reconstruction and correlation measures in mixed states: Modular flows versus quantum recovery channels

In this work we study the nature of correlations among mixed states in the setup of two symmetric strips. We use various tools to determine how the bulk geometry could be reconstructed from the boundary mixed information. These tools would be the modular Hamiltonian and modular flow, OPE blocks, quantum recovery channels such as Petz map, Uhlmann holonomy and Wilson lines. We comment on the similarities and connections between these approaches in our symmetric setup of a mixed system. Specially, we use parameters such as dissipation which is being modeled by the mass of graviton, and also the same sign charge of the two strips to find connections between these different approaches. Then, using Uhlmann fidelity as the correlation measure, we look into the various types of correlations in mixed systems such as discord. Next, we use simple results of modular Hamiltonian for fermions to get insights about the relations between modular flow and entanglement and complexity of purification (EoP/CoP), and also behavior of modular flows in confining geometries. Finally, we study the dynamics of correlations using various information speeds and also model of void formation in CFTs and again we comment on their relationships with the behavior of EoP and CoP.

(Co)isotropic triples and poset representations

We study triples of coisotropic or isotropic subspaces in symplectic vector spaces; in particular, we classify indecomposable structures of this kind. The classification depends on the ground field, which we assume only to be perfect and not of characteristic 2. Our work uses the theory of representations of partially ordered sets with (order reversing) involution; for (co)isotropic triples, the relevant poset is “2 + 2 + 2” consisting of three independent ordered pairs, with the involution exchanging the members of each pair. A key feature of the classification is that any indecomposable (co)isotropic triple is either “split” or “non-split.” The latter is the case when the poset representation underlying an indecomposable (co)isotropic triple is itself indecomposable. Otherwise, in the “split” case, the underlying representation is decomposable and necessarily the direct sum of a dual pair of indecomposable poset representations; the (co)isotropic triple is a “symplectification.” In the course of the paper we develop the framework of “symplectic poset representations,” which can be applied to a range of problems of symplectic linear algebra. The classification of linear Hamiltonian vector fields, up to conjugation, is an example; we briefly explain the connection between these and (co)isotropic triples.

On Symmetric Brackets Induced by Linear Connections

In this note, we discuss symmetric brackets on skew-symmetric algebroids associated with metric or symplectic structures. Given a pseudo-Riemannian metric structure, we describe the symmetric brackets induced by connections with totally skew-symmetric torsion in the language of Lie derivatives and differentials of functions. We formulate a generalization of the fundamental theorem of Riemannian geometry. In particular, we obtain an explicit formula of the Levi-Civita connection. We also present some symmetric brackets on almost Hermitian manifolds and discuss the first canonical Hermitian connection. Given a symplectic structure, we describe symplectic connections using symmetric brackets. We define a symmetric bracket of smooth functions on skew-symmetric algebroids with the metric structure and show that it has properties analogous to the Lie bracket of Hamiltonian vector fields on symplectic manifolds.

Noncommutative Kepler Dynamics: symmetry groups and bi-Hamiltonian structures

Integrals of motion are constructed from noncommutative (NC ) Kepler dynamics, generating $$\mathrm{SO}(3)$$ , $$\mathrm{SO}(4)$$ , and $$\mathrm{SO}(1,3)$$ dynamical symmetry groups. The Hamiltonian vector field is derived in action–angle coordinates, and the existence of a hierarchy of bi-Hamiltonian structures is highlighted. Then, a family of Nijenhuis recursion operators is computed and discussed.

Symplectic reduction of Yang-Mills theory with boundaries: from superselection sectors to edge modes, and back

I develop a theory of symplectic reduction that applies to bounded regions in electromagnetism and Yang--Mills theories. In this theory gauge-covariant superselection sectors for the electric flux through the boundary of the region play a central role: within such sectors, there exists a natural, canonically defined, symplectic structure for the reduced Yang--Mills theory. This symplectic structure does not require the inclusion of any new degrees of freedom. In the non-Abelian case, it also supports a family of Hamiltonian vector fields, which I call ``flux rotations,'' generated by smeared, Poisson-non-commutative, electric fluxes. Since the action of flux rotations affects the total energy of the system, I argue that flux rotations fail to be dynamical symmetries of Yang--Mills theory restricted to a region. I also consider the possibility of defining a symplectic structure on the union of all superselection sectors. This in turn requires including additional boundary degrees of freedom aka ``edge modes.'' However, I argue that a commonly used phase space extension by edge modes is inherently ambiguous and gauge-breaking.

Dynamical reduced basis methods for Hamiltonian systems

We consider model order reduction of parameterized Hamiltonian systems describing nondissipative phenomena, like wave-type and transport dominated problems. The development of reduced basis methods for such models is challenged by two main factors: the rich geometric structure encoding the physical and stability properties of the dynamics and its local low-rank nature. To address these aspects, we propose a nonlinear structure-preserving model reduction where the reduced phase space evolves in time. In the spirit of dynamical low-rank approximation, the reduced dynamics is obtained by a symplectic projection of the Hamiltonian vector field onto the tangent space of the approximation manifold at each reduced state. A priori error estimates are established in terms of the projection error of the full model solution onto the reduced manifold. For the temporal discretization of the reduced dynamics we employ splitting techniques. The reduced basis satisfies an evolution equation on the manifold of symplectic and orthogonal rectangular matrices having one dimension equal to the size of the full model. We recast the problem on the tangent space of the matrix manifold and develop intrinsic temporal integrators based on Lie group techniques together with explicit Runge–Kutta (RK) schemes. The resulting methods are shown to converge with the order of the RK integrator and their computational complexity depends only linearly on the dimension of the full model, provided the evaluation of the reduced flow velocity has a comparable cost.