Narrow Results

By using the Atiyah–Singer theorem through some similarities with the instanton and the antiinstanton moduli spaces, the dimension of the moduli space for two- and four-dimensional BF theories valued in different background manifolds and gauge groups scenarios is determined. Additionally, we develop Dirac's canonical analysis for a four-dimensional modified BF theory, which reproduces the topological YM theory. This framework will allow us to understand the local symmetries, the constraints, the extended Hamiltonian and the extended action of the theory.

A pure Dirac's method for Abelian and non-Abelian massive theories in three dimensions is performed. Our analysis is developed on the extended phase space, reporting the relevant structure of the theories, namely, the extended action, the extended Hamiltonian, the full structure of the constraints and the counting of degrees of freedom. In addition, we compare our results with those found in the literature.

The global time in geometrodynamics is defined in a covariant under diffeomorphisms form. An arbitrary static background metric is taken in the tangent space. The global intrinsic time is identified with the mean value of the logarithm of the square root of the ratio of the metric determinants. The procedures of the Hamiltonian reduction and deparametrization of dynamical systems are implemented. The reduced Hamiltonian equations of motion of gravitational field in semi-geodesic coordinate system are written.

This study aims to tackle the generalized coupled nonlinear Schrödinger ($𝔾ℂ$ – $\mathbb{N}𝕃𝕊$) equations, with a focus on understanding their physical significance and stability, especially in the realm of plasma physics. These equations are crucial for grasping the complex dynamics of wave interactions within plasma systems, which are fundamental for phenomena like wave-particle interactions, turbulence, and magnetic confinement.

We employ analytical methods such as the generalized rational ($𝔾ℝ$at) and Khater II ($𝕂$hat.II) techniques, along with characterizing the system using Hamiltonian principles, to carefully examine the stability of solutions. The relevance of this model extends across various plasma phenomena, including electromagnetic wave propagation, Langmuir wave dynamics, and plasma instabilities.

By applying these analytical techniques, we derive solutions and investigate their stability using Hamiltonian dynamics, providing valuable insights into the fundamental behavior of nonlinear plasma waves. Our findings reveal the existence of stable solutions under specific conditions, thus advancing our understanding of plasma dynamics significantly.

This research carries significant implications for fields such as plasma physics, astrophysics, and fusion research, where a deep understanding of plasma wave stability and dynamics is crucial. Essentially, our study represents a scholarly effort to offer fresh perspectives on the behavior of $𝔾ℂ$ – $\mathbb{N}𝕃𝕊$ equations within plasma systems, contributing to the academic discourse on plasma wave phenomena.

The numerical behavior of the truncated 3-particle Toda lattice (3pTL) is reviewed and studied in more detail (than in previous papers) and at higher energies (at odd-orders n ≤ 9). We further extended our study to higher truncations at odd-orders, n = 2k + 1, k = 1, …, 9. We have located the majority of the families of periodic orbits along with their main bifurcations. By using: (a) the method of Poincaré surface of section, (b) the maximum Lyapunov characteristic number and (c) the ratio of the families of ordered periodic orbits, we studied the topology of the nine odd-order Hamiltonians with respect to their order of truncation.

We complete the study of the numerical behavior of the truncated 3-particle Toda lattice (3pTL) with even truncations at orders n = 2k, k = 2, …, 10. We use (as in Part I): (a) the method of Poincaré surface of section, (b) the maximum Lyapunov characteristic number and (c) the ratio of the families of ordered periodic orbits. We derived some similarities and quite many differences between the odd and even order expansions.

A Rikitake type system with quadratic control having a negative tuning parameter is considered and some of the geometrical and dynamical properties are analyzed.

Applying parametric controls to the 3D real-valued Maxwell–Bloch equations, we obtain a Hamilton–Poisson system, a dissipative system with chaotic behavior, and a transitional system between the aforementioned states, which is a conservative system that has only one constant of motion. In the Hamiltonian case, we present some connections of the energy-Casimir mapping with the equilibrium states and the existence of the homoclinic orbits. We study the stability of the equilibrium points of the transitional system and the dissipative system. Furthermore, we point out the chaotic behavior of the introduced system.

In this paper we are analyzing the stability of equilibria and also the existence of periodic solutions of three dimensional Hamiltonian systems using Darboux–Weinstein coordinates.

In this paper we analyze the quadratic and homogeneous Hamiltonian systems on (𝔰𝔬(3))* from the Poisson dynamics and geometry point of view.

In this paper we will give a formula for computing conservation laws for a Hamiltonian system that admits non-Noether infinitesimal symmetry. The formula involves the differential operator associated with the dual Lefschetz operator corresponding to a symplectic form.

We perform Dirac's canonical analysis for a four-dimensional BF and for a generalized four-dimensional BF theory depending on a connection valued in the Lie algebra of SO(3, 1). This analysis is developed by considering the corresponding complete set of variables that define these theories as dynamical, and we find out the relevant symmetries, the constraints, the extended Hamiltonian, the extended action, gauge transformations and the counting of physical degrees of freedom. The results obtained are compared with other approaches found in the literature.

Some dynamical and geometrical properties of controls dynamic for a drift-free left invariant control system from the Poisson geometry point of view are described. The integrability of such system are also studied.

A pure Dirac's canonical analysis, defined in the full phase space for the Husain–Kuchar (HK) model is discussed in detail. This approach allows us to determine the extended action, the extended Hamiltonian, the complete constraint algebra and the gauge transformations for all variables that occur in the action principle. The complete set of constraints defined on the full phase space allow us to calculate the Dirac algebra structure of the theory and a local weighted measure for the path integral quantization method. Finally, we discuss briefly the necessary mathematical structure to perform the canonical quantization program within the framework of the loop quantum gravity approach.

A detailed Hamiltonian analysis for a 5D Kalb–Ramond, massive Kalb–Ramond and Stüeckelberg Kalb–Ramond theories with an extra compact dimension is performed. We develop a complete constraint program, then we quantize the theory by constructing the Dirac brackets. From the gauge transformations of the theories, we fix a particular gauge and we find pseudo-Goldstone bosons in Kalb–Ramond and Stüeckelberg Kalb–Ramond systems. Finally we discuss some remarks and prospects.

A covariant Hamiltonian formalism for the dynamics of compact spinning bodies in curved space-time in the test-particle limit is described. The construction allows a large class of Hamiltonians accounting for specific properties and interactions of spinning bodies. The dynamics for a minimal and a specific non-minimal Hamiltonian is discussed. An independent derivation of the equations of motion from an appropriate energy–momentum tensor is provided. It is shown how to derive constants of motion, both background-independent and background-dependent ones.

In this paper, we prove the fiberwise convexity of the regularized Hill’s lunar problem below the critical energy level. This allows us to see Hill’s lunar problem of any energy level below the critical value as the Legendre transformation of a geodesic problem on $S^2$ with a family of Finsler metrics. Therefore the compactified energy hypersurfaces below the critical energy level have the unique tight contact structure on $ℝP^3$. Also one can apply the systolic inequality of Finsler geometry to the regularized Hill’s lunar problem.

We consider two disjoint and homotopic non-contractible embedded loops on a Riemann surface and prove the existence of a non-contractible orbit for a Hamiltonian function on the surface whenever it is sufficiently large on one of the loops and sufficiently small on the other. This gives the first example of an estimate from above for a generalized form of the Biran–Polterovich–Salamon capacity for a closed symplectic manifold.

An area-preserving diffeomorphism of an annulus has an “action function” which measures how the diffeomorphism distorts curves. The average value of the action function over the annulus is known as the Calabi invariant of the diffeomorphism, while the average value of the action function over a periodic orbit of the diffeomorphism is the mean action of the orbit. If an area-preserving annulus diffeomorphism is a rotation near the boundary, and if its Calabi invariant is less than the maximum boundary value of the action function, then we show that the infimum of the mean action over all periodic orbits of the diffeomorphism is less than or equal to its Calabi invariant.

There are a number of known constructions of quasimorphisms on Hamiltonian groups. We show that on surfaces many of these quasimorphisms are not compatible with the Hofer norm in a sense they are not continuous and not Lipschitz. The only exception known to the author is the Calabi quasimorphism on a sphere [M. Entov and L. Polterovich, Calabi quasimorphism and quantum homology, Int. Math. Res. Not. 2003 (2003) 1635–1676] and induced quasimorphisms on genus-zero surfaces (e.g. [P. Biran, M. Entov and L. Polterovich, Calabi quasimorphisms for the symplectic ball, Commun. Contemp. Math. 6 (2004) 793–802]).