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We present a kinetic version of the Watanabe-Strogatz (WS) transform for vector models in this paper. From the generalized WS-transform, we can reduce the kinetic vector model into an ODE system. We also obtain the cross-ratio type constant of motion functionals for kinetic vector models under suitable conditions. We present the sufficient and necessary conditions for the existence of the suggested constant of motion functionals. As an application of the constant of motion functional, we provide the instability of bipolar states of the kinetic swarm sphere model. We also provide the WS-transform and constant of motion functionals for non-identical kinetic vector models. [ABSTRACT FROM AUTHOR]

The concept of extended Hamiltonian systems allows a geometric interpretation of several integrable and superintegrable systems with polynomial first integrals of a degree depending on a rational parameter. Until now, the extension procedure has been applied only in the case of natural Hamiltonians. We give several examples of application to nonnatural Hamiltonians, such as the Hamiltonian of a system of two point-vortices, the Hamiltonian of the Lotka–Volterra model, and some Hamiltonians quartic in the momenta. We effectively obtain extended Hamiltonians in some cases, fail in other cases, and briefly discuss the reasons for these results. [ABSTRACT FROM AUTHOR]

We present a classical and a quantum constant of motion used to study the behavior of an electron moving in the vicinity of a tridimensional charged nanostructure with electric dipole. It is the generalization of the bidimensional version studied in a previous work. The classical case is found by using two approaches: the Hamilton Jacobi equation and a Newtonian treatment. The quantum case is obtained by separating the Schrödinger equation in spherical coordinates. We use three quantum numbers to classify the states. The angular part is solved by using an expansion of spherical harmonics to get an eigenvalue equation for the new constant of motion. The solution shows that the probability density of the electron shifts toward the values where 𝜃 < π 2 . The eigenvalue λ is near the value l(l + 1) of the angular momentum, but their difference grows if the electric dipole increases, so that the effect is small for a cluster like (GaAs)3 but very important for fullerenes like RbC60 and LiC60. The radial part is solved using the shooting method and we find the ground state, and first and second excited states. [ABSTRACT FROM AUTHOR]

Based on fundamental principles of quantum mechanics, we present a method for a rigorous analytic construction of the spectra of the one- and two-photon Jaynes-Cummings models in a Kerr medium. To obtain an idea of the method, we consider a first generalized Jaynes-Cummings model with a real, linear superpotential using techniques of supersymmetric quantum mechanics. The Hamiltonian of this model is written as a combination of operators generating the underlying superalgebra whose elements are defined as differential matrix operators. Based on the formalism of supersymmetric quantum mechanics and the properties of sets of common observables, we derive solutions of the one- and two-photon models expressed in terms of confluent hypergeometric functions. Finally, using numerical analysis, we study the influence of the Kerr effect on the energy spectra of the physical system. [ABSTRACT FROM AUTHOR]

We develop an analytic formalism using basic quantum mechanics techniques to successfully solve the multiphoton Jaynes–Cummings and the generalized Dicke Hamiltonians. For this, we split the Hamiltonians of these models into two operators that have the properties of constants of motion for these systems. We then use some well-known operator properties to obtain complete analytic spectra for the considered models. [ABSTRACT FROM AUTHOR]

We present a three-dimensional differential equation, which robustly displays a degenerate Andronov–Hopf bifurcation of infinite codimension, leading to a center, i.e., an invariant two-dimensional surface that is filled with periodic orbits surrounding an equilibrium. The system arises from a three-species bimolecular chemical reaction network consisting of four reactions. In fact, it is, up to a natural equivalence, the only such mass-action system that admits a center via an Andronov–Hopf bifurcation. [ABSTRACT FROM AUTHOR]

The integrable deformation method for a three-dimensional Hamilton–Poisson system consists in alteration of its constants of motion in order to obtain a new Hamilton–Poisson system. We assume that a three-dimensional system of differential equations has a Hamilton–Poisson part and a nonconservative part. We give integrable deformations of the Hamilton–Poisson part and, adding the nonconservative part, we obtain integrable deformations of the considered three-dimensional system of differential equations. In particular, applying this method to chaotic systems may lead to new systems with chaotic behavior. We use this method to obtain integrable deformations of Lorenz, Chen, and Rössler systems. Using particular deformation functions, we have pointed out some deformations of the above-mentioned attractors. [ABSTRACT FROM AUTHOR]

In this paper, we consider variational problems with delayed argument. Simple proofs of the first and second Noether theorems are given. Applying our results to some concrete problems we obtain conservation laws for systems with delay. [ABSTRACT FROM AUTHOR]

Abstract: We discuss a minimalist model of assortative mating for sexually reproducing haploid individuals with two biallelic loci. Assortativeness is introduced in the model by preventing mating between individuals whose alleles differ at both loci. Using methods of dynamical systems and population genetics we provide a full description of the evolution of the system for the case of very large populations. We derive the equations governing the evolution of haplotype frequencies and study the equilibrium solutions, stability, and speed of convergence to equilibrium. We find a constant of motion which allows us to introduce a geometrical construction that makes it straightforward to predict the fate of initial conditions. Finally, we discuss the consequences of this class of assortative mating models, including their possible extensions and implications for sympatric and topopatric speciation. [Copyright &y& Elsevier]

Integrability of systems in N = 2 dimensions possessing a non-trivial scaling symmetry is reformulated in the context of dynamical symmetries and solvable Lie algebras. A method to construct integrable (Lagrangians) from a dynamical vector field subjected to a scaling symmetry is proposed. [ABSTRACT FROM AUTHOR]

Abstract: A unifying method based on factorization properties is introduced for finding symmetries of quantum and classical superintegrable systems using the example of the Tremblay–Turbiner–Winternitz (TTW) model. It is shown that the symmetries of the quantum system can be implemented in a natural way to its classical version. Besides, by this procedure we get also other type of constants of motion depending explicitly on time that allow to find directly the motion of the system whose corresponding trajectories coincide with those obtained previously by using its symmetries. [Copyright &y& Elsevier]

The geometric formulation of Hamilton–Jacobi theory for systems with nonholonomic constraints is developed, following the ideas of the authors in previous papers. The relation between the solutions of the Hamilton–Jacobi problem with the symplectic structure defined from the Lagrangian function and the constraints is studied. The concept of complete solutions and their relationship with constants of motion, are also studied in detail. Local expressions using quasivelocities are provided. As an example, the nonholonomic free particle is considered. [ABSTRACT FROM AUTHOR]

The Lagrangian, the Hamiltonian and the constant of motion of the gravitational attraction of two bodies when one of them has variable mass is considered. The relative and center of mass coordinates are not separated, and choosing the reference system in the body with much higher mass, it is possible to reduce the system of equations to 1-D problem. Then, a constant of motion, the Lagrangian, and the Hamiltonian are obtained. The trajectories found in the space position-velocity,( x, v), are qualitatively different from those on the space position-momentum,( x, p). [ABSTRACT FROM AUTHOR]

For a relativistic particle under a constant force and a linear velocity dissipation force, a constant of motion is found. Problems are shown for getting the Hamiltonian of this system. Thus, the quantization of this system is carried out through the constant of motion and using the quantization on the velocity variable. The dissipative relativistic quantum bouncer is outlined within this quantization approach. [ABSTRACT FROM AUTHOR]

Given a constant of motion for the one-dimensional harmonic oscillator with linear dissipation in the velocity, the problem to get the Hamiltonian for this system is pointed out, and the quantization up to second order in the perturbation approach is used to determine the modification on the eigenvalues when dissipation is taken into consideration. This quantization is realized using the constant of motion instead of the Hamiltonian. [ABSTRACT FROM AUTHOR]

The constants of motion of the following systems are deduced: a relativistic particle with linear dissipation; a no-relativistic particle with a time explicitly depending force; a no-relativistic particle with a constant force and time depending mass; and a relativistic particle under a conservative force with position depending mass. The Hamiltonian for these systems, which is determined by getting the velocity as a function of position and generalized linear momentum, can be found explicitly at first approximation for the first system. The Hamiltonians for the other systems are kept implicitly in their expressions for their constants of motion. [ABSTRACT FROM AUTHOR]

An equation is obtained to find the Lagrangian for a one-dimensional autonomous system. The continuity of the first derivative of its constant of motion is assumed. This equation is solved for a generic nonconservative autonomous system that has certain quasi-relativistic properties. A new method based on a Taylor series expansion is used to obtain the associated Hamiltonian for this system. These results have the usual expression for a conservative system when the dissipation parameter goes to zero. An example of this approach is given. [ABSTRACT FROM AUTHOR]

A structuring and eventual exclusion of water surrounding hydrogen bonds takes place during protein folding as desolvation shells are formed. Taken as an average over all hydrogen bonds, the extent of protection determined by the desolvation shell constitutes very nearly a constant of motion, as revealed by re-examination of the longest all-atom trajectory with explicit solvent [Y. Duan, P.A. Kollman, Science 282 (1998) 740]. [Copyright &y& Elsevier]

We present constants of motion for the finite-dimensional Lohe type aggregation models with frustration, and apply them to the analysis of collective behaviors. The Lohe type models have been proposed as possible non-abelian and higher-dimensional generalizations of the Kuramoto model, which is a prototype phase model for synchronization. In this paper, we study the emergent collective dynamics of these models under the effect of (interaction) frustration, which generalizes phase-shift frustrations in the Kuramoto model. To this end, we provide constants of motion, i.e., conserved quantities along the flow generated by the models under consideration. From the perspective of the low-dimensional dynamics, we derive several results on the emergent asymptotic patterns of the Kuramoto and Lohe sphere models. • Constant of motion for the Lohe sphere model and its applications. • Emergent dynamics for the Lohe models with frustration. • Reduction to low dimensional dynamics from the Kuramoto model with frustration. [ABSTRACT FROM AUTHOR]