Complete set of quasi-conserved quantities for spinning particles around Kerr

We revisit the conserved quantities of the Mathisson-Papapetrou-Tulczyjew equations describing the motion of spinning particles on a fixed background. Assuming Ricci-flatness and the existence of a Killing-Yano tensor, we demonstrate that besides the two non-trivial quasi-conserved quantities, i.e. conserved at linear order in the spin, found by Rüdiger, non-trivial quasi-conserved quantities are in one-to-one correspondence with non-trivial mixed-symmetry Killing tensors. We prove that no such stationary and axisymmetric mixed-symmetry Killing tensor exists on the Kerr geometry. We discuss the implications for the motion of spinning particles on Kerr spacetime where the quasi-constants of motion are shown not to be in complete involution.

One dimensional reduced model for ITER relevant energetic particle transport

We set up a mapping procedure able to translate the evolution of the radial profile of fast ions, interacting with Toroidal Alfvén Eigenmodes, into the dynamics of an equivalent one dimensional bump-on-tail system. We apply this mapping technique to reproduce ITER relevant simulations, which clearly outlined deviations from the diffusive quasi-linear model. Our analysis demonstrates the capability of the one-dimensional beam-plasma dynamics to predict the relevant features of the non-linear hybrid LIGKA/HAGIS simulations. In particular, we clearly identify how the deviation from the quasi-linear evolutive profiles is due to the presence of avalanche processes. A detailed analysis regarding the reduced dimensionality is also addressed, by means of phase-space slicing based on constants of motion. In the conclusions, we outline the main criticalities and outcomes of the procedure, which must be satisfactorily addressed to make quantitative prediction on the observed outgoing fluxes in a Tokamak device.

Noether symmetries, dynamical constants of motion, and spectrum generating algebras

When discussing consequences of symmetries of dynamical systems based on Noether’s first theorem, most standard textbooks on classical or quantum mechanics present a conclusion stating that a global continuous Lie symmetry implies the existence of a time-independent conserved Noether charge which is the generator of the action on phase space of that symmetry, and which necessarily must as well commute with the Hamiltonian. However this need not be so, nor does that statement do justice to the complete scope and reach of Noether’s first theorem. Rather a much less restrictive statement applies, namely, that the corresponding Noether charge as an observable over phase space may in fact possess an explicit time dependency, and yet define a constant of the motion by having a commutator with the Hamiltonian which is nonvanishing, thus indeed defining a dynamical conserved quantity. Furthermore, and this certainly within the Hamiltonian formulation, the converse statement is valid as well, namely, that any dynamical constant of motion is necessarily the Noether charge of some symmetry leaving the system’s action invariant up to some total time derivative contribution. This contribution revisits these different points and their consequences, straightaway within the Hamiltonian formulation which is the most appropriate for such issues. Explicit illustrations are also provided through three general but simple enough classes of systems.

Exploring Alternatives to the Hamiltonian Calculation of the Ashtekar-Olmedo-Singh Black Hole Solution

In this article, we reexamine the derivation of the dynamical equations of the Ashtekar-Olmedo-Singh black hole model in order to determine whether it is possible to construct a Hamiltonian formalism where the parameters that regulate the introduction of quantum geometry effects are treated as true constants of motion. After arguing that these parameters should capture contributions from two distinct sectors of the phase space that had been considered independent in previous analyses in the literature, we proceed to obtain the corresponding equations of motion and analyze the consequences of this more general choice. We restrict our discussion exclusively to these dynamical issues. We also investigate whether the proposed procedure can be reconciled with the results of Ashtekar, Olmedo, and Singh, at least in some appropriate limit.

Superintegrability of (2n + 1)-body choreographies, n = 1,2,3,…,∞ on the algebraic lemniscate by Bernoulli (inverse problem of classical mechanics)

For one 3-body and two 5-body planar choreographies on the same algebraic lemniscate by Bernoulli we found explicitly a maximal possible set of (particular) Liouville integrals, 7 and 15, respectively, (including the total angular momentum), which Poisson commute with the corresponding Hamiltonian along the trajectory. Thus, these choreographies are particularly maximally superintegrable. It is conjectured that the total number of (particular) Liouville integrals is maximal possible for any odd number of bodies [Formula: see text] moving choreographically (without collisions) along given algebraic lemniscate, thus, the corresponding trajectory is particularly, maximally superintegrable. Some of these Liouville integrals are presented explicitly. The limit [Formula: see text] is studied: it is predicted that one-dimensional liquid with nearest-neighbor interactions occurs, it moves along algebraic lemniscate and it is characterized by infinitely many constants of motion.

On the Integrable Deformations of the Maximally Superintegrable Systems

In this paper, we present the integrable deformations method for a maximally superintegrable system. We alter the constants of motion, and using these new functions, we construct a new system which is an integrable deformation of the initial system. In this manner, new maximally superintegrable systems are obtained. We also consider the particular case of Hamiltonian mechanical systems. In addition, we use this method to construct some deformations of an arbitrary system of first-order autonomous differential equations.

The Stäckel theorem in the Lagrangian formalism and the use of local times

We show that the conditions for the separability of the Hamilton-Jacobi equation given by the Stäckel theorem imply that, making use of the elementary Lagrangian formalism, one can find $n$ functionally independent constants of motion, where $n$ is the number of degrees of freedom. We also show that this result can be linked to the fact that the Lagrangian for a system of this class is related to the sum of $n$ one-dimensional Lagrangians, if one makes use of multiple local times.

Noncompact CPN as a phase space of superintegrable systems

We propose the description of superintegrable models with dynamical [Formula: see text] symmetry, and of the generic superintegrable deformations of oscillator and Coulomb systems in terms of higher-dimensional Klein model (the noncompact analog of complex projective space) playing the role of phase space. We present the expressions of the constants of motion of these systems via Killing potentials defining the [Formula: see text] isometries of the Kähler structure.

Signature of Generalized Gibbs Ensemble Deviation from Equilibrium: Negative Absorption Induced by a Local Quench

When a parameter quench is performed in an isolated quantum system with a complete set of constants of motion, its out of equilibrium dynamics is considered to be well captured by the Generalized Gibbs Ensemble (GGE), characterized by a set {λα} of coefficients related to the constants of motion. We determine the most elementary GGE deviation from the equilibrium distribution that leads to detectable effects. By quenching a suitable local attractive potential in a one-dimensional electron system, the resulting GGE differs from equilibrium by only one single λα, corresponding to the emergence of an only partially occupied bound state lying below a fully occupied continuum of states. The effect is shown to induce optical gain, i.e., a negative peak in the absorption spectrum, indicating the stimulated emission of radiation, enabling one to identify GGE signatures in fermionic systems through optical measurements. We discuss the implementation in realistic setups.