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

2021 ◽  
pp. 2150116
Author(s):  
Alexander V. Turbiner ◽  
Juan Carlos Lopez Vieyra
Keyword(s):  
Inverse Problem ◽  
Angular Momentum ◽  
Total Angular Momentum ◽  
Nearest Neighbor ◽  
Classical Mechanics ◽  
One Dimensional ◽  
Limit Formula ◽  
Constants Of Motion ◽  
Poisson Commute

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.

scholarly journals Particular superintegrability of 3-body (modified) Newtonian gravity

2020 ◽  
Vol 35 (22) ◽  
pp. 2050185
Author(s):  
Alexander V. Turbiner ◽  
Juan Carlos Lopez Vieyra
Keyword(s):  
Angular Momentum ◽  
Potential Theory ◽  
Total Angular Momentum ◽  
Newtonian Gravity ◽  
Constants Of Motion ◽  
Body Potential ◽  
Poisson Commute

It is found explicitly 5 Liouville integrals in addition to total angular momentum which Poisson commute with Hamiltonian of 3-body Newtonian Gravity in [Formula: see text] along the remarkable figure-8-shape trajectory discovered by Moore-Chenciner-Montgomery. It is verified that they become constants of motion along this trajectory. Hence, 3-body choreographic motion on figure-8-shape trajectory in [Formula: see text] Newtonian gravity (Moore, 1993), as well as in [Formula: see text] modified Newtonian gravity by Fujiwara et al., is maximally superintegrable. It is conjectured that any 3-body potential theory that admits Figure-8-shape choreographic motion is superintegrable along the trajectory.

scholarly journals Spectral Statistics of the Triaxial Rigid Rotator: Semiclassical Origin of Their Pathological Behavior

2003 ◽  
Vol 17 (15) ◽  
pp. 803-812
Author(s):  
V. R. Manfredi ◽  
V. Penna ◽  
L. Salasnich
Keyword(s):  
Angular Momentum ◽  
Spectral Properties ◽  
Total Angular Momentum ◽  
Energy Levels ◽  
Rigid Rotator ◽  
One Dimensional ◽  
Spectral Statistics ◽  
Rotational Motions ◽  
Pathological Behavior ◽  
The One

In this paper we investigate the local and global spectral properties of the triaxial rigid rotator. We demonstrate that, for a fixed value of the total angular momentum, the energy spectrum can be divided into two sets of energy levels, whose classical analogs are librational and rotational motions. By using diagonalization, semiclassical and algebric methods, we show that the energy levels follow the anomalous spectral statistics of the one-dimensional harmonic oscillator.

scholarly journals Effective dynamics of a conditioned generalized linear Glauber model

2018 ◽  
Vol 32 (22) ◽  
pp. 1850243
Author(s):  
Sara Kaviani ◽  
Farhad H. Jafarpour
Keyword(s):  
Nearest Neighbor ◽  
Reaction Diffusion ◽  
Glauber Model ◽  
Transition Rates ◽  
Original Process ◽  
One Dimensional ◽  
Effective Dynamics ◽  
Physical Observable ◽  
Constants Of Motion ◽  
Classical Particles

In order to study the stochastic Markov processes conditioned on a specific value of a time-integrated observable, the concept of ensembles of trajectories has been recently used extensively. In this paper, we consider a generic reaction–diffusion process consisting of classical particles with nearest-neighbor interactions on a one-dimensional lattice with periodic boundary conditions. By introducing a time-integrated current as a physical observable, we have found certain constraints on the microscopic transition rates of the process under which the effective process contains local interactions; however, with rescaled transition rates comparing to the original process. A generalization of the linear Glauber model is then introduced and studied in detail as an example. Associated effective dynamics of this model is investigated and constants of motion are obtained.

scholarly journals From one-dimensional fields to Vlasov equilibria: theory and application of Hermite polynomials

2016 ◽  
Vol 82 (3) ◽  
Author(s):  
O. Allanson ◽  
T. Neukirch ◽  
S. Troscheit ◽  
F. Wilson
Keyword(s):  
Inverse Problem ◽  
Distribution Function ◽  
Solution Method ◽  
Hermite Polynomials ◽  
Distribution Functions ◽  
Pressure Tensor ◽  
One Dimensional ◽  
Function Solution ◽  
Candidate Solution ◽  
Constants Of Motion

We consider the theory and application of a solution method for the inverse problem in collisionless equilibria, namely that of calculating a Vlasov–Maxwell equilibrium for a given macroscopic (fluid) equilibrium. Using Jeans’ theorem, the equilibrium distribution functions are expressed as functions of the constants of motion, in the form of a Maxwellian multiplied by an unknown function of the canonical momenta. In this case it is possible to reduce the inverse problem to inverting Weierstrass transforms, which we achieve by using expansions over Hermite polynomials. A sufficient condition on the pressure tensor is found which guarantees the convergence and the boundedness of the candidate solution, when satisfied. This condition is obtained by elementary means, and it is clear how to put it into practice. We also argue that for a given pressure tensor for which our method applies, there always exists a positive distribution function solution for a sufficiently magnetised plasma. Illustrative examples of the use of this method with both force-free and non-force-free macroscopic equilibria are presented, including the full verification of a recently derived distribution function for the force-free Harris sheet (Allansonet al.,Phys. Plasmas, vol. 22 (10), 2015, 102116). In the effort to model equilibria with lower values of the plasma${\it\beta}$, solutions for the same macroscopic equilibrium in a new gauge are calculated, with numerical results presented for${\it\beta}_{pl}=0.05$.

Study of the one-dimensional Holstein model with next-nearest-neighbor hopping

2010 ◽  
Vol 23 (2) ◽  
pp. 025601 ◽  
Author(s):  
Monodeep Chakraborty ◽  
A N Das ◽  
Atisdipankar Chakrabarti
Keyword(s):  
Nearest Neighbor ◽  
One Dimensional ◽  
Holstein Model ◽  
The One

scholarly journals Vortex states in an acoustic Weyl crystal with a topological lattice defect

2021 ◽  
Vol 12 (1) ◽  
Author(s):  
Qiang Wang ◽  
Yong Ge ◽  
Hong-xiang Sun ◽  
Haoran Xue ◽  
Ding Jia ◽  
...  
Keyword(s):  
Angular Momentum ◽  
Orbital Angular Momentum ◽  
Lattice Defect ◽  
Three Dimensional ◽  
Surface States ◽  
Real Space ◽  
Lattice Defects ◽  
One Dimensional ◽  
Topological Features ◽  
Topological Lattice

AbstractCrystalline materials can host topological lattice defects that are robust against local deformations, and such defects can interact in interesting ways with the topological features of the underlying band structure. We design and implement a three dimensional acoustic Weyl metamaterial hosting robust modes bound to a one-dimensional topological lattice defect. The modes are related to topological features of the bulk bands, and carry nonzero orbital angular momentum locked to the direction of propagation. They span a range of axial wavenumbers defined by the projections of two bulk Weyl points to a one-dimensional subspace, in a manner analogous to the formation of Fermi arc surface states. We use acoustic experiments to probe their dispersion relation, orbital angular momentum locked waveguiding, and ability to emit acoustic vortices into free space. These results point to new possibilities for creating and exploiting topological modes in three-dimensional structures through the interplay between band topology in momentum space and topological lattice defects in real space.

scholarly journals Total Angular Momentum Dichroism of the Terahertz Vortex Beams at the Antiferromagnetic Resonances

2021 ◽  
Vol 126 (15) ◽  
Author(s):  
A. A. Sirenko ◽  
P. Marsik ◽  
L. Bugnon ◽  
M. Soulier ◽  
C. Bernhard ◽  
...  
Keyword(s):  
Angular Momentum ◽  
Total Angular Momentum ◽  
Vortex Beams
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