arXiv subjects

Himalaya Senapati

Publications and source records attributed to Himalaya Senapati.

7 recordsLinked to original sources

arXiv2024

Quantum three-rotor problem in the identity representation

The quantum three-rotor problem concerns the dynamics of 3 equally massive particles moving on a circle subject to pairwise attractive cosine potentials and can model coupled Josephson junctions. Classically, it displays order-chaos-order behavior with increasing energy. The quantum system admits a dimensionless coupling with semiclassical behavior at strong coupling. We study stationary states with periodic `relative' wave functions. Perturbative and harmonic approximations capture the spectrum at weak coupling and that of low-lying states at strong coupling. More generally, the cumulative distribution of energy levels obtained by numerical diagonalization is well-described by a Weyl-like semiclassical estimate. However, the system has an $S_3 \times Z_2$ symmetry that is obscured when working with relative angles. By exploiting a basis for invariant states, we obtain the spectrum restricted to the identity representation. To uncover universal quantum hallmarks of chaos, we partition the spectrum into energy windows where the classical motion is regular, mixed or chaotic and unfold each separately. At strong coupling, we find striking signatures of transitions between regularity and chaos: spacing distributions morph from Poisson to Wigner-Dyson while the number variance shifts from linear to logarithmic behavior at small lengths. Some nonuniversal features are also examined. For instance, for strong coupling, the number variance saturates and oscillates at large lengths while the spectral form factor displays a nonuniversal peak at short times. Moreover, deviations from Poisson spacings at asymptotically low and high energies are well-explained by quantum harmonic and free-rotor spectra projected to the identity representation at strong and weak coupling. Interestingly, the degeneracy of free-rotor levels admits an elegant formula that we deduce using properties of Eisenstein primes.

nlin.CD↗

arXiv2020

Instabilities and chaos in the classical three-body and three-rotor problems

This thesis studies instabilities and singularities in a geometrical approach to the planar 3-body problem as well as instabilities, chaos and ergodicity in the 3-rotor problem. Trajectories of the planar 3-body problem are expressed as geodesics of the Jacobi-Maupertuis (JM) metric on the configuration space $C^3$. Translation, rotation and scaling isometries lead to reduced dynamics on quotients of $C^3$ that encode information on the full dynamics. Riemannian submersions are used to find the quotient metrics and to show that the geodesic formulation regularizes collisions for the $1/r^2$ but not for the $1/r$ potential. Extending work of Montgomery, we show the negativity of the scalar curvature on the center of mass configuration space and certain quotients for equal masses and zero energy. Sectional curvatures are also found to be largely negative indicating widespread geodesic instabilities. In the 3-rotor problem, 3 equal masses move on a circle subject to attractive cosine inter-particle potentials. This problem arises as the classical limit of a model of coupled Josephson junctions. The energy E serves as a control parameter. We find analogues of the Euler-Lagrange family of periodic solutions: pendula and breathers at all E and choreographies up to moderate E. The model displays order-chaos-order behavior and undergoes a fairly sharp transition to chaos at a critical energy E$_c$ with several manifestations: (a) a dramatic rise in the fraction of Poincaré surfaces occupied by chaotic sections, (b) spontaneous breaking of discrete symmetries, (c) a geometric cascade of stability transitions in pendula and (d) a change in the sign of the JM curvature. Poincaré sections indicate global chaos in a band of energies slightly above E$_c$ where we provide evidence for ergodicity and mixing with respect to the Liouville measure and study the statistics of recurrence times.

nlin.CD↗

arXiv2020

Ergodicity, mixing and recurrence in the three rotor problem

In the classical three rotor problem, three equal point masses move on a circle subject to attractive cosine potentials of strength g. In the center of mass frame, energy E is the only known conserved quantity. In earlier work [Krishnaswami and Senapati, arXiv:1810.01317, Oct. 2018, arXiv:1811.05807, Nov. 2018], an order-chaos-order transition was discovered in this system along with a band of global chaos for 5.33g < E < 5.6g. Here, we provide numerical evidence for ergodicity and mixing in this band. The distributions of relative angles and angular momenta along generic trajectories are shown to approach the corresponding distributions over constant energy hypersurfaces (weighted by the Liouville measure) as a power-law in time. Moreover, trajectories emanating from a small volume are shown to become uniformly distributed over constant energy hypersurfaces, indicating that the dynamics is mixing. Outside this band, ergodicity and mixing fail, though the distributions of angular momenta over constant energy hypersurfaces show interesting phase transitions from Wignerian to bimodal with increasing energy. Finally, in the band of global chaos, the distribution of recurrence times to finite size cells is found to follow an exponential law with the mean recurrence time satisfying a scaling law involving an exponent consistent with global chaos and ergodicity.

nlin.CD↗

arXiv2019

Classical three rotor problem: periodic solutions, stability and chaos

This paper concerns the classical dynamics of three coupled rotors: equal masses moving on a circle subject to attractive cosine inter-particle potentials. It is a simpler variant of the gravitational three-body problem and also arises as the classical limit of a model of coupled Josephson junctions. Unlike in the gravitational problem, there are no singularities (neither collisional nor non-collisional), leading to global existence and uniqueness of solutions. In appropriate units, the non-negative energy $E$ of the relative motion is the only free parameter. We find analogues of the Euler-Lagrange family of periodic solutions: pendulum and isosceles solutions at all energies and choreographies up to moderate energies. The model displays order-chaos-order behavior: it is integrable at zero and infinitely high energies but displays a fairly sharp transition from regular to chaotic behavior as $E$ is increased beyond $E_c \approx 4$ and a more gradual return to regularity. The transition to chaos is manifested in a dramatic rise of the fraction of the area of the Hill region of Poincaré surfaces occupied by chaotic sections and also in the spontaneous breaking of discrete symmetries of Poincaré sections present at lower energies. Interestingly, the above pendulum solutions alternate between being stable and unstable, with the transition energies cascading geometrically from either side at $E = 4$. The transition to chaos is also reflected in the curvature of the Jacobi-Maupertuis metric that ceases to be everywhere positive when $E$ exceeds four. Examination of Poincaré sections also indicates global chaos in a band of energies $(5.33 \lesssim E \lesssim 5.6)$ slightly above this transition.

nlin.CD↗

arXiv2019

Stability and chaos in the classical three rotor problem

We study the equal-mass classical three rotor problem, a variant of the three body problem of celestial mechanics. The quantum $N$-rotor problem has been used to model chains of coupled Josephson junctions and also arises via a partial continuum limit of the Wick-rotated XY model. In units of the coupling, the energy serves as a control parameter. We find periodic 'pendulum' and 'breather' orbits at all energies and choreographies at relatively low energies. They furnish analogs of the Euler-Lagrange and figure-8 solutions of the planar three body problem. Integrability at very low energies gives way to a rather marked transition to chaos at $E_c \approx 4$, followed by a gradual return to regularity as $E \to \infty$. We find four signatures of this transition: (a) the fraction of the area of Poincaré surfaces occupied by chaotic sections rises sharply at $E_c$, (b) discrete symmetries are spontaneously broken at $E_c$, (c) $E=4$ is an accumulation point of stable to unstable transitions in pendulum solutions and (d) the Jacobi-Maupertuis curvature goes from being positive to having both signs above $E=4$. Moreover, Poincaré plots also reveal a regime of global chaos slightly above $E_c$.

nlin.CD↗

arXiv2019

An introduction to the classical three-body problem: From periodic solutions to instabilities and chaos

The classical three-body problem arose in an attempt to understand the effect of the Sun on the Moon's Keplerian orbit around the Earth. It has attracted the attention of some of the best physicists and mathematicians and led to the discovery of chaos. We survey the three-body problem in its historical context and use it to introduce several ideas and techniques that have been developed to understand classical mechanical systems.

nlin.CD↗

arXiv2016

Curvature and geodesic instabilities in a geometrical approach to the planar three-body problem

The Maupertuis principle allows us to regard classical trajectories as reparametrized geodesics of the Jacobi-Maupertuis (JM) metric on configuration space. We study this geodesic reformulation of the planar three-body problem with both Newtonian and attractive inverse-square potentials. The associated JM metrics possess translation and rotation isometries in addition to scaling isometries for the inverse-square potential with zero energy E. The geodesic flow on the full configuration space $C^3$ (with collision points excluded) leads to corresponding flows on its Riemannian quotients: the center of mass configuration space $C^2$ and shape space $R^3$ (as well as $S^3$ and the shape sphere $S^2$ for the inverse-square potential when E = 0). The corresponding Riemannian submersions are described explicitly in `Hopf' coordinates which are particularly adapted to the isometries. For equal masses subject to inverse-square potentials, Montgomery shows that the zero-energy `pair of pants' JM metric on the shape sphere is geodesically complete and has negative gaussian curvature except at Lagrange points. We extend this to a proof of boundedness and strict negativity of scalar curvatures everywhere on $C^2, R^3$ & $S^3$ with collision points removed. Sectional curvatures are also found to be largely negative, indicating widespread geodesic instabilities. We obtain asymptotic metrics near collisions, show that scalar curvatures have finite limits and observe that the geodesic reformulation `regularizes' pairwise and triple collisions on $C^2$ and its quotients for arbitrary masses and allowed energies. For the Newtonian potential with equal masses and E=0, we find that the scalar curvature on $C^2$ is strictly negative though it could have either sign on $R^3$. However, unlike for the inverse-square potential, geodesics can encounter curvature singularities at collisions in finite geodesic time.

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