arXiv subjects

Govind S. Krishnaswami

Publications and source records attributed to Govind S. Krishnaswami.

At least 19 recordsLinked to original sources

arXiv2026

Lax pairs and $r$-matrices for some two-dimensional isotropic oscillators

This paper concerns Lax pairs for circularly symmetric harmonic, Fock-Darwin-type and quartic anharmonic oscillators in two dimensions. Although the 2d isotropic harmonic oscillator is bi-Hamiltonian, its recursion operator does not lead to a Lax pair, nor do we obtain such a pair by taking a limit of the harmonic Calogero model. On the other hand, we show that this superintegrable harmonic oscillator admits a $4 \times 4$ block-form Lax pair with spectral parameter giving two conserved mode energies in involution and a corresponding dynamical $r$-matrix. Interestingly, we also find $2 \times 2$ Lax pairs with spectral parameter that give all three independent conserved quantities satisfying a nonabelian Poisson algebra, thereby providing a simple example of a Lax pair whose conserved quantities are not all in involution. Next, we construct a family of $su(2)$ Lax pairs and $r$-matrices for the quadratic+quartic isotropic anharmonic oscillator. This is then extended to an isotropic oscillator with a rotational energy, which may be viewed as the Fock-Darwin oscillator with a quartic potential. With a change of variables, these Lax pairs and $r$-matrices also apply to the Rajeev-Ranken model, although its noncanonical Poisson structure is distinct from that of the anharmonic oscillator.

nlin.SI↗

arXiv2026

Symbol sequences from three-rotor coincidences and their word-complexity

In the three-rotor problem, three equally massive point particles move on a circle interacting via attractive pairwise cosine potentials. Rotors can represent superconducting phases of distinct metallic segments in a chain of coupled Josephson junctions. We propose a digitization of the classical dynamics that records successive pair and triple coincidences of rotors using four symbols. Rotor coincidences correspond to boundaries in a disjoint partition of the configuration torus into cells where the rotors are ordered clockwise and anticlockwise. It is shown that isolated rotor coincidences must be crossings. Despite being a rather coarse digitization, we find that replacing trajectories by coincidence symbol sequences captures significant qualitative features of the dynamics through word statistics. Word-complexity $C_n$ measures the diversity of $n$-letter words in the symbol sequence while topological entropy governs asymptotic exponential growth of $C_n$. Sequences from periodic orbits have a word-complexity that saturates at the period. Ultra-high-energy trajectories with irrational 'slope' are quasiperiodic. We show that they have zero entropy and $C_n = n+3$ by examining limiting slopes and by a mapping to Sturmian sequences. We examine their grammar rules and propose how their right-special words may be identified. On the other hand, numerical investigation of sequences from chaotic orbits in the band of global chaos leads us to conjecture an exponentially growing word-complexity $C_n = 3 \times 2^{n-1}$, corresponding to a topological entropy $\log 2$. We identify their grammar rules and model them by a subshift of finite type, unlike the quasiperiodic ultra-high-energy sequences which cannot be modeled as a topological Markov shift.

nlin.CD↗

arXiv2024

Level crossing instabilities in inviscid isothermal compressible Couette flow

We study the linear stability of inviscid steady parallel flow of an ideal gas in a channel of finite width. Compressible isothermal two-dimensional monochromatic perturbations are considered. The eigenvalue problem governing density and velocity perturbations is a compressible version of Rayleigh's equation and involves two parameters: a flow Mach number $M$ and the perturbation wavenumber $k$. For an odd background velocity profile, there is a $\mathbb{Z}_2 \times \mathbb{Z}_2$ symmetry and growth rates $γ$ come in symmetrically placed 4-tuples in the complex eigenplane. Specializing to uniform background vorticity Couette flow, we find an infinite tower of noninflectional eigenmodes and derive stability theorems and bounds on growth rates. We show that eigenmodes are neutrally stable for small $k$ and small $M$ but that they otherwise display an infinite sequence of stability transitions with increasing $k$ or $M$. Using a search algorithm based on the Fredholm alternative, we find that the transitions are associated to level crossings between neighboring eigenmodes. Repeated level crossings result in windows of instability. For a given eigenmode, they are arranged in a zebra-like striped pattern on the $k$-$M$ plane. A canonical square-root power law form for $γ(k,M)$ in the vicinity of a stability transition is identified. In addition to the discrete spectrum, we find a continuous spectrum of eigenmodes that are always neutrally stable but fail to be smooth across critical layers.

physics.flu-dyn↗

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↗

arXiv2024

Screwon spectral statistics and dispersion relation in the quantum Rajeev-Ranken model

The Rajeev-Ranken (RR) model is a Hamiltonian system describing screw-type nonlinear waves (screwons) of wavenumber $k$ in a scalar field theory pseudodual to the 1+1D SU(2) principal chiral model. Classically, the RR model based on a quadratic Hamiltonian on a nilpotent/Euclidean Poisson algebra is Liouville integrable. Upon adopting canonical variables in a slightly extended phase space, the model was interpreted as a novel 3D cylindrically symmetric quartic oscillator with a rotational energy. Here, we examine the spectral statistics and dispersion relation of quantized screwons via numerical diagonalization validated by variational and perturbative approximations. We also derive a semiclassical estimate for the cumulative level distribution which compares favorably with the one from numerical diagonalization. The spectrum shows level crossings typical of an integrable system. The $i^{\rm th}$ unfolded nearest neighbor spacings are found to follow Poisson statistics for small $i$. Nonoverlapping spacing ratios also indicate that successive spectral gaps are independently distributed. After displaying universal linear behavior over energy windows of short lengths, the spectral rigidity saturates at a length and value that scales with the square-root of energy. For strong coupling $λ$ and intermediate $k$, we argue that reduced screwon energies can depend only on the product $λk$. Numerically, we find power law dependences on $λ$ and $k$ with an approximately common exponent $2/3$ provided the angular momentum quantum number $l$ is small compared to the number of nodes $n$ in the radial wavefunction. On the other hand, for the ground state $n = l = 0$, the common exponent becomes 1.

nlin.SI↗

arXiv2020

The Added Mass Effect and the Higgs Mechanism

A rigid body accelerated through an inviscid, incompressible fluid appears to gain mass, which is encoded in an added mass tensor. Swimmers, air bubbles, submarines and airships are slowed down by the associated `added mass' force proportional to their acceleration, which is distinct from viscous drag and buoyancy. In particle physics, otherwise massless electrons, quarks, $W$ and $Z$ bosons, moving through the Higgs scalar field acquire masses encoded in a mass matrix. In this expository article we give an introduction to the fluid mechanical added mass effect through examples of potential flows in various dimensions and exploit a correspondence with the Higgs mechanism [introduced in Proc. R. Soc. A471: 20140803 (2015)] to relate this to how the $W$ and $Z$ bosons can get their masses, while leaving the photon massless. The correspondence relates the Higgs scalar field to the fluid, the vacuum expectation value of the Higgs field to a constant fluid density and quantum fluctuations around the Higgs condensate to compressional waves in the fluid. The shape of the rigid body encodes the pattern of gauge symmetry breaking through the eigenvalues of the corresponding mass matrices. Possible directions of acceleration of the rigid body are related to directions in the gauge Lie algebra with a `flat' direction in the body corresponding to a massless photon. Moreover, symmetries of the body are related to those of the scalar vacuum manifold. The Higgs boson may be viewed as the analog of a long wavelength fluid mode around an accelerated body.

physics.flu-dyn↗

arXiv2020

An introduction to Lax pairs and the zero curvature representation

Lax pairs are a useful tool in finding conserved quantities of some dynamical systems. In this expository article, we give a motivated introduction to the idea of a Lax pair of matrices $(L,A)$, first for mechanical systems such as the linear harmonic oscillator, Toda chain, Eulerian rigid body and the Rajeev-Ranken model. This is then extended to Lax operators for one-dimensional field theories such as the linear wave and KdV equations and reformulated as a zero curvature representation via a $(U,V)$ pair which is illustrated using the nonlinear Schrödinger equation. The key idea is that of realizing a (possibly) nonlinear evolution equation as a compatibility condition between a pair of linear equations. The latter could be an eigenvalue problem for the Lax operator $L$ and a linear evolution equation generated by $A$, for the corresponding eigenfunction. Alternatively, they could be the first order linear system stating the covariant constancy of an arbitrary vector with respect to the 1+1 dimensional gauge potential $(V,U)$. The compatibility conditions are then either the Lax equation $\dot L = [L, A]$ or the flatness condition $U_t - V_x + [U, V] = 0$ for the corresponding gauge potential. The conserved quantities then follow from the isospectrality of the Lax and monodromy matrices.

nlin.SI↗

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

Invariant tori, action-angle variables and phase space structure of the Rajeev-Ranken model

We study the classical Rajeev-Ranken model, a Hamiltonian system with three degrees of freedom describing nonlinear continuous waves in a 1+1-dimensional nilpotent scalar field theory pseudodual to the SU(2) principal chiral model. While it loosely resembles the Neumann and Kirchhoff models, its equations may be viewed as the Euler equations for a centrally extended Euclidean algebra. The model has a Lax pair and r-matrix leading to four generically independent conserved quantities in involution, two of which are Casimirs. Their level sets define four-dimensional symplectic leaves on which the system is Liouville integrable. On each of these leaves, the common level sets of the remaining conserved quantities are shown, in general, to be 2-tori. The non-generic level sets can only be horn tori, circles and points. They correspond to measure zero subsets where the conserved quantities develop relations and solutions degenerate from elliptic to hyperbolic, circular or constant functions. A geometric construction allows us to realize each common level set as a bundle with base determined by the roots of a cubic polynomial. A dynamics is defined on the union of each type of level set, with the corresponding phase manifolds expressed as bundles over spaces of conserved quantities. Interestingly, topological transitions in energy hypersurfaces are found to occur at energies corresponding to horn tori, which support purely homoclinic orbits. The dynamics on each horn torus is non-Hamiltonian, but expressed as a gradient flow. Finally, we discover a family of action-angle variables for the system that apply away from horn tori.

nlin.SI↗

arXiv2019

On the Hamiltonian formulation, integrability and algebraic structures of the Rajeev-Ranken model

The integrable 1+1-dimensional SU(2) principal chiral model (PCM) serves as a toy-model for 3+1-dimensional Yang-Mills theory as it is asymptotically free and displays a mass gap. Interestingly, the PCM is 'pseudodual' to a scalar field theory introduced by Zakharov and Mikhailov and Nappi that is strongly coupled in the ultraviolet and could serve as a toy-model for non-perturbative properties of theories with a Landau pole. Unlike the 'Euclidean' current algebra of the PCM, its pseudodual is based on a nilpotent current algebra. Recently, Rajeev and Ranken obtained a mechanical reduction by restricting the nilpotent scalar field theory to a class of constant energy-density classical waves expressible in terms of elliptic functions, whose quantization survives the passage to the strong-coupling limit. We study the Hamiltonian and Lagrangian formulations of this model and its classical integrability from an algebraic perspective, identifying Darboux coordinates, Lax pairs, classical r-matrices and a degenerate Poisson pencil. We identify Casimirs as well as a complete set of conserved quantities in involution and the canonical transformations they generate. They are related to Noether charges of the field theory and are shown to be generically independent, implying Liouville integrability. The singular submanifolds where this independence fails are identified and shown to be related to the static and circular submanifolds of the phase space. We also find an interesting relation between this model and the Neumann model allowing us to discover a new Hamiltonian formulation of the latter.

hep-th↗

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↗

arXiv2018

Conservative regularization of compressible dissipationless two-fluid plasmas

This paper extends our earlier approach [cf. Phys. Plasmas 17, 032503 (2010), 23, 022308 (2016)] to obtaining a priori bounds on enstrophy in neutral fluids (R-Euler) and ideal magnetohydrodynamics (R-MHD). This results in a far-reaching local, three-dimensional, non-linear, dispersive generalization of a KdV-type regularization to compressible/incompressible dissipationless two-fluid plasmas and models derived therefrom (quasi-neutral, Hall and ideal MHD). It involves the introduction of vortical and magnetic `twirl' terms $λ_l^2 ({\bf w}_l + \frac{q_l}{m_l} {\bf B}) \times (\nabla \times {\bf w}_l)$ in the ion/electron velocity equations ($l = i,e$) where ${\bf w}_l = \nabla \times {\bf v}_l$ are vorticities. The cut-off lengths $λ_l$ must be inversely proportional to the square-roots of the number densities $(λ_l^2 n_l = C_l)$ and may be taken as Debye lengths or skin-depths. A novel feature is that the `flow' current $\sum_l q_l n_l {\bf v}_l$ in Ampere's law is augmented by a solenoidal `twirl' current $\sum_l \nabla \times \nabla \times λ_l^2 {\bf j}_{{\rm flow},l}$. The resulting equations imply conserved linear and angular momenta and a positive definite swirl energy density ${\cal E}^*$ which includes an enstrophic contribution $\sum_l (1/2) λ_l^2 ρ_l {\bf w}_l^2$. It is shown that the equations admit a Hamiltonian-Poisson bracket formulation. Furthermore, singularities in $\nabla \times {\bf B}$ are conservatively regularized by adding $(λ_B^2/2 μ_0) (\nabla \times {\bf B})^2$ to ${\cal E}^*$. Finally, it is proved that among regularizations that admit a Hamiltonian formulation and preserve the continuity equations along with the symmetries of the ideal model, the twirl term is unique and minimal in non-linearity and space derivatives of velocities.

physics.plasm-ph↗

arXiv2016

Conservative regularization of compressible flow and ideal magnetohydrodynamics

Ideal systems like MHD and Euler flow may develop singularities in vorticity (w = curl v). Viscosity and resistivity are dissipative regularizations. We propose a minimal, local, conservative, nonlinear, dispersive regularization of compressible flow and ideal MHD, in analogy with the KdV regularization of the 1D Hopf equation. This work significantly extends earlier work on incompressible Euler and ideal MHD. It involves a cut-off lambda inversely proportional to square-root of density rho, which is like a position-dependent mean free path. In MHD, lambda can be taken of order the electron collisionless skin depth. The regularizing `twirl' term is - lambda w x curl w. Such a term could be important in high speed flows with vorticity and arise in an expansion of kinetic equations in Knudsen number. A magnetic analogue of the twirl term - (B x curl B)/(rho mu_0), arises as the Lorentz force in ideal MHD. Our regularization preserves symmetries of the ideal systems, and with appropriate boundary conditions, implies associated conservation laws. Energy and enstrophy are subject to a priori bounds determined by initial data. A Hamiltonian and Poisson bracket formulation is developed and used to generalize the constitutive relation to bound higher moments of w and curl w. A `swirl' velocity field is shown to transport w/rho and B/rho, generalizing the Kelvin-Helmholtz and Alfvén theorems. The steady regularized equations are used to model a rotating vortex, MHD pinch, plane vortex sheet, channel flow, plane flow and propagating spherical and cylindrical vortices; solutions are more regular than corresponding Eulerian ones. The proposed regularization could facilitate simulations of fluid/MHD equations and provide a consistent statistical mechanics of vortices/current filaments in 3D, without blowup of enstrophy. Implications for detailed analyses of fluid and plasma systems are discussed.

physics.flu-dyn↗

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.

math-ph↗

arXiv2016

Algebra and geometry of Hamilton's quaternions

Inspired by the relation between the algebra of complex numbers and plane geometry, William Rowan Hamilton sought an algebra of triples for application to three dimensional geometry. Unable to multiply and divide triples, he invented a non-commutative division algebra of quadruples, in what he considered his most significant work, generalizing the real and complex number systems. We give a motivated introduction to quaternions and discuss how they are related to Pauli matrices, rotations in three dimensions, the three sphere, the group SU(2) and the celebrated Hopf fibrations.

math.HO↗

arXiv2016

Local conservative regularizations of compressible magnetohydrodynamic and neutral flows

Ideal systems like MHD and Euler flow may develop singularities in vorticity (w = curl v). Viscosity and resistivity provide dissipative regularizations of the singularities. In this paper we propose a minimal, local, conservative, nonlinear, dispersive regularization of compressible flow and ideal MHD, in analogy with the KdV regularization of the 1D kinematic wave equation. This work extends and significantly generalizes earlier work on incompressible Euler and ideal MHD. It involves a micro-scale cutoff length lambda which is a function of density, unlike in the incompressible case. In MHD, it can be taken to be of order the electron collisionless skin depth c/omega_pe. Our regularization preserves the symmetries of the original systems, and with appropriate boundary conditions, leads to associated conservation laws. Energy and enstrophy are subject to a priori bounds determined by initial data in contrast to the unregularized systems. A Hamiltonian and Poisson bracket formulation is developed and applied to generalize the constitutive relation to bound higher moments of vorticity. A `swirl' velocity field is identified, and shown to transport w/rho and B/rho, generalizing the Kelvin-Helmholtz and Alfven theorems. The steady regularized equations are used to model a rotating vortex, MHD pinch and a plane vortex sheet. The proposed regularization could facilitate numerical simulations of fluid/MHD equations and provide a consistent statistical mechanics of vortices/current filaments in 3D, without blowup of enstrophy. Implications for detailed analyses of fluid and plasma dynamic systems arising from our work are briefly discussed.

physics.plasm-ph↗