Superintegrable systems on curved manifolds found to have Hessian structures.
problem Characterizing superintegrable systems on curved manifolds.
method Identifying and computing Hessian coordinates for superintegrable systems.
result Examples of superintegrable systems in 2D and 3D have natural Hessian coordinates.
New superintegrable systems derived from Frobenius structures.
problem Constructing second-order superintegrable systems.
method Using conification and direct product construction, applying to semi-simple and nilpotent algebras.
result Explicitly constructed second-order superintegrable systems in three dimensions.
Study finds Stäckel equivalence for superintegrable systems via invariant quadrics.
problem Understanding Stäckel equivalence in superintegrable systems.
method Using invariant quadrics to determine Stäckel classes of superintegrable systems.
result Stäckel classes of superintegrable systems can be derived from associated invariant quadrics.
Study reveals geometric context of second-order superintegrable systems.
problem Understanding second-order superintegrable systems and their Weylian geometry.
method Re-examined second-order maximally conformally superintegrable Hamiltonian systems, revealing their Weyl structure.
result Extended conformal superintegrability to Weyl structures, interpreting systems as semi-Weyl structures.
Superintegrable systems on surfaces are classified geometrically.
problem Classifying superintegrable systems on conformal surfaces.
method Geometric structures on conformal surfaces, conformal covariant structural equations.
result Explicit set of algebraic equations defining superintegrable systems on all constant curvature surfaces.
New algebraic approach classifies conformally superintegrable systems in arbitrary dimensions.
problem Classifying conformally superintegrable systems in arbitrary dimensions.
method Algebraic geometric approach extended to conformally superintegrable systems.
result An algebraic equation governs the classification under conformal equivalence for a prolific class of second order conformally superintegrable systems.
New algebraic-geometric method classifies superintegrable systems in any dimension.
problem Classifying superintegrable systems in arbitrary dimensions is challenging.
method Algebraic-geometric approach based on quasi-projective varieties.
result Established foundations for classification in arbitrary dimensions.
New product structures encode superintegrable Hamiltonian systems in Euclidean spaces.
problem Encoding superintegrable Hamiltonian systems using product structures.
method Introducing commutative and associative product structures on Euclidean spaces of dimension at least three, satisfying specific conditions.
result All abundant superintegrable Hamiltonian systems on Euclidean space of dimension at least three arise from these product structures.
Study on Haantjes tensors for superintegrable systems, focusing on vanishing properties.
problem Understanding the vanishing of Haantjes tensors in superintegrable systems.
method Investigating Killing tensor fields associated with second-order superintegrable systems.
result Characterization of Haantjes-zero Killing tensor fields.
This paper combines two classical theories, namely metric projective differential geometry and superintegrability. We study superintegrable systems on 2-dimensional geometries that share the same geodesics, viewed as unparametrized curves. We give a definition of projective equivalence of such systems, which may be con…
New connections share geodesics with superintegrable systems.
problem Understanding geodesics in affine connections related to superintegrable systems.
method Analyzing dual-geodesics and comparing them across different connections.
result Certain torsion-free affine connections associated with second order superintegrable systems share the same dual-geodesics.
The paper studies connections in superintegrable systems, revealing geometric insights.
problem Understanding non- and semi-degenerate superintegrable systems.
method Analyzes two torsion-free connections associated with superintegrable systems.
result Semi-degenerate secondary structure tensor is the Ricci curvature of a natural torsion-free connection.
The paper classifies second-order superintegrable systems with torsion and semi-degeneracy.
problem Classifying second-order superintegrable systems with torsion and semi-degeneracy.
method Information-geometric structure and geometric conditions for non-degeneracy.
result A (n+1)-parameter potential is non-degenerate if a certain trace-free tensor field vanishes. New findings on magnetic geodesic flows and periodic motions.
problem Characterizing superintegrable systems in magnetic geodesic flows.
method Analyzing rotationally symmetric magnetic geodesic flows.
result All sufficiently slow motions in a central magnetic field are periodic under specific curvature and homogeneity conditions.
The paper proves real-analyticity of superintegrable metrics and solves two conjectures.
problem Proving real-analyticity of superintegrable metrics and solving conjectures.
method Analyzing Poisson brackets and constructing new superintegrable systems.
result Proves real-analyticity of superintegrable metrics and solves two conjectures.
Study of superintegrable systems linked to affine hypersurfaces.
problem Understanding superintegrable systems through geometric structures.
method Established a correspondence between superintegrable systems and affine hypersurfaces, defining conformal equivalence.
result Identified conformal classes of abundant manifolds with abundant hypersurface immersions.
Superintegrable systems are classical and quantum Hamiltonian systems which enjoy much symmetry and structure that permit their solubility via analytic and even, algebraic means. They include such well-known and important models as the Kepler potential, Calogero-Moser model, and harmonic oscillator, as well as its inte…
We prove that the set of non-degenerate second order maximally superintegrable systems in the complex Euclidean plane carries a natural structure of a projective variety, equipped with a linear isometry group action. This is done by deriving the corresponding system of homogeneous algebraic equations. We then solve the…
We prove that for Matveev and Shevchishin superintegrable system, with a linear and a cubic integral, the metrics defined on S^2 and on Tannery's orbifold T^2 are either Zoll or Tannery metrics.
We describe all local Riemannian metrics on surfaces whose geodesic flows are superintegrable with one integral linear in momenta and one integral cubic in momenta. We also show that some of these metrics can be extended to the 2-sphere. This gives us new examples of Hamiltonian systems on the sphere with integrals of …
The paper finds new metrics for geodesic flows with rational integrals.
problem Finding Riemannian metrics with rational integrals for geodesic flows.
method Explicit construction of metrics and integrals.
result New examples of metrics with rational integrals are provided.
We review the fundamentals of coupling constant metamorphosis (CCM) and the Stäckel transform, and apply them to map integrable and superintegrable systems of all orders into other such systems on different manifolds. In general, CCM does not preserve the order of constants of the motion or even take polynomials in the…
We describe a procedure naturally associating relativistic Klein-Gordon equations in static curved spacetimes to non-relativistic quantum motion on curved spaces in the presence of a potential. Our procedure is particularly attractive in application to (typically, superintegrable) problems whose energy spectrum is give…
A Carter like constant for the geodesic motion in the Y(p,q) Einstein-Sasaki geometries is presented. This constant is functionally independent with respect to the five known constants for the geometry. Since the geometry is five dimensional and the number of independent constants of motion is at least six, the geode…
Curved Frobenius manifolds link to Hessian metrics in geometry.
problem Understanding curved Frobenius manifolds and their relation to Hessian metrics.
method Analyzing the relationship between curved Frobenius structures and Hessian metrics on spaces with non-vanishing curvature.
result Consistent curved Frobenius structures on constant curvature spaces are linked to Hessian metrics.
Study geodesic flows on cones over Riemannian manifolds, showing superintegrability.
problem Behavior of geodesics on cones over arbitrary Riemannian manifolds.
method Show existence of first integrals uniquely determining geodesics.
result Geodesic flow on cones is superintegrable and Liouville--Arnold integrable for non-radial trajectories.
Develops Kleinian Sphere Packings and Bugs, proving their arithmetic origins.
problem Understanding sphere packings and their arithmetic origins in various dimensions.
method Introduces Kleinian Sphere Packings and Bugs, extending Arithmeticity Theorem.
result Kleinian packings and Bugs come from Q-arithmetic lattices of simplest type.
We prove that a surface carries a hexagonal 3-web of geodesics if and only if the geodesic flow on the surface admits a cubic first integral and show that the system of partial differential equations, governing metrics on such surfaces, is integrable by generalized hodograph transform method. We present some new local …
We consider the Kepler problem on surfaces of revolution that are homeomorphic to S2 and have constant Gaussian curvature. We show that the system is maximally superintegrable, finding constants of motion that generalize the Runge-Lentz vector. Then, using such first integrals, we determine the class of surfaces tha…
We employ the language of Cartan's geometry to present a model for studying vector spaces of Killing two-tensors defined in pseudo-Riemannian spaces of constant curvature under the action of the corresponding isometry group. We also discuss geometric properties of joint invariants of Killing two-tensors defined in the …
Projective connections arise from equivalence classes of affine connections under the reparametrization of geodesics. They may also be viewed as quotient systems of the classical geodesic equation. After studying the link between integrals of the (classical) geodesic flow and its associated projective connection, we tu…
We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces and prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces
We introduce the notion of a "crystallographic sphere packing," defined to be one whose limit set is that of a geometrically finite hyperbolic reflection group in one higher dimension. We exhibit for the first time an infinite family of conformally-inequivalent such with all radii being reciprocals of integers. We then…
We describe all pseudo-Riemannian metrics on closed surfaces whose geodesic flows admit nontrivial integrals quadratic in momenta. As an application, we solve the Beltrami problem on closed surfaces, prove the nonexistence of quadratically-superintegrable metrics of nonconstant curvature on closed surfaces, and prove t…
The paper solves the existence problem of sphere packings in higher dimensions.
problem Existence of crystallographic sphere packings in certain higher dimensions.
method Geometric doubling procedure and computations with Lorentzian quadratic forms.
result Solves the existence problem of crystallographic sphere packings in higher dimensions.
For a fundamental solution of Laplace's equation on the R-radius d-dimensional hypersphere, we compute the azimuthal Fourier coefficients in closed form in two and three dimensions. We also compute the Gegenbauer polynomial expansion for a fundamental solution of Laplace's equation in hyperspherical geometry in geo…
The harmonic oscillator as a distinguished dynamical system can be defined not only on the Euclidean plane but also on the sphere and on the hyperbolic plane, and more generally on any configuration space with constant curvature and with a metric of any signature, either Riemannian (definite positive) or Lorentzian (in…
Overview of integrable systems with symmetries, focusing on toric and semitoric systems.
problem Classifying and understanding integrable systems with symmetries.
method Using decorated polygons and controlled bifurcations in one-parameter families of systems.
result Construction of explicit semitoric systems with prescribed invariants.
New method to derive integrable systems from existing Lax systems.
problem Deriving new integrable systems from existing ones.
method Systematic method of deriving new integrable systems from a given one.
result Examples of new integrable systems derived, including the dispersionless Hirota equation, the general heavenly equation, and the web equations.
Learning to control linear systems is statistically hard, especially for underactuated systems.
problem Statistical difficulty of learning to control linear systems, especially underactuated ones.
method Utilized minimax lower bounds and structural assumptions to prove learning complexity can be exponential.
result Learning complexity can be at most exponential with the controllability index of the system.
Discrete-time systems can be characterized by simple flat coordinates and their shifts.
problem Characterizing flatness of discrete-time systems.
method Developed a map from flat coordinates and their shifts to system state and input, fulfilling system equations identically.
result Derived necessary conditions for a system to be flat, without requiring differential geometry methods.
The paper explores when linear system identification is hard or easy, especially for under-actuated systems.
problem Statistical hardness of learning linear systems, especially under-actuated or under-excited systems.
method Using tools from minimax theory and recent statistical tools for finite sample analysis of system identification.
result The controllability index of linear systems affects the sample complexity of identification, making some systems hard to learn.
This paper improves system identification by reducing sample complexity for high-dimensional linear dynamical systems.
problem High sample complexity for learning partially observed linear dynamical systems in high dimensions.
method Introduces an ℓ1-regularized estimation method that reduces sample complexity from linear to logarithmic with system dimension. result Markov parameters can be learned with logarithmic number of samples relative to system dimension, improving sample complexity.
In integrable hydrodynamic systems, coordinates exist where generators and symmetries are simple.
problem Existence of Riemannian invariants for integrable systems of hydrodynamic type.
method Finding coordinates where the generator and all symmetries are diagonal.
result In integrable hydrodynamic systems, there exist coordinates where the generator and all symmetries are diagonal.
This paper studies nonholonomic constraints in Hamiltonian systems, deriving equations and theorems.
problem Analyzing nonholonomic constraints in Hamiltonian systems.
method Deriving distributional RCH systems, geometric constraint conditions, and Hamilton-Jacobi theorems.
result Derives precise geometric constraint conditions and Hamilton-Jacobi theorems for nonholonomic systems.
New method models unknown systems with hidden parameters using neural networks.
problem Modeling unknown dynamical systems with hidden parameters.
method Training a deep neural network (DNN) model using trajectory data of the unknown system.
result DNN model accurately predicts unknown dynamical systems with new initial conditions.
Study absolute equivalence for Pfaffian systems, applying to control systems.
problem Absolute equivalence of Pfaffian systems with specific independence conditions.
method Structural results for Pfaffian systems of corank 3, applied to control systems.
result Dynamic feedback linearization of control systems with 2 inputs.
Estimates input from output of nonlinear systems using ANN.
problem Estimating unknown compositional input from system output.
method Artificial Neural Networks (ANNs) for nonlinear system inversion.
result ANNs can compete with optimal bounds for linear systems and demonstrate promising results for nonlinear systems.