Paper studies Lie-Trotter integrator for symmetric free rigid body dynamics.
problem Understanding dynamics of symmetric free rigid body using numerical methods.
method Examines Lie-Trotter integrator applied to Euler equations.
result Lie-Trotter integrator results in a Poisson integrator for symmetric free rigid body dynamics.
We consider an SO(4) Euler rigid body with two 'inertia momenta' coinciding. We study it from the point of view of bihamiltonian geometry. We show how to algebraically integrate it by means of the method of separation of variables.
Solves Christoffel-Minkowski problem for axially symmetric bodies.
problem Necessary and sufficient conditions for mixed area measures of axially symmetric convex bodies.
method Introduced a new method to transform mixed area measures and mixed volumes of axially symmetric bodies, refining Firey's classification and improving estimates.
result Complete solution to the mixed Christoffel-Minkowski problem for axially symmetric bodies without regularity assumptions.
Paper studies how discrete space curves with constant torsion deform to model linkage motions.
problem Modeling and understanding the motion of discrete space curves with constant torsion.
method Using semi-discrete mKdV equations to describe the motion of discrete space curves.
result The motion of discrete space curves is governed by semi-discrete mKdV equations.
The paper proves convex bodies are minimal fillings and have Lipschitz-volume rigidity.
problem Finding minimal fillings of convex bodies.
method Analyzing integral current spaces and proving rigidity properties.
result Convex bodies are the unique minimal fillings of their boundary metrics among integral current spaces and enjoy Lipschitz-volume rigidity.
Study on symmetries and equilibria in Poisson manifolds, with applications to rigid body dynamics.
problem Characterizing conformal relative equilibria on Poisson manifolds.
method Introducing conformally Poisson actions and momentum maps, establishing algebraic criteria.
result Classification of nontrivial conformal relative equilibria in Lie algebras, with applications to rigid body dynamics.
New equations for rigid body motion on infinite-dimensional spaces of operators.
problem Integrating rigid body dynamics on infinite-dimensional spaces of operators.
method Introducing pseudo-Riemannian metrics and adapting classical integrability theory.
result Existence of geodesics and integrals of motion for the rigid body equations.
Paper compares Lagrangian reduction methods for rigid body systems.
problem Modeling and reduction of rigid body systems with rotors.
method Euler-Poincaré reduction by the whole group and reduction by stages.
result Equivalence of equations and conservation laws are tracked.
Paper provides closed-form time derivatives for rigid body systems.
problem Need for time derivatives of equations of motion in robotics.
method Lie group formulation for rigid body systems to derive closed-form derivatives up to second-order.
result Closed-form equations provide direct insight into system dynamics.
New model simplifies symmetry handling in generative AI.
problem Symmetry handling in generative models for scientific tasks.
method Quotient-space diffusion models, viewing symmetry as quotient space.
result Improves performance over existing methods for molecular structure generation.
New surface area measures defined for ball-convex bodies, leading to entropy and inequalities.
problem Defining and analyzing surface area measures for ball-convex bodies.
method Introducing Lp relative surface areas, proving invariance and inequalities, and using geometric interpretations. result Established inequalities and a new notion of entropy for ball-convex bodies.
Geometrically interpolates rigid body motions with initial and terminal twists.
problem Finding spatial trajectories between prescribed initial and terminal poses.
method Derives solutions for k-IV-TIP and k-BV-TIP for k=1,...,4.
result Automatic cubic interpolation identical to minimum acceleration curve when twists are zero.
A modular object in a symmetric monoidal bicategory is a Frobenius algebra object whose product and coproduct are biadjoint, equipped with a braided structure and a compatible twist, satisfying rigidity, ribbon, pivotality, and modularity conditions. We prove that the oriented 3-dimensional bordism bicategory of 1-, 2-…
Develops computational methods for simulating rigid body dynamics on SO(3).
problem Simulating rotational dynamics of rigid bodies on SO(3).
method Discrete Mechanics, Variational Integrators, Newton-Raphson algorithm.
result Preserves symplectic structure of SO(3) manifold dynamics.
A sliding surface stabilizes rigid body attitude control.
problem Stabilizing rigid body attitude control robustly.
method Proposed sliding surface as Lie subgroup, designed sliding-mode controller.
result Closed-loop system is robust against disturbances and unwinding.
The paper studies rigidity and continuity in nonlinear elasticity on manifolds and hypersurfaces.
problem Rigidity and continuity properties of elastic bodies in non-Euclidean settings.
method Geometric rigidity estimates, asymptotic rigidity of elastic membranes, simplified geometric proof of continuous dependence.
result Established geometric rigidity estimate and proved asymptotic rigidity of elastic membranes.
The paper proves rigidity results for manifolds with special holonomy.
problem Proving rigidity results for compact Riemannian manifolds with special holonomy.
method Using divergence free Weyl tensors and curvature operators, the paper proves similar results for manifolds with special holonomy.
result The paper proves that manifolds with special holonomy are locally symmetric or conformally equivalent to a quotient of the sphere.
Rigidity of spectral data for spherical manifolds with boundary.
problem Determining the length spectrum of spherically symmetric manifolds with boundary.
method Proving a trace formula and using it to show spectral rigidity.
result The Neumann spectrum uniquely determines the length spectrum for spherically symmetric manifolds with boundary.
We give a criterion of (micro-)kroneckerity of the linear Poisson pencil on g∗ related to an algebraic Nijenhuis operator N:g→g on a finite-dimensional Lie algebra g. As an application we get a series of examples of completely integrable systems on semisimple Lie algebras related t…
Proves inequality for special 3D shapes, generalizing to non-symmetric ones.
problem Proving a mathematical inequality for specific 3D shapes.
method Operator theoretic approach combined with spherical function decomposition.
result Generalized inequality for non-symmetric bodies of revolution.
Meyer and Reisner had proved the Mahler conjecture for rovelution bodies. In this paper, using a new method, we prove that among origin-symmetric bodies of revolution in R^3, cylinders have the minimal Mahler volume. Further, we prove that among parallel sections homothety bodies in R^3, 3-cubes have the minimal Mahler…
This paper proves a curvature entropy inequality for non-symmetric convex bodies.
problem Proving a curvature entropy inequality for non-symmetric convex bodies.
method Demonstrated the log-Minkowski inequality of curvature entropy for general convex bodies in 2D.
result Equivalence of cone-volume measure uniqueness, log-Minkowski volume inequality, and curvature entropy inequality for general convex bodies in 2D.
Paper presents derivative-free methods for online inverse dynamics modeling.
problem Online learning of inverse dynamics models without numerical differentiation.
method Derivative-free framework for rigid body dynamics, data-driven, and semiparametric models.
result Proposed `derivative-free' methods outperform existing methodologies in real data experiments.
Solves a complex geometric problem for symmetric convex bodies.
problem Conditions for a measure to be the dual curvature measure of a symmetric convex body.
method Variational approach using entropy and quermassintegrals, with estimates on entropy and curvature measures.
result Explicit conditions for the measure concentration, leading to a full solution for 1<q<n. The article considers the problem of existence and uniqueness of centrally symmetrical convex body for which the projection curvature radius function coincides with a given flag function. A necessary and sufficient condition is found that ensures a positive answer. An algorithm for construction the body in question is …
Paper proves rigidity of metrics near hyperbolic ones in 3D.
problem Proving rigidity of metrics near hyperbolic ones in 3D.
method Introducing marked Poincaré determinant and proving local rigidity.
result Lichnerowicz Laplacian is injective in negative curvature.
The Hamilton-Jacobi problem is revisited bearing in mind the consequences arising from a possible bi-Hamiltonian structure. The problem is formulated on the tangent bundle for Lagrangian systems in order to avoid the bias of the existence of a natural symplectic structure on the cotangent bundle. First it is developed …
Sharp inequalities for star bodies in 2D space.
problem Understanding star bodies in 2D space.
method Sharp inequalities for star bodies in R2. result New inequalities and proofs for star bodies.
Paper learns particle dynamics for versatile object manipulation.
problem Challenges in traditional rigid-body physics engines for complex scenes.
method Combines learning with particle-based systems for versatile object manipulation.
result Robots achieve complex manipulation tasks using the learned simulator.
Log-Brunn-Minkowski inequality proven for ball and symmetric convex bodies.
problem Proving the log-Brunn-Minkowski inequality for specific geometric shapes.
method Analyzing the inequality for a ball and symmetric convex bodies in a C2 neighborhood. result Log-Brunn-Minkowski inequality holds for a ball and symmetric convex bodies.
We define \emph{piecewise rank 1} manifolds, which are aspherical manifolds that generally do not admit a nonpositively curved metric but can be decomposed into pieces that are diffeomorphic to finite volume, irreducible, locally symmetric, nonpositively curved manifolds with π1-injective cusps. We prove smooth (sel…
New proof for 2D lens rigidity from billiard trajectories.
problem Determining obstacles in 2D space from billiard trajectories.
method Separate proof for 2D case, using billiard trajectories.
result Proves rigidity for 2D strictly convex bodies.
Suppose that the initial triangle formed by the three moving masses of the three-body problem is similar to the triangle formed at some later time. We derive a simple integral formula for the overall rotation relating the two triangles. The formula is based on the fact that the space of similarity classes of triangles …
Study of rigid body displacements in a projective space over dual numbers with geometric interpretations.
problem Understanding rigid body displacements in a novel geometric space.
method Projective differential geometry over the ring of dual numbers.
result Existence of non-straight curves with multiple osculating tangents.
A Minkowski class is a closed subset of the space of convex bodies in Euclidean space Rn which is closed under Minkowski addition and non-negative dilatations. A convex body in Rn is universal if the expansion of its support function in spherical harmonics contains non-zero harmonics of all orders. If K is universal, t…
We prove complete integrability of the Manakov-type SO(n)-invariant geodesic flows on homogeneous spaces SO(n)/SO(k1)×...×SO(kr), for any choice of k1,...,kr, k1+...+kr≤n. In particular, a new proof of the integrability of a Manakov symmetric rigid body motion around a fixed point is presented…
New proof of Wulff-Gage inequality with applications.
problem Proving the Wulff-Gage isoperimetric inequality.
method Provided a new proof of the inequality for origin-symmetric convex bodies.
result Uniqueness of log-Minkowski problem and new proof of log-Minkowski inequality.
Proves gap rigidity theorem for Hermitian symmetric spaces.
problem Gap rigidity problems in compact Hermitian symmetric spaces.
method Dual analogy to Mok's noncompact case theorem, theorem on higher dimensional submanifolds.
result Proves gap rigidity theorem for diagonal curves in tube type spaces.
Theory of symmetric rigidity in hyperbolic geometry.
problem Symmetric rigidity in hyperbolic frameworks.
method Gain graphs and orbit rigidity matrix, matroidal sparsity conditions.
result Characterization of infinitesimal rigidity for Gamma-symmetric frameworks.
The possibility of the global Lagrangian reduction of a mechanical system with symmetry is shown to be connected with the characteristic class of a principal fiber bundle of the configuration space over the factor manifold. It is proved that the reduced system is globally Lagrangian if and only if the product of the mo…
In the paper, some concepts of modern differential geometry are used as a basis to develop an invariant theory of mechanical systems, including systems with gyroscopic forces. An interpretation of systems with gyroscopic forces in the form of flows of a given geodesic curvature is proposed. For illustration, the proble…
Study contact geometry of energy hypersurface in symmetric 3-body problem on S^2.
problem Contact geometry of energy hypersurface in symmetric 3-body problem on S^2.
method Contact geometry analysis, Moser regularization, Weinstein conjecture.
result Components of energy hypersurface are contactomorphic to R^3 or its connected sum for specific energies.
The paper examines the failure of Brunn-Minkowski inequality for certain convex bodies.
problem Brunn-Minkowski inequality for q-th dual quermassintegrals with q>n. method Second variation argument, dimension reduction, Hadwiger's inequality, singular weighted Reilly formula, coordinate-slice Hardy inequality.
result Established the inequality for unconditional convex bodies in the full range 0<q≤n+1. We use the reflection group trick to glue manifolds with corners that are Borel-Serre compactifications of locally symmetric spaces of noncompact type and obtain aspherical manifolds. We call these \emph{piecewise locally symmetric} manifolds. This class of spaces provide new examples of aspherical manifolds whose fund…
Study on spherical Finsler metrics with isotropic curvature rigidity.
problem Characterizing and understanding spherically symmetric Finsler metrics with isotropic E-curvature. method Provided the correct formula for mean Berwald curvature, established differential equations for projective and dual flatness, and derived a rigidity result.
result Rigidity result on spherically symmetric Finsler metrics with isotropic E-curvature. Compact rank one symmetric spaces are rigid under certain curvature conditions.
problem Rigidity of compact rank one symmetric spaces under curvature constraints.
method Examined compact symmetric spaces with metric g0 of rank one, and another metric g with sectional curvature bounded by 0 to 1. result If g equals g0 outside a convex subset, then g is isometric with g0. For i=1,2, let Gi be cocompact groups of isometries of hyperbolic space $\Hyp^n$ of real dimension n, n≥3. Let Hi⊂Gi be infinite index quasiconvex subgroups satisfying one of the following conditions: 1) limit set of Hi is a codimension one topological sphere. 2) limit set of Hi is an e…
Paper derives and applies a parallel transport equation on Lie groups.
problem Efficiently solving parallel transport on Lie groups with left-invariant metrics.
method Derives a parallel transport equation in Lie algebra, applies it to SE(3), and compares to existing methods.
result Stable and efficient parallel transport implementation on Lie groups.