A new integrable system on the sphere
英文摘要：
We present a new Liouville-integrable natural Hamiltonian system on the (cotangent bundle of the) two-dimensional sphere. The second integral is cubic in the momenta.
机构、总结、中文摘要均由AI分析而来，可能存在错误，请以论文正文为准。
继续探索
相关论文
Rigidity of cotangent lifts and integrable systems
In this article we generalize a theorem by Palais on the rigidity of compact group actions to cotangent lifts. We use this result to prove rigidity for integrable systems on symplectic manifolds including sytems with degenerate singularities which are invariant under a torus action.
math.DSmath.SG
Cotangent models for integrable systems
We associate cotangent models to a neighbourhood of a Liouville torus in symplectic and Poisson manifolds focusing on a special class called $b$-Poisson/$b$-symplectic manifolds. The semilocal equivalence with such models uses the corresponding action-angle coordinate theorems in these settings: the theorem of Liouville-Mineur-Arnold [A74] for symplectic manifolds and an action-angle theorem for regular Liouville tori in Poisson manifolds [LMV11]. Our models comprise regular Liouville tori of Poisson manifolds but also consider the Liouville tori on the singular locus of a $b$-Poisson manifold…
math-phmath.DSmath.MP
Equivalent Integrable Metrics on the Sphere with Quartic Invariants
We discuss canonical transformations relating well-known geodesic flows on the cotangent bundle of the sphere with a set of geodesic flows with quartic invariants. By adding various potentials to the corresponding geodesic Hamiltonians, we can construct new integrable systems on the sphere with quartic invariants.
math-phmath.DSmath.MP
Integrable systems on the sphere, ellipsoid and hyperboloid
Affine transformations in Euclidean space generates a correspondence between integrable systems on cotangent bundles to the sphere, ellipsoid and hyperboloid embedded in $R^n$. Using this correspondence and the suitable coupling constant transformations we can get real integrals of motion in the hyperboloid case starting with real integrals of motion in the sphere case. We discuss a few such integrable systems with invariants which are cubic, quartic and sextic polynomials in momenta.
math-phmath.DSmath.MP
The Maslov cocycle, smooth structures and real-analytic complete integrability
This paper studies smooth obstructions to integrability and proves two main results. First, it is shown that if a smooth topological n-torus admits a real-analytically completely integrable convex hamiltonian on its cotangent bundle, then the torus is diffeomorphic to the standard n-torus. This is the first known result where the smooth structure of a manifold obstructs complete integrability. Second, it is proven that each one of the Witten-Kreck-Stolz 7-manifolds admit a real-analytically completely integrable geodesic flow on its cotangent bundle. This gives examples of topological manifold…
math.DSmath.SG
New integrable two-centre problem on sphere in Dirac magnetic field
We present a new family of integrable versions of the Euler two-centre problem on two-dimensional sphere in the presence of the Dirac magnetic monopole of arbitrary charge. The new systems have very special algebraic potential and additional integral quadratic in momenta, both in classical and quantum versions.
math-phmath.DSmath.MP
更多相关论文收起更多论文
The $C^0$ integrability of symplectic twist maps without conjugate points
We prove that symplectic twist maps defined on the cotangent bundle of the d-dimensional torus that have no conjugate points are $C^0$ integrable, i.e. the cotangent bundle is continuously foliated by a family of invariant Lagrangian graphs.
math.DS
On the Routh sphere problem
We discuss an embedding of a vector field for the nonholonomic Routh sphere into a subgroup of commuting Hamiltonian vector fields on six dimensional phase space. The corresponding Poisson brackets are reduced to the canonical Poisson brackets on the Lie algebra e(3). It allows us to relate nonholonomic Routh system with the Hamiltonian system on cotangent bundle to the sphere with canonical Poisson structure.
math-phmath.DSmath.MP
On the existence of periodic orbits for magnetic systems on the two-sphere
We prove that there exist periodic orbits on almost all compact regular energy levels of a Hamiltonian function defined on a twisted cotangent bundle over the two-sphere. As a corollary, given any Riemannian two-sphere and a magnetic field on it, there exists a closed magnetic geodesic for almost all kinetic energy levels.
math.DSmath.SG
Bäcklund transformations for the nonholonomic Veselova system
We present auto and hetero Bäcklund transformations of the nonholonomic Veselova system using standard divisor arithmetic on the hyperelliptic curve of genus two. As a by-product one gets two natural integrable systems on the cotangent bundle to the unit two-dimensional sphere whose additional integrals of motion are polynomials in the momenta of fourth order.
math-phmath.DSmath.MP
Integrable Mechanical Billiards in Higher Dimensional Space Forms
In this article, we consider mechanical billiard systems defined with Lagrange's integrable extension of Euler's two-center problems in the Euclidean space, on the sphere, and in the hyperbolic space of arbitrary dimension $n \ge 3$. In the three-dimensional Euclidean space, we show that the billiard systems with any finite combination of quadrics having two foci at the Kepler centers are integrable. The same holds for the projections of these systems on the three-dimensional sphere and the hyperbolic space by means of central projection. With the same approach, we also extend these results to…
math-phmath.DSmath.MP
A Reeb flow on the three-sphere without a disk-like global surface of section
We show that there are Reeb flows on the standard, tight three-sphere that do not admit global surfaces of section with one boundary component. In particular, the Reeb flows that we construct do not admit disk-like global surfaces of section. These Reeb flows are constructed using integrable systems, and a connected sum construction that extends the integrable system.
math.DSmath.SG
Tonelli Hamiltonians without conjugate points and $C^0$ integrability
We prove that all the Tonelli Hamiltonians defined on the cotangent bundle $T^*\T^n$ of the $n$-dimensional torus that have no conjugate points are $C^0$ integrable, i.e. $T^*\T^n$ is $C^0$ foliated by a family $\Fc$ of invariant $C^0$ Lagrangian graphs. Assuming that the Hamiltonian is $C^\infty$, we prove that there exists a $G_δ$ subset $\Gc$ of $\Fc$ such that the dynamics restricted to every element of $\Gc$ is strictly ergodic. Moreover, we prove that the Lyapunov exponents of every $C^0$ integrable Tonelli Hamiltonian are zero and deduce that the metric and topological entropies vanish.
math.DGmath.DS
球面与平面滚珠轴承——可积情形的若干研究
Spherical and Planar Ball Bearings -- a Study of Integrable Cases
本文研究球面与平面上多个均质球体及一个动力学非对称球体或平面构成的无滑动滚动非完整系统，推导运动方程并证明不变测度存在，对 n=1 情形找到两类可用求积法积分的可积模型，推广了 Chaplygin 球可积问题，并显式求解了相应的平面问题。
math-phmath.DSmath.MP
余切丛中哈密顿作用的莫尔斯同调
Morse homology for the Hamiltonian action in cotangent bundles
本文利用梯度流方程在余切丛的混合正则性环路空间上为哈密顿作用构造了Morse复形，通过引入强可积(0)-本质子丛保证紧致性，并利用伪梯度向量场扰动实现横截性，最终证明所得Morse同调与Floer同调和自由环路空间的奇异同调同构。
math.DSmath.SG
二体问题在均匀磁场中的球面运动
Two body problem on a sphere in the presence of a uniform magnetic field
研究了在均匀磁场中球面上二体问题的运动特性，发现两种类型的相对平衡态及其稳定性。
math-phmath.DSmath.MP
Magnetic Katok Examples on the two-sphere
We show that there exist non-exact magnetic flows on the two-sphere having an energy level whose double cover is strictly contactomorphic to an irrational ellipsoid in $\mathbb C^2$. Our construction generalizes the examples of integrable Finsler metrics on the two-sphere with only two closed geodesics due to A. Katok.
math.DSmath.SG
球面上双体问题的混沌动力学
Chaotic dynamics for the two body problem on a sphere
本文研究了球面上双体问题中混沌轨迹的存在性，通过构造近碰撞轨迹并结合反可积极限方法，证明了混沌动力学的出现。
math.DS
