Alexei Tsygvintsev
We study the planar three-body problem and prove the absence of a complete set of complex meromorphic first integrals in a neighborhood of the Lagrangian solution.
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
We study the planar three-body problem and prove the absence of a complete set of complex meromorphic first integrals in a neighborhood of the Lagrangian solution. We use the Ziglin's method and study the monodromy group of the corresponding normal variational equations.
Let us consider a vector field $X$ meromorphic on a neighbourhood of an algebraic curve $\barΓ\subset \mathbb{P}^n$ such that $Γ$ is a particular solution of $X$. The vector field $X$ is $(l,n-l)$ integrable if it there exists $Y_1,\dots,Y_{l-1},X$ vector fields commuting pairwise, and $F_1,\dots,F_{n-l}$ common first integrals. The Ayoul-Zung Theorem gives necessary conditions in terms of Galois groups for meromorphic integrability of $X$ in a neighbourhood of $Γ$. Conversely, if these conditions are satisfied, we prove that if the first normal variational equation $NVE_1$ has a virtually dia…
We show that the main theorem of Morales--Ramis--Simo about Galoisian obstructions to meromorphic integrability of Hamiltonian systems can be naturally extended to the non-Hamiltonian case. Namely, if a dynamical system is meromorphically integrable in the non-Hamiltonian sense, then the differential Galois groups of the variational equations (of any order) along its solutions must be virtually Abelian
In this paper we explore the two dimensional system describing trapped ionic system in the quadrapole field with a superposition of rationally symmetric hexapole and octopole fields for meromorphic integrability. We use the Lyapunov's and Ziglin-Morales-Ramis classical methods for the proofs. The inaccuracies from the previous such paper have been removed.
We prove a meromorphic integrability condition at order 2 near a homothetic orbit for a meromorphic homogeneous potential of degree -1, which extend the Morales Ramis conditions of order 1. Conversely, we prove that if this criterion is satisfied, then the Galois group of second order variational equation is abelian and we compute explicitly the Galois group and the Picard-Vessiot extension.
Integrability of vector fields and meromorphic solutions
本文证明，若复射影流形上的叶状结构具有足够多由亚纯函数给出的积分曲线，则其限制在不变二维解析集上具有刘维尔型首次积分；特别地，$\mathbb{C}^3$ 上的有理向量场若解为亚纯函数，则存在二维不变解析集使其限制为刘维尔可积叶状结构。
We prove an integrability criterion and a partial integrability criterion for homogeneous potentials of degree -1 which are invariant by rotation. We then apply it to the proof of the meromorphic non-integrability of the n body problem with Newtonian interaction in the plane on a surface of equation $(H,C)=(H_0,C_0)$ with $(H_0,C_0) \neq (0,0)$ where C is the angular momentum and H the energy, in the case where the n masses are equal.
On the Meromorphic Integrability of the Critical Systems for Optimal Sums of Eigenvalues
本文提出并研究了一族由非线性常微分方程组成的临界系统，用于探索Sturm-Liouville算子前m个特征值和的最优界，并利用微分Galois理论对系统的亚纯可积性进行了完整分类。
Non-integrability of the Critical Systems for Optimal Sums of Eigenvalues of Sturm-Liouville Operators
本文利用微分伽罗瓦理论证明Sturm-Liouville算子特征值最优和对应的临界哈密顿系统除两个已知情形外均不可积，揭示了原问题的复杂性。
We prove that the minimum-time controlled Kepler problem is not Liouville integrable in the class of meromorphic functions, via the Moralès-Ramis theory.
We show that any dynamics on any planar set $S$ discrete in some domain $D$ can be realized by the postcritical dynamics of a function holomorphic in $D$, up to a small perturbation. A key step in the proof, and a result of independent interest, is that any planar domain $D$ can be equilaterally triangulated with triangles whose diameters $\rightarrow0$ (at any prescribed rate) near $\partial D$.
We study a necessary condition for the integrability of the polynomials fields in the plane by means of the differential Galois theory. More concretely, by means of the variational equations around a particular solution it is obtained a necessary condition for the existence of a rational first integral. The method is systematic starting with the first order variational equation. We illustrate this result with several families of examples. A key point is to check wether a suitable primitive is elementary or not. Using a theorem by Liouville, the problem is equivalent to the existence of a ratio…
In this work, we consider rational ordinary differential equations dy/dx = Q(x,y)/P(x,y), with Q(x,y) and P(x,y) coprime polynomials with real coefficients. We give a method to construct equations of this type for which a first integral can be expressed from two independent solutions of a second-order homogeneous linear differential equation. This first integral is, in general, given by a non Liouvillian function. We show that all the known families of quadratic systems with an irreducible invariant algebraic curve of arbitrarily high degree and without a rational first integral can be constru…
We prove that the motion of a triaxial Riemann ellipsoid of homogeneous liquid without angular momentum does not possess an additional first integral which is meromorphic in position, impulsions, and the elliptic functions which appear in the potential, and thus is not integrable. We prove moreover that this system is not integrable even on a fixed energy level hypersurface.
We prove the non integrability of the colinear $3$ and $4$ body problem, for any masses positive masses. To deal with resistant cases, we present strong integrability criterions for $3$ dimensional homogeneous potentials of degree $-1$, and prove that such cases cannot appear in the $4$ body problem. Following the same strategy, we present a simple proof of non integrability for the planar $n$ body problem. Eventually, we present some integrable cases of the $n$ body problem restricted to some invariant vector spaces.
We show that any dynamics on any discrete planar sequence $S$ can be realized by the postsingular dynamics of some transcendental meromorphic function, provided we allow for small perturbations of $S$. This work was influenced by an analogous result of DeMarco, Koch and McMullen for finite $S$ in the rational setting. The proof contains a method for constructing meromorphic functions with good control over both the postsingular set of $f$ and the geometry of $f$, using the Folding Theorem of Bishop and a classical fixpoint theorem of Tychonoff.
Algebraic integrability of planar polynomial vector fields by extension to Hirzebruch surfaces
本文通过将复平面多项式向量场扩展到Hirzebruch曲面来研究其代数可积性，利用由此产生的无穷向量场族在有理首次积分下的临界性强性质，推导出代数可积性的新必要条件并界定了通用不变曲线单项式的参数区域。
We prove an integrability criterion of order 3 for a homogeneous potential of degree -1 in the plane. Still, this criterion depends on some integer and it is impossible to apply it directly except for families of potentials whose eigenvalues are bounded. To address this issue, we use holonomic and asymptotic computations with error control of this criterion and apply it to the potential of the form V(r,θ)=r^{-1} h(\exp(iθ)) with h a polynomial of degree less than 3. We find then all meromorphically integrable potentials of this form.
We consider the Newtonian planar three--body problem with positive masses $m_1$, $m_2$, $m_3$. We prove that it does not have an additional first integral meromorphic in the complex neighborhood of the parabolic Lagrangian orbit besides three exceptional cases $ \sum m_i m_j/(\sum m_k)^2= 1/3$, $2^3/3^3$, $2/3^2$ where the linearized equations are shown to be partially integrable. This result completes the non-integrability analysis of the three-body problem started in our previous papers and based of the Morales-Ramis-Ziglin approach.
In this paper we study the integrability of the Hamiltonian system associated to the fourth Painlevé equation. We prove that one two parametric family of this Hamiltonian system is not integrable in the sense of the Liouville-Arnold theorem. Computing explicitly the Stokes matrices and the formal invariants of the second variational equations we deduce that the connected component of the unit element of the corresponding Galois grou is not Abelian. Thus the Morales-Ramis-Simó theory leads to a non-integrable result. Moreover, combining the new result with our previous one we establish that for…
Non-integrability of the problem of a rigid satellite in gravitational and magnetic fields
本文利用 Morales-Ruiz 与 Ramis 推广的 Ziglin 理论证明，除感生磁矩与主转动惯量满足特殊关系外，受引力和磁场作用的对称卫星旋转系统不存在额外实亚纯第一积分。
Non-integrability of the axisymmetric Bianchi IX cosmological model via Differential Galois Theory
本研究针对含物质和宇宙学常数的 Bianchi IX 各向异性宇宙模型，通过分析各向同性不变平面的二阶变分方程的微分伽罗瓦群，应用相关可积性定理，证明该模型无法通过亚纯函数积分。
Legendre Functions and the Non-Integrability of a Hamiltonian System
本文研究了一个二维Hamilton系统在六次齐次势场下的可微分可积性，通过Ziglin-Moralez-Ruiz-Ramis-Simo理论，将问题转化为线性微分方程的微分Galois群分析，利用Legendre方程简化了计算，并探讨了非零对数项的条件。
We consider the problem of characterizing, for certain natural number $m$, the local $\mathcal{C}^m$-non-integrability near elliptic fixed points of smooth planar measure preserving maps. Our criterion relates this non-integrability with the existence of some Lie Symmetries associated to the maps, together with the study of the finiteness of its periodic points. One of the steps in the proof uses the regularity of the period function on the whole period annulus for non-degenerate centers, question that we believe that is interesting by itself. The obtained criterion can be applied to prove the…
Non-integrability of n-centre billiards
本文研究受n-中心势影响的力学台球系统，证明了在至少两个中心且墙壁足够远时系统必然不可积并产生混沌，并在经典双中心问题中给出了保证混沌的墙壁几何条件。
扫码添加微信好友，提出您的宝贵建议 👇
💡 备注请填写：网站反馈
发表机构由 AI 从论文文本抽取，如发现错误，欢迎反馈修正建议。