A Note on the Rotationally Symmetric SO(4) Euler Rigid Body
Gregorio Falqui
本文考虑SO（4）欧拉刚体的两个“惯性矩”重合。我们从双哈密顿几何的观点来研究它。我们展示了如何通过分离变量的方法代数积分它。
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.
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
A unified formulation of rigid body dynamics based on Gauss principle is proposed. The Lagrange, Kirchhoff and Newton-Euler equations are seen to arise from different choices of the quasicoordinates in the velocity space. The group-theoretical aspects of the method are discussed.
On tensor invariants for integrable cases of Euler, Lagrange and Kovalevskaya rigid body motion
本文研究了 Euler、Lagrange 和 Kovalevskaya 刚体运动可积情形的全局张量不变量，通过将含三次项的张量场代入不变性方程并用计算机代数系统求解，基于 Poincaré-Cartan 理论探讨了利用这些不变几何结构研究动力学的问题。
The classical Euler--Poinsot case of the rigid body dynamics admits a class of simple but non-trivial integrable generalizations, which modify the Poisson equations describing the motion of the body in space. These generalizations possess first integrals which are polynomial in the angular momenta. We consider the modified Poisson equations as a system of linear equations with elliptic coefficients and show that all the solutions of it are single-valued. By using the vector generalization of the Picard theorem, we derive the solutions explicitly in terms of sigma functions of the corresponding…
We prove that one cannot construct, for arbitrary initial data, global-in-time physical classical solutions to Euler's equations of continuum rigid body mechanics when the constituent rigid bodies are not perfect spheres. By 'physical' solutions, we mean those that conserve the total linear momentum, angular momentum and kinetic energy of any given initial datum. The reason for absence of classical solutions is due to the non-existence of velocity scattering maps which resolve a collision between two non-spherical rigid bodies in such a way that (i) they do not interpenetrate, and (ii) total l…
It is a classical result of Euler that the rotation of a torque-free three-dimensional rigid body about the short or the long axis is stable, whereas the rotation about the middle axis is unstable. This result is generalized to the case of a multidimensional body.
We show that a symplectic reduction of the symmetric representation of the generalized $n$-dimensional rigid body equations yields the $n$-dimensional Euler equation. This result provides an alternative to the more elaborate relationship between these equations established by Bloch, Crouch, Marsden, and Ratiu. Specifically, we exploit the inherent $\mathsf{Sp}(2n,\mathbb{R})$-symmetry in the symmetric representation to present its relationship with the Euler equation via symplectic reduction facilitated by the dual pair recently developed by Skerritt and Vizman.
Geodesic motion on the symplectic leaf of $SO(3)$ with distorted $e(3)$ algebra and Liouville integrability of a free rigid body
该研究计算了$SO(3)$流形对应辛叶上不依赖惯性张量的Poisson结构的显式形式，确认其异于$e(3)$代数，并用所得Poisson括号证明自由刚体的刘维尔可积性，给出Euler-Poisson方程的通解形式。
Rigid body with rotors is a widespread mechanical system modeled after the direct product $SO(3)\times S^1\times S^1\times S^1$, which under mild assumptions is the symmetry group of the system. In this paper, the authors present and compare different Lagrangian reduction procedures: Euler-Poincaré reduction by the whole group and reduction by stages in different orders or using different connections. The exposition keeps track of the equivalence of equations as well as corresponding conservation laws.
General solution to Euler-Poisson equations of a free symmetric body by direct summation of power series
本文利用运动积分并直接对幂级数逐项求和，以初等函数给出自由对称刚体 Euler–Poisson 方程的闭式通解，避免使用 Euler 角参数化。
The Euler top describes a free rotation of a rigid body about its center of mass and provides an important example of a completely integrable system. A salient feature of its first integrals is that, up to a reparametrization of time, they uniquely determine the dynamical equations themselves. In this note, this property is used to construct integrable deformations of the Euler top.
Heavy rigid body with a gyroscope in $\mathbb R^n$
基于重刚体系统的多维可积推广（欧拉陀螺、拉格朗日陀螺、拉格朗日双陀螺及全对称情形），为每个系统添加陀螺仪，给出多项式矩阵Lax表示并证明Liouville可积性。
General solution to the Euler-Poisson equations of a free Lagrange top directly for the rotation matrix
本文针对自由拉格朗日陀螺，直接求解欧拉-泊松方程得到旋转矩阵的一般解析解，并与庞索运动图像比较。
Motion of four-dimensional rigid body around a fixed point: an elementary approach. I
本文的目的是用初等方法给出Manakov四维情形下Euler-Frahm方程的显式解。为此，我们使用了Schottky [Sch 1891]、Koetter [Koe 1892]、Weber [We 1878]和Caspary [Ca 1893]原始论文中的一些结果。我们希望这种方法对解决$n$ -维顶点问题有一定的帮助。
Discrete Spinning Tops -- Difference equations for Euler, Lagrange, and Kowalevski tops
本文研究 Euler、Lagrange 和 Kowalevski 陀螺等可积刚体模型的时间离散化方法，并为 Kowalevski 陀螺提出一种能精确保持 $\oldsymbolγ^2=1$ 和 Kowalevski 积分守恒的新离散格式，数值实验验证有效。
We study global controllability of 'rotating' multidimensional rigid body (MRB) controlled by application of few torques. Study by methods of geometric control requires analysis of algebraic structure introduced by the quadratic term of Euler-Frahm equation. We discuss problems, which arise in the course of this analysis, and establish several global controllability criteria for damped and non damped cases.
The Lie-Trotter integrator in the dynamics of the symmetric free rigid body
研究对称自由刚体动力学中李-特罗特积分器的性质，证明其为泊松积分器，并提出重要结论。
An L-A pair for the Apel'rot system and a new integrable case for the Euler-Poisson equations on so(4)xso(4)
我们提出了一个L-A对的Apel'rot的情况下，一个沉重的刚性三维机构。利用它我们给出了一个代数几何积分过程。推广这个L-A对，我们得到了四维Euler-Poisson方程的一个新的完全可积情形。给出了对合积分的显式公式。这个系统是一个反例一个著名的Ratiu定理。证明了这个分类定理的一个修正版本。
We equip the regular Fréchet Lie group of invertible, odd-class, classical pseudodifferential operators $Cl^{0,*}_{odd}(M,E)$ -- in which $M$ is a compact smooth manifold and $E$ a (complex) vector bundle over $M$ -- with pseudo-Riemannian metrics, and we use these metrics to introduce a large class of rigid body equations. We adapt to our infinite-dimensional setting Manakov's classical observation on the integrability of Euler's equations for the rigid body, and we show that our equations can be written in Lax form (with parameter) and that they admit an infinite number of integrals of motio…
Optimal Mass Transport over the Euler Equation
研究在Euler方程角速度动力学下联合状态概率分布的有限时域最优导引问题，将其视为具有双线性先验漂移的最优质量传输实例，推导了静态与动态版本并给出最优控制器的解析与数值结果。
Lorenz integrable system moves à la Poinsot
本文导出了一个变换，它把Lorenz可积系统转化为著名的有自由力矩的刚体带固定点的欧拉方程，即著名的Poinsot运动。证明是基于李群分析应用到两个三阶常微分方程承认相同的二维李对称代数。利用平面二维对称代数的李分类。若对任意参数的Lorenz系统作同样的变换，则可得到刚体受一个依赖于时间和角速度的扭转作用时的Euler方程。该系统的数值解产生了一个三维图像，看起来像一个“龙卷风”，其横截面具有蝴蝶形状。因此，洛伦兹的{ \em蝴蝶}已转化为{ \em龙卷风}。
Equilibrium measures for higher dimensional rotationally symmetric Riesz gases
研究高维里斯气体在径向对称外场下的平衡测度，确定支持于单位球的约束势，并通过幂级数构造外势，应用超几何函数证明关键恒等式，推导外势与高斯超几何函数的关系，以及纯幂型外势下的平衡测度。
We consider three-dimensional elastic frames constructed out of Euler--Bernoulli beams and describe a simple process of generating joint conditions out of the geometric description of the frame. The corresponding differential operator is shown to be self-adjoint. In the special case of planar frames, the operator decomposes into a direct sum of two operators, one coupling out-of-plane displacement to angular (torsional) displacement and the other coupling in-plane displacement with axial displacement (compression). Detailed analysis of two examples is presented. We actively exploit the symmetr…
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。