The goal of this note is to give the explicit solution of Euler-Frahm equations for the Manakov four-dimensional case by elementary means. For this, we use some results from the original papers by Schottky [Sch 1891], Koetter [Koe 1892], Weber [We 1878], and Caspary [Ca 1893]. We hope that such approach will be useful for the solution of the problem of $n$-dimensional top.
The equations governing the flow of a viscous incompressible fluid around a rigid body that performs a prescribed time-periodic motion with constant axes of translation and rotation are investigated. Under the assumption that the period and the angular velocity of the prescribed rigid-body motion are compatible, and that the mean translational velocity is non-zero, existence of a time-periodic solution is established. The proof is based on an appropriate linearization, which is examined within a setting of absolutely convergent Fourier series. Since the corresponding resolvent problem is ill-p…
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.
Motion of a Rigid Body in a Special Lorentz Gas: Loss of Memory Effect
对刚体在一种特殊的洛仑兹气体中的直线运动进行了数学分析。刚体根据牛顿方程克服气体阻力运动。气体模型是由气体分子和背景障碍物组成的一种特殊的洛仑兹气体，它是在~（Tsuji and Aoki：J. Stat.物理\textbf{146}，620- 645，2012）。镜面边界条件施加在所得的动力学方程。这项研究补充了Tsuji和Aoki的数值研究-虽然本文的设置与他们略有不同，但定性地证明了相同的渐近行为：如果障碍物发生热运动，则刚体的速度$V(t)$按指数衰减;如果障碍物不动，则速度$V(t)$以独立于空间维度的速率$t^{-5}$代数衰减。这证明了分子与背景障碍物的相互作用破坏了由于共振而产生的记忆效应的想法。
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.
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…
