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Note on the integration of Euler’s equations of the dynamics of an \(n\)-dimensional rigid body. (Eine Bemerkung über die Integration der Eulerschen Gleichungen der Dynamik eines \(n\)-dimensionalen festen Körpers.) (Russian) Zbl 0343.70003
From the text: An interesting example of Euler’s equations in the group \(O(n)\) is provided by the equation of free rotation of an \(n\)-dimensional rigid body. In this case, \(A\Omega = J\cdot\Omega + \Omega\cdot J\), where \(J\) is a symmetric positive-definite matrix (inertia tensor), which can always be regarded as diagonal, and the Euler-Arnold equation can be rewritten as
\[ J\dot \Omega + \dot\Omega\cdot J = [J, n\Omega^2]. \tag{1} \]
Equation (2) with arbitrary \(n\) was first considered by S. Mishchenko [Funct. Anal. Appl. 4, 232–235 (1970); translation from Funkts. Anal. Prilozh. 4, No. 3, 73–77 (1970; Zbl 0241.22022)] who discovered a series of nontrivial quadratic integrals of (2) of the form \(C_s =\sum_{k=0}^s \operatorname{tr}(\Omega J^k\Omega J^{s-k+1})\) \((0\le s \le n- 2\), \(s\ne 1)\). The \(C_s\) are functionally independent, and were shown by Dikii [Funct. Anal. Appl. 6, 326–327 (1973); translation from Funkts. Anal. Prilozh. 6, No. 4, 83–84 (1972; Zbl 0288.58004)] to be involutive.
By Liouville’s theorem, there are sufficient Mishchenko integrals in the case \(n = 4\) for proving the complete integrability of Euler’s equations of a four-dimensional rigid body. But no effective solution of the problem, i.e., no description of the motion by explicit expressions, is available, even in the case \(n = 4\). We have, however:
Theorem. Given any \(n\), (2) has \(N(n) = \frac12 \left[\frac{n}2\right] + \frac{n(n-1)}{4}\) single-valued integrals of motion, and its general solution is expressible in terms of \(\theta\) functions of Riemann surfaces.
The proof is based on the following basic lemma.
Lemma. Euler’s equations (2) of the dynamics of an \(n\)-dimensional rigid body have, for any \(n\), a representation in Lax form in matrices, linearly dependent on arbitrary \(\lambda\in\mathbb C\) \[ \frac{d}{dt} (M+J^2\lambda) = [M+J^2\lambda, \Omega+J\lambda]. \]

70E15 Free motion of a rigid body
70H99 Hamiltonian and Lagrangian mechanics