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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 $$\lambda\in\mathbb C$$ $\frac{d}{dt} (M+J^2\lambda) = [M+J^2\lambda, \Omega+J\lambda].$

##### MSC:
 70E15 Free motion of a rigid body 70H99 Hamiltonian and Lagrangian mechanics
##### Keywords:
Euler-Arnold equation