Various aspects of 𝑛-dimensional rigid body dynamics

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“…This is the reason that the integration theory of constrained mechanical systems is less developed then for unconstrained ones. However, in some solvable nonholonomic systems with an invariant measure, the phase space is foliated by invariant tori, placing these systems together with integrable Hamiltonian systems (see [6,1,9,20,14,8]). …”

“…For a motion g(t) ∈ SO(n), the angular velocity and momentum (in body coordinates) are Ω(t) = g −1 · g(t) and M = J(Ω) respectively. J : so(n) → so(n) * is the inertia tensor and has the form: M = J(Ω) = IΩ + ΩI, where I is symmetric n×n matrix called mass tensor (see [8]). Here we identified so(n) and so(n) * by the Killing scalar product.…”

“…They are reduced equations of nonholonomic systems on Lie groups with left-invariant kinetic energy and left-invariant constraints (see [14,13]). Integrable examples can be found in [8,10,11]. Since in the three-dimensional case, only infinitesimal rotations in the planes e 1 ∧ e 3 and e 2 ∧ e 3 are allowed, Fedorov and Kozlov suggested that multidimensional analogue of Suslov's conditions can be define in the following way: only infinitesimal rotations in the planes e 1 ∧ e n , .…”

“…. , e n−1 ∧ e n , i.e., in the planes containing the vector e n are allowed [8]. Therefore, we have the following constraints imposed to the components of the angular velocity matrix:…”

“…If D is an eigen space of the inertia operator J then the solutions are simply Ω = const. In general, all trajectories lying on the energy ellipsoid E = 1 2 JΩ, Ω = h are double asymptotic: they tend to the diametrically opposed points w + and w − as t → ±∞ (see [8]). …”