The rotational motions of mechanical systems

Periodic motions of a reversible second-order mechanical system

“…34 In the case of the functions U and V, which are 2-periodic in v, the fixed set will be M * = {u, v, t : sin v = 0, sin t = 0}. 34 The functions U and V can also satisfy the conditions System (1.1) then has two fixed sets.…”

“…34 In the case of the functions U and V, which are 2-periodic in v, the fixed set will be M * = {u, v, t : sin v = 0, sin t = 0}. 34 The functions U and V can also satisfy the conditions System (1.1) then has two fixed sets. We will call the solution u = (t), v = (t) of system (1.1) a 2k-periodic motion, k ∈ N, if (1.2) Note that, in mechanical systems, the solution (1.2) describes both the oscillatory motion (m = 0) and the rotational motions (m = 0).…”

“…We will call the solution u = (t), v = (t) of system (1.1) a 2k-periodic motion, k ∈ N, if (1.2) Note that, in mechanical systems, the solution (1.2) describes both the oscillatory motion (m = 0) and the rotational motions (m = 0). 34 Here, a rotation can be a forward rotation (m > 0) or a reverse rotation (m < 0). The motion (1.2) in the problem of the rotation of a satellite in an elliptic orbit (the Beletskii problem 3 ) has been called a a generalized periodic solution (also, see Ref.…”

“…The corresponding necessary and sufficient conditions for existence of the motions (1.2) in an autonomous reversible system are well known. 34,37 The formulation of the problem of the continuation of periodic motions with respect to a parameter and the method of solving it belong to Poincaré; 38 the method of continuation with respect to a parameter, proposed for the first time for problems in celestial mechanics, 38 was subsequently developed in detail for analytical systems of general form. 35 Two cases arise when solving a problem: 39 (a) the structurally stable case, when a property of the system of having periodic motions is solely determined by the generating system (it is called structurally stable in the sense of periodic motions) and is independent of the actual form of the perturbations, (b) the non-structurally stable case when, to solve the problem, an examination of the perturbation is necessary.…”

“…34. The stability of the SPMs is then investigated on the basis of the variational equations (1.5) In the non-degenerate case, Lyapunov stability follows from the existence of pure imaginary roots if there are no resonances of up to the fourth order inclusive.…”