A completely integrable case in the dynamics of a four-dimensional rigid body in a non-conservative field

Abstract

The paper continues the search for new integrable cases in the dynamics of a fourdimensional rigid body in R×so(4) in a non-conservative force field [1]–[3]. The author [1] has previously found a case of complete integrability for the equations of motion of a dynamically symmetric rigid body under the condition that I1 = I2 = I3 = I4. In the present paper, a different case of logically possible dynamical symmetry is investigated in full. Suppose that a four-dimensional rigid body Ξ with smooth boundary ∂Ξ is dynamically symmetric (the inertia operator can be written as diag{I1, I1, I3, I3} in some coordinate system Dx1x2x3x4). Hence, the two-dimensional planes Dx1x2 and Dx3x4 are planes of dynamical symmetry of the body. The dynamical part of the equations of motion on R× so(4) can be obtained from the non-conservative force (resistance) S = {S1, S2, 0, 0} and the coordinates of the point of its application N = (0, 0, x3N , x4N ) in the system Dx1x2x3x4 together with the relations S1 = σ sin γ, S1 = −S cos γ, γ = const, x3N = R cos β1, x4N = R sin β1 (γ is the angle in the Dx1x2-plane and β1 is the angle in the Dx3x4-plane). If the line CD (C being the centre of mass) lies in the Dx1x2-plane and the vector DC governs the location of the centre of mass, DC = {σ sin γ,−σ cos γ, 0, 0}, then the velocity vector vD of the point D can be written as: