Classical models of affinely-rigid bodies with "thickness" in degenerate dimension

On Classical Dynamics of Affinely-Rigid Bodies Subject to the Kirchhoff-Love Constraints

“…And finally, three main branches of special solutions (stationary ellipsoids) for our strongly nonlinear equations of motion are gathered in the form of Proposition 1. Additionally some remarks about the complementarity of the obtained results to those described in our previous work [6] are presented in the Summary.…”

“…It is interesting to note that the special solutions obtained for the polar decomposition case are conceptually different from those obtained for the two-polar one [6] because here the Green deformation tensor G = S 2 has a constant value (i.e., Ġ = 2S Ṡ = 0) contrary to the situation described in [6] when the Green deformation tensor G = Φ T Φ = U D 2 U −1 , as well as the Cauchy one C = Φ −1T Φ −1 = RD 2 R −1 , depended on time explicitly through the time dependence of U and R respectively, i.e.,…”

“…and performed the stationary rotations around their principal axes, whereas the deformation invariants λ, µ, ̺ had the constant values. So, if we additionally keep in mind that in [6] we obtained the stationary solutions only for the isotropic model J 1 = J 2 = J and here the general situation J 1 = J 2 is allowed, then we can compare the four (one in [6] and three here) studied cases according to the following scheme:…”

“…Then introducing some modelled potentials in [6] we obtained the Hamiltonian (total energy) and calculated the corresponding equations of motion for the isotropic case with the help of the Poisson brackets. In the present article we concentrate mainly on the alternative decomposition, i.e., the polar one.…”

“…In the case of the two-polar (singular value) decomposition (1) we obtained the abovementioned special solutions just putting the deformation invariants λ, µ, ̺ and the angular velocities ω, ϑ equal to some constant values [6]. But now, in the case of the polar decomposition (4), we see that the Green deformation tensor G, therefore the deformation matrix S, and the angular velocity ν of the L-top have to be constant [15], i.e.,…”