The dynamics of coupled planar rigid bodies. II. Bifurcations,periodic solutions,and chaos

We give a complete bifurcation and stability analysis for the relative equilibria of the dynamics of three coupled planar rigid bodies. We also use the equivariant Weinstein-Moser theorem to show the existence of two periodic orbits distinguished by symmetry type near the stable equilibrium. Finally we prove that the dynamics is chaotic in the sense of Poincaré-Birkhoff-Smale horseshoes using the version of Melnikov's method suitable for systems with symmetry due to Holmes and Marsden.     

Hamiltonian structures and stability for rigid bodies with flexible attachments

The dynamics of a rigid body with flexible attachments is studied. A general framework for problems of this type is established in the context of Poisson manifolds and reduction. A simple model for a rigid body with an attached linear extensible shear beam is worked out for illustration. Second, the Energy-Casimir method for proving nonlinear stability is recalled and specific stability criteria for our model example are worked out. The Poisson structure and stability results take into account vibrations of the string, rotations of the rigid body, their coupling at the point of attachment, and centrifugal and Coriolis forces.     

A novel Sr_{3}Pb_{6}Ce_{2}Ti_{12}O_{36} ferroelectric thin film grown by pulsed laser ablation

Complex perovskite oxide ferroelectric thin films are of great technological interest because of their high dielectric constant and large tunability. In this paper, we report the structural and electrical properties of Sr \(_{3}\) Pb \(_{6}\) Ce \(_{2}\) Ti \(_{12}\) O \(_{36}\) (SPCTO) thin films grown by pulsed laser deposition. The role of oxygen pressure and substrate temperature on the microstructure, dielectric properties and leakage current mechanism of SPCTO thin films was investigated. Strong oxygen partial pressure dependence on the microcrystalline properties and leakage current conduction mechanism was observed. Both Raman spectra and C-V characteristics show a ferroelectric phase rather than paraelectric phase for the deposited thin films. Investigations on the leakage current showed that SPCTO thin films deposited at different oxygen pressure have different dominant conduction mechanism at various electric fields. The low field conduction mechanism is governed by Ohmic and space charge limited conduction mechanisms, whereas at high fields, the conduction process is dominated by Schottky emission mechanism. The dielectric constant as well as the tunability is found to increase with increase in the crystallite size.     

Gyroscopic control and stabilization

Summary In this paper, we consider the geometry of gyroscopic systems with symmetry, starting from an intrinsic Lagrangian viewpoint. We note that natural mechanical systems with exogenous forces can be transformed into gyroscopic systems, when the forces are determined by a suitable class of feedback laws. To assess the stability of relative equilibria in the resultant feedback systems, we extend the energy-momentum block-diagonalization theorem of Simo, Lewis, Posbergh, and Marsden to gyroscopic systems with symmetry. We illustrate the main ideas by a key example of two coupled rigid bodies with internal rotors. The energy-momentum method yields computationally tractable stability criteria in this and other examples.     

Novel silaproline (Sip)-incorporated close structural mimics of potent antidepressant peptide drug rapastinel (GLYX-13)

Rapastinel (GLYX-13) is a C-amidated tetrapeptide drug under clinical development for adjunctive treatment of major depressive disorder (MDD). Rapastinel features two consecutive proline residues centered at the peptide sequence (Thr-Pro-Pro-Thr-NH₂), which are detrimental to its biological activity. In this communication, we report the synthesis of very close structural analogues of rapastinel comprising silaproline (Sip) as proline surrogate. By virtue of its enhanced lipophilicity and metabolic stability, Sip introduction in the native rapastinel sequence is expected to improve its pharmacokinetic profiles.     

The Euler-Poincaré equations and double bracket dissipation

This paper studies the perturbation of a Lie-Poisson (or, equivalently an Euler-Poincaré) system by a special dissipation term that has Brockett's double bracket form. We show that a formally unstable equilibrium of the unperturbed system becomes a spectrally and hence nonlinearly unstable equilibrium after the perturbation is added. We also investigate the geometry of this dissipation mechanism and its relation to Rayleigh dissipation functions. This work complements our earlier work (Bloch, Krishnaprasad, Marsden and Ratiu [1991, 1994]) in which we studied the corresponding problem for systems with symmetry with the dissipation added to the internal variables; here it is added directly to the group or Lie algebra variables. The mechanisms discussed here include a number of interesting examples of physical interest such as the Landau-Lifschitz equations for ferromagnetism, certain models for dissipative rigid body dynamics and geophysical fluids, and certain relative equilibria in plasma physics and stellar dynamics.Research partially supported by the National Science Foundation PYI grant DMS-91-57556, and AFOSR grant F49620-93-1-0037.Research partially supported by the AFOSR University Research Initiative Program under grants AFOSR-87-0073 and AFOSR-90-0105 and by the National Science Foundation's Engineering Research Centers Program NSFD CDR 8803012.Research partially supported by, DOE contract DE-FG03-92ER-25129, a Fairchild Fellowship at Caltech, and the Fields Institute for Research in the Mathematical Sciences.Research partially supported by NSF Grant DMS 91-42613, DOE contract DE-FG03-92ER-25129, the Fields Institute, the Erwin Schrödinger Institute, and the Miller Institute of the University of California.