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A new musculoskeletal model for the lumbar spine is described in this paper. This model features a rigid pelvis and sacrum, the five lumbar vertebrae, and a rigid torso consisting of a lumped thoracic spine and ribcage. The motion of the individual lumbar vertebrae was defined as a fraction of the net lumbar movement about the three rotational degrees of freedom: flexion-extension lateral bending, and axial rotation. Additionally, the eight main muscle groups of the lumbar spine were incorporated using 238 muscle fascicles with prescriptions for the parameters in the Hill-type muscle models obtained with the help of an extensive literature survey. The features of the model include the abilities to predict joint reactions, muscle forces, and muscle activation patterns. To illustrate the capabilities of the model and validate its physiological similarity, the model's predictions for the moment arms of the muscles are shown for a range of flexion-extension motions of the lower back. The model uses the OpenSim platform and is freely available on https://www.simtk.org/home/lumbarspine to other spinal researchers interested in analyzing the kinematics of the spine. The model can also be integrated with existing OpenSim models to build more comprehensive models of the human body.

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The dynamics of disks which move with one point in contact with a fixed horizontal surface are investigated. Two cases are considered: one where the disk rolls without slipping and a second where the disk slides. A two-parameter family of integrable second order differential equations is established for both of these systems by exploiting classical integrability results. Using these families of dynamical systems, complete descriptions of the bifurcations and stability of the steady motions of the disks are presented. One of these families is also used to establish the existence of motions which result in the rolling disk colliding with the horizontal surface. These motions occur for arbitrarily large angular velocities and in the absence of dissipation. The paper closes with some comments on non-integrability.

A material momentum balance law is presented in this paper where it is also specialized for a variety of rod and string theories. The local form of the law is assumed to be identically satisfied, while the jump condition provides an extra equation which is often needed to solve problems involving the application of rod and string theories. The balance law is also related to several existing conservation laws for strings and rods, including Kelvin's circulation theorem. A novel identity for the singular sources at a discontinuity is also established.

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