Abraham and Shaw: Dynamics: Classical Applications

by Carson Reynolds

Celestial mechanics, conservative mechanics and classical dissipative systems are the topic of Chapter 2. Pendula are first described (using Newton’s Formulation). The “eye” diagram presented in Chapter 3 of SICM is explicated in greater detail, and with slightly different vocabulary. With a frictionless system, we come to understand the phase space of trajectories drawn out. We see that centers, or vortex points become attractors or focal points when the system is dissipative. A variaiton of the pendulum incorporate magnets is explored to show a system with two attractors.

Buckling elastic columns are looked into next. The idealization used is a hinged connected to springs at a single joint. In the frictionless model, we have concentric rings representing greater amplitude oscillations. When friction is added, the center again becomes a (focal | rest | equalibrium | limit) point. Under heavier weights, the fricitonless system divides into two centers of oscillation. With friction, the whole mess develops into something that looks like a yin-yang. Two inserts with equal probability that lead to do different attractor states.

A similar model for percussion instruments is looked at next. The model is a single mass moving on a surface attached to a spring. Assuming Hook’s law (linear spring) and no friction, the phase portrait looks like concentric rings. Various distortions of this picture are the presented: soft springs (poor freq response) hard springs (better frequency response). Lastly, linear damping is explored. This again turns the concentric rings into a spiral toward an attractor. When the amount of friction is varied then the rate of spiraling increases.

A predator prey model is looked at next. Lotka and Volterra’s model describes static equilibria and periodic limit cycles that correspond to unchanging stable points and sustainable predator-prey oscillations respectively.