### 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.