Part 1 Geometric Mechanics; 1 Physics and Geometry; 1.1 Newton and geometry; 1.2 State space and flows; 1.3 Coordinate transformations; 1.4 Non-inertial transformations; 1.5 Uniformly rotating frames; 1.6 Rigid-body motion; 1.7 Summary; 1.8 Bibliography; 1.9 Homework exercises; 2 Hamiltonian Dynamics and Phase Space; 2.1 Hamilton's principle; 2.2 Conservation laws; 2.3 The Hamiltonian function; 2.4 Central force motion; 2.5 Phase space; 2.6 Integrable systems and action-angle variables; 2.7 Summary; 2.8 Bibliography; 2.9 Homework exercises.
Part 2 Nonlinear Dynamics3 Nonlinear Dynamics and Chaos; 3.1 One-variable dynamical systems; 3.2 Two-variable dynamical systems; 3.3 Discrete iterative maps; 3.4 Three-dimensional state space and chaos; 3.5 Fractals and strange attractors; 3.6 Hamiltonian chaos; 3.7 Summary and glossary; 3.8 Bibliography; 3.9 Homework exercises; 4 Coupled Oscillators and Synchronization; 4.1 Coupled linear oscillators; 4.2 Simple models of synchronization; 4.3 External synchronization of an autonomous phase oscillator; 4.4 External synchronization of a van der Pol oscillator.
4.5 Mutual synchronization of two autonomous oscillators4.6 Summary; 4.7 Bibliography; 4.8 Homework exercises; 5 Network Dynamics; 5.1 Network structures; 5.2 Random network topologies; 5.3 Diffusion and epidemics on networks; 5.4 Linear synchronization of identical oscillators; 5.5 Nonlinear synchronization of coupled phase oscillators on regular graphs; 5.6 Summary; 5.7 Bibliography; 5.8 Homework exercises; Part 3 Complex Systems; 6 Neurodynamics and Neural Networks; 6.1 Neuron structure and function; 6.2 Neuron dynamics; 6.3 Network nodes: artificial neurons.
6.4 Neural network architectures6.5 Hopfield neural network; 6.6 Content-addressable (associative) memory; 6.7 Summary; 6.8 Bibliography; 6.9 Homework exercises; 7 Evolutionary Dynamics; 7.1 Population dynamics; 7.2 Virus infection and immune deficiency; 7.3 The replicator equation; 7.4 The quasi-species equation; 7.5 The replicator-mutator equation; 7.6 Dynamics of finite numbers (optional); 7.7 Summary; 7.8 Bibliography; 7.9 Homework exercises; 8 Economic Dynamics; 8.1 Micro- and macroeconomics; 8.2 Supply and demand; 8.3 Business cycles; 8.4 Consumer market competition; 8.5 Macroeconomics.
8.6 Stochastic dynamics and stock prices (optional)8.7 Summary; 8.8 Bibliography; 8.9 Homework exercises; Part 4 Relativity and Space-Time; 9 Metric Spaces and Geodesic Motion; 9.1 Manifolds and metric tensors; 9.2 Reciprocal spaces in physics; 9.3 Derivative of a tensor; 9.4 Geodesic curves in configuration space; 9.5 Geodesic motion; 9.6 Summary; 9.7 Bibliography; 9.8 Homework exercises; 10 Relativistic Dynamics; 10.1 The special theory; 10.2 Lorentz transformations; 10.3 Metric structure of Minkowski space; 10.4 Relativistic dynamics; 10.5 Linearly accelerating frames (relativistic); 10.6 Summary; 10.7 Bibliography; 10.8 Homework exercises.
11 The general theory of relativity and gravitation; 11.1 Riemann curvature tensor; 11.2 The Newtonian correspondence; 11.3 Einstein's field equations; 11.4 Schwarzschild space-time; 11.5 Kinematic consequences of gravity; 11.6 The deflection of light by gravity; 11.7 Planetary orbits; 11.8 Orbits near a black hole; 11.9 Summary; 11.10 Bibliography; 11.11 Homework exercises.
