From Classical Mechanics to Complex Systems

By David D. Nolte

Part I Geometric Mechanics

Chapter 1.  Physics and Geometry

            1.1 State space and dynamical flows

            1.2 Coordinate transformation

            1.3 Uniformly rotating frames

            1.4 Rigid-bodies

            1.5 Euler’s Equations

            1.6 Summary

            1.7 Bibliography

            1.8 Homework problems

Chapter 2.  Lagrangian Mechanics

            2.1 Calculus of variations

            2.2 Lagrangian applications

            2.3 Dynamics with Constraints

            2.4 Conservation laws

            2.5 Central force motion

            2.6 Summary

            2.7 Bibliography

            2.8 Homework problems

Chapter 3.  Hamiltonian Dynamics and Phase Space

            3.1 The Hamiltonian function

            3.2 Phase space

            3.3 Integrable systems and action-angle variables

            3.4 Adiabatic invariants

            3.5 Many-body Physics

            3.6 Summary

            3.7 Bibliography

            3.8 Homework problems

Part II Nonlinear Dynamics

Chapter 4.  Nonlinear Dynamics and Chaos

            4.1 One-variable dynamical systems

            4.2 Two-variable dynamical systems

            4.3 Limit cycles

            4.4 Discrete iterative maps

            4.5 Three-dimensional state space and chaos

            4.6 Non-autonomous (driven) flows

            4.7 Summary and glossary

            4.8 Bibliography

            4.9 Homework problems

Chapter 5.  Hamiltonian Chaos

            5.1 Perturbed Hamiltonian systems and separatrix chaos

            5.2 Nonintegrable Hamiltonian systems

            5.3 The Chirikov Standard Map

            5.4 KAM theory

            5.5 Degeneracy and the web map

            5.6 Quantum chaos [optional]

            5.7 Summary

            5.8 Bibliography

            5.9 Homework problems

Chapter 6.  Stochastic Dynamics

            6.1 Flipping Coins and Random Walks

            6.2 probabilities and Distributions

            6.3 Langevin Dynamics

            6.4 Stochastic Chaos

            6.5 Summary

            6.6 Bibliography

            6.7 Homework problems

Part III Complex Systems

Chapter 7.  Coupled Oscillators and Synchronization

            7.1 Simple models of synchronization

            7.2 Rational resonances

            7.3 External synchronization

            7.4 Synchronization of chaos

            7.5 Summary

            7.6 Bibliography

            7.7 Homework problems

Chapter 8. Dynamics on Network

            8.1 Network structures

            8.2 Random network topologies

            8.3 Network Synchronization

            8.4 Mean-field Theory

            8.5 Percolation Through Networks

            8.6 Diffusion on networks

            8.7 Network Growth and Decay

            8.8 Summary

            8.9 Bibliography

            8.10 Homework problems

Chapter 9.  Evolutionary Dynamics

            9.1 Population dynamics

            9.2 Viral infection

            9.3 Replicator dynamics

            9.4 Quasispecies

            9.5 Summary

            9.6 Bibliography

            9.7 Homework problems

Chapter 10.  Neurodynamics and Neural Networks

            10.1 Neuron structure and function

            10.2 Neuron dynamics

            10.3 Network nodes: artificial neurons

            10.4 Deep learning

            10.5 Summary

            10.6 Bibliography

            10.7 Homework problems

Part IV Relativity and Space-Time

Chapter 11.  Metric Spaces and Geodesic Motion

            11.1 Manifolds

            11.2 Derivative of a tensor

            11.3 Geodesic curves in configuration space

            11.4 Geodesic motion

            11.5 Summary

            11.6 Bibliography

            11.7 Homework problems

Chapter 12.  Relativistic Dynamics

            12.1 The special theory

            12.2 Lorentz transformations

            12.3 Metric structure of Minkowski space

            12.4 Relativistic trajectories

            12.5 Relativistic dynamics

            12.6 Linearly accelerating frames (relativistic)

            12.7 Summary

            12.8 Bibliography

            12.9 Homework problems

Chapter 13.  The General Theory of Relativity and Gravitation

            13.1 The Equivalence Principle

            13.2 Warped Spacetime

            13.3 the Deflection of Light by Gravity

            13.4 Schwarzschild Dynamics

            13.5 Summary

            13.6 Bibliography

            13.7 Homework problems