Introduction to Modern Dynamics: Chaos, Networks, Space and Time

The second edition of Introduction to Modern Dynamics: Chaos, Networks, Space and Time publishes this week (Novermber 18, 2019), available from Oxford University Press and Amazon.

Most physics majors will use modern dynamics in their careers: nonlinearity, chaos, network theory, econophysics, game theory, neural nets, geodesic geometry, among many others.

The first edition of Introduction to Modern Dynamics (IMD) was an upper-division junior-level mechanics textbook at the level of Thornton and Marion (Classical Dynamics of Particles and Systems) and Taylor (Classical Mechanics).  IMD helped lead an emerging trend in physics education to update the undergraduate physics curriculum.  Conventional junior-level mechanics courses emphasized Lagrangian and Hamiltonian physics, but notably missing from the classic subjects are modern dynamics topics that most physics majors will use in their careers: nonlinearity, chaos, network theory, econophysics, game theory, neural nets, geodesic geometry, among many others.  These are the topics at the forefront of physics that drive high-tech businesses and start-ups, which is where more than half of all physicists work. IMD introduced these modern topics to junior-level physics majors in an accessible form that allowed them to master the fundamentals to prepare them for the modern world.

The second edition (IMD2) continues that trend by expanding the chapters to include additional material and topics.  It rearranges several of the introductory chapters for improved logical flow and expands them to include key conventional topics that were missing in the first edition (e.g., Lagrange undetermined multipliers and expanded examples of Lagrangian applications).  It is also an opportunity to correct several typographical errors and other errata that students have identified over the past several years.  The second edition also has expanded homework problems.

The goal of IMD2 is to strengthen the sections on conventional topics (that students need to master to take their GREs) to make IMD2 attractive as a mainstream physics textbook for broader adoption at the junior level, while continuing the program of updating the topics and approaches that are relevant for the roles that physicists play in the 21st century.

(New Chapters and Sections highlighted in red.)

New Features in Second Edition:

Second Edition Chapters and Sections

Part 1 Geometric Mechanics

• Expanded development of Lagrangian dynamics

• Lagrange multipliers

• More examples of applications

• Connection to statistical mechanics through the virial theorem

• Greater emphasis on action-angle variables

• The key role of adiabatic invariants

Part 1 Geometric Mechanics

Chapter 1 Physics and Geometry

1.1 State space and dynamical flows

1.2 Coordinate representations

1.3 Coordinate transformation

1.4 Uniformly rotating frames

1.5 Rigid-body motion

Chapter 2 Lagrangian Mechanics

2.1 Calculus of variations

2.2 Lagrangian applications

2.3 Lagrange’s undetermined multipliers

2.4 Conservation laws

2.5 Central force motion

2.6 Virial Theorem

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

Part 2 Nonlinear Dynamics

• New section on non-autonomous dynamics

• Entire new chapter devoted to Hamiltonian mechanics

• Added importance to Chirikov standard map

• The important KAM theory of “constrained chaos” and solar system stability

• Degeneracy in Hamiltonian chaos

• A short overview of quantum chaos

• Rational resonances and the relation to KAM theory

• Synchronized chaos

Part 2 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 Fractals and strange attractors

Chapter 5 Hamiltonian Chaos

5.1 Perturbed Hamiltonian systems

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

Chapter 6 Coupled Oscillators and Synchronization

6.1 Coupled linear oscillators

6.2 Simple models of synchronization

6.3 Rational resonances

6.4 External synchronization

6.5 Synchronization of Chaos

Part 3 Complex Systems

• New emphasis on diffusion on networks

• Epidemic growth on networks

• A new section of game theory in the context of evolutionary dynamics

• A new section on general equilibrium theory in economics

Part 3 Complex Systems

Chapter 7 Network Dynamics

7.1 Network structures

7.2 Random network topologies

7.3 Synchronization on networks

7.4 Diffusion on networks

7.5 Epidemics on networks

Chapter 8 Evolutionary Dynamics

81 Population dynamics

8.2 Virus infection and immune deficiency

8.3 Replicator Dynamics

8.4 Quasi-species

8.5 Game theory and evolutionary stable solutions

Chapter 9 Neurodynamics and Neural Networks

9.1 Neuron structure and function

9.2 Neuron dynamics

9.3 Network nodes: artificial neurons

9.4 Neural network architectures

9.5 Hopfield neural network

9.6 Content-addressable (associative) memory

Chapter 10 Economic Dynamics

10.1 Microeconomics and equilibrium

10.2 Macroeconomics

10.3 Business cycles

10.4 Random walks and stock prices (optional)

Part 4 Relativity and Space–Time

• Relativistic trajectories

• Gravitational waves

Part 4 Relativity and Space–Time

Chapter 11 Metric Spaces and Geodesic Motion

11.1 Manifolds and metric tensors

11.2 Derivative of a tensor

11.3 Geodesic curves in configuration space

11.4 Geodesic motion

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)

Chapter 13 The General Theory of Relativity and Gravitation

13.1 Riemann curvature tensor

13.2 The Newtonian correspondence

13.3 Einstein’s field equations

13.4 Schwarzschild space–time

13.5 Kinematic consequences of gravity

13.6 The deflection of light by gravity

13.7 The precession of Mercury’s perihelion

13.8 Orbits near a black hole

13.9 Gravitational waves

Synopsis of 2nd Ed. Chapters

Chapter 1. Physics and Geometry (Sample Chapter)

This chapter has been rearranged relative to the 1st edition to provide a more logical flow of the overarching concepts of geometric mechanics that guide the subsequent chapters.  The central role of coordinate transformations is strengthened, as is the material on rigid-body motion with expanded examples.

Chapter 2. Lagrangian Mechanics (Sample Chapter)

Much of the structure and material is retained from the 1st edition while adding two important sections.  The section on applications of Lagrangian mechanics adds many direct examples of the use of Lagrange’s equations of motion.  An additional new section covers the important topic of Lagrange’s undetermined multipliers

Chapter 3. Hamiltonian Dynamics and Phase Space (Sample Chapter)

The importance of Hamiltonian systems and dynamics merits a stand-alone chapter.  The topics from the 1st edition are expanded in this new chapter, including a new section on adiabatic invariants that plays an important role in the development of quantum theory.  Some topics are de-emphasized from the 1st edition, such as general canonical transformations and the symplectic structure of phase space, although the specific transformation to action-angle coordinates is retained and amplified.

Chapter 4. Nonlinear Dynamics and Chaos

The first part of this chapter is retained from the 1st edition with numerous minor corrections and updates of figures.  The second part of the IMD 1st edition, treating Hamiltonian chaos, will be expanded into the new Chapter 5.

Chapter 5. Hamiltonian Chaos

This new stand-alone chapter expands on the last half of Chapter 3 of the IMD 1st edition.  The physical character of Hamiltonian chaos is substantially distinct from dissipative chaos that it deserves its own chapter.  It is also a central topic of interest for complex systems that are either conservative or that have integral invariants, such as our N-body solar system that played such an important role in the history of chaos theory beginning with Poincaré.  The new chapter highlights Poincaré’s homoclinic tangle, illustrated by the Chirikov Standard Map.  The Standard Map is an excellent introduction to KAM theory, which is one of the crowning achievements of the theory of dynamical systems by Komogorov, Arnold and Moser, connecting to deeper aspects of synchronization and rational resonances that drive the structure of systems as diverse as the rotation of the Moon and the rings of Saturn.  This is also a perfect lead-in to the next chapter on synchronization.  An optional section at the end of this chapter briefly discusses quantum chaos to show how Hamiltonian chaos can be extended into the quantum regime.

Chapter 6. Synchronization

This is an updated version of the IMD 1st ed. chapter.  It has a reduced initial section on coupled linear oscillators, retaining the key ideas about linear eigenmodes but removing some irrelevant details in the 1st edition.  A new section is added that defines and emphasizes the importance of quasi-periodicity.  A new section on the synchronization of chaotic oscillators is added.

Chapter 7. Network Dynamics

This chapter rearranges the structure of the chapter from the 1st edition, moving synchronization on networks earlier to connect from the previous chapter.  The section on diffusion and epidemics is moved to the back of the chapter and expanded in the 2nd edition into two separate sections on these topics, adding new material on discrete matrix approaches to continuous dynamics.

Chapter 8. Neurodynamics and Neural Networks

This chapter is retained from the 1st edition with numerous minor corrections and updates of figures.

Chapter 9. Evolutionary Dynamics

Two new sections are added to this chapter.  A section on game theory and evolutionary stable solutions introduces core concepts of evolutionary dynamics that merge well with the other topics of the chapter such as the pay-off matrix and replicator dynamics.  A new section on nearly neutral networks introduces new types of behavior that occur in high-dimensional spaces which are counter intuitive but important for understanding evolutionary drift.

Chapter 10.  Economic Dynamics

This chapter will be significantly updated relative to the 1st edition.  Most of the sections will be rewritten with improved examples and figures.  Three new sections will be added.  The 1st edition section on consumer market competition will be split into two new sections describing the Cournot duopoly and Pareto optimality in one section, and Walras’ Law and general equilibrium theory in another section.  The concept of the Pareto frontier in economics is becoming an important part of biophysical approaches to population dynamics.  In addition, new trends in economics are drawing from general equilibrium theory, first introduced by Walras in the nineteenth century, but now merging with modern ideas of fixed points and stable and unstable manifolds.  A third new section is added on econophysics, highlighting the distinctions that contrast economic dynamics (phase space dynamical approaches to economics) from the emerging field of econophysics (statistical mechanics approaches to economics).

Chapter 11. Metric Spaces and Geodesic Motion

 This chapter is retained from the 1st edition with several minor corrections and updates of figures.

Chapter 12. Relativistic Dynamics

This chapter is retained from the 1st edition with minor corrections and updates of figures.  More examples will be added, such as invariant mass reconstruction.  The connection between relativistic acceleration and Einstein’s equivalence principle will be strengthened.

Chapter 13. The General Theory of Relativity and Gravitation

This chapter is retained from the 1st edition with minor corrections and updates of figures.  A new section will derive the properties of gravitational waves, given the spectacular success of LIGO and the new field of gravitational astronomy.

Homework Problems:

All chapters will have expanded and updated homework problems.  Many of the homework problems from the 1st edition will remain, but the number of problems at the end of each chapter will be nearly doubled, while removing some of the less interesting or problematic problems.

Bibliography

D. D. Nolte, Introduction to Modern Dynamics: Chaos, Networks, Space and Time, 2nd Ed. (Oxford University Press, 2019)