Coopersmith J. The Lazy Universe: An Introduction to the Principle of Least Action

Oxford: Oxford University Press, 2017. — 280 p.This is a rare book on a rare topic: it is about 'action' and the Principle of Least Action. A surprisingly well-kept secret, these ideas are at the heart of physical science and engineering. Physics is well known as being concerned with grand conservatory principles (e.g. the conservation of energy) but equally important is the optimization principle (such as getting somewhere in the shortest time or with the least resistance). The book explains: why an optimization principle underlies physics, what action is, what `the Hamiltonian' is, and how new insights into energy, space, and time arise. It assumes some background in the physical sciences, at the level of undergraduate science, but it is not a textbook. The requisite derivations and worked examples are given but may be skim-read if desired..
The author draws from Cornelius Lanczos's book "The Variational Principles of Mechanics" (1949 and 1970). Lanczos was a brilliant mathematician and educator, but his book was for a postgraduate audience. The present book is no mere copy with the difficult bits left out - it is original, and a popularization. It aims to explain ideas rather than achieve technical competence, and to show how Least Action leads into the whole of physics.Antecedents.
Mathematics and physics preliminaries: of hills and plains and other things.
The Principle of Virtual Work.
D’Alembert’s Principle.
Lagrangian Mechanics.
Hamiltonian Mechanics.
The whole of physics.
Final words.
Appendxs.
Newton’s Laws of Motion.
Portraits of the physicists.
Reversible displacements.
Worked examples in Lagrangian Mechanics.
Proof that T is a function of v².
Energy conservation and the homogeneity of time.
The method of Lagrange Multipliers.
Generalized Forces.
Hamilton’s Transformation, examples.
Demonstration that the pi s are independent coordinates.
Worked examples in Hamiltonian Mechanics.
Incompressibility of the phase fluid.
Energy conservation in extended phase space.
Link between the action, S, and the ‘circulation’.
Transformation equations linking p and q via S.
Infinitesimal canonical transformations.
Perpendicularity of wavefronts and rays.
Problems solved using the Hamilton-Jacobi Equation.
Quasi refractive index in mechanics.
Einstein’s link between Action and the de Broglie waves.
Bibliography and Further Reading.