This is a textbook on classical Hamiltonian dynamics designed primarily for students commencing graduate studies in physics. The aim is to cover all essential topics in a relatively concise format, without sacrificing the intellectual coherence of the subject, or the conceptual precision which is the sine qua non of advanced education in physics.

Encouraged by my colleagues at New York University, I have taken it as a pedagogical challenge to create a textbook suitable for a twenty-first-century course of duration no more than one semester (at NYU, the material is covered in about two-thirds of a semester). To do so, I have chosen to limit the scope of the book in certain important ways. It is assumed that the student has already had a course in which Newtonian mechanics, in both F = ma and Lagrangian versions, has been systematically developed and applied to a standard array of soluble examples: the harmonic oscillator, the simple pendulum, the Kepler problem, small oscillations (normal modes), and rigid-body motion. In the present book, the Hamiltonian formulation in phase space is introduced at the outset and applied directly to the same familiar systems.

Topics usually found in more encyclopedic textbooks, but omitted from the present treatment, include dissipative systems, nonholonomic constraints, special and general theories of relativity, continuum mechanics, and classical field theory. A further choice I have made is to limit the use of advanced differential geometry.

In the preface to his book Statistical Mechanics Made Simple Professor Daniel Mattis writes:

This frank but discouraging admission suggests that thermodynamics may not be a course eagerly anticipated by many students – not even physics, chemistry or engineering majors – and at completion I would suppose that few are likely to claim it was an especially inspiring experience. With such open aversion, the often disappointing performance on GRE questions covering the subject should not be a surprise. As a teacher of the subject I have often conjectured on reasons for this lack of enthusiasm.

Apart from its subtlety and perceived difficulty, which are probably immutable, I venture to guess that one problem might be that most curricula resemble the thermodynamics of nearly a century ago.

Another might be that, unlike other areas of physics with their epigrammatic equations – Newton's, Maxwell's or Schrödinger's, which provide accessibility and direction – thermal physics seems to lack a comparable unifying principle. Students may therefore fail to see conceptual or methodological coherence and experience confusion instead.

With those assumptions I propose in this book alternatives which try to address the disappointing experience of Professor Mattis and undoubtedly others.

Thermodynamics, the set of rules and constraints governing interconversion and dissipation of energy in macroscopic systems, can be regarded as having begun with Carnot's (1824) pioneering paper on heat-engine efficiency.

The beginning

Thermodynamics has exceeded the scope and applicability of its utile origins in the industrial revolution to a far greater extent than other subjects of physics' classical era, such as mechanics and electromagnetism. Unquestionably this results from over a century of synergistic development with quantum mechanics, to which it has given and from which it has gained clarification, enhancement and relevance, earning for it a vital role in the modern development of physics as well as chemistry, biology, engineering, and even aspects of philosophy.

Introduction

Particles with half-integer angular momentum obey the Pauli exclusion principle (PEP) – a restriction that a non-degenerate single-particle quantum state can have occupation number of only 0 or 1. This restriction was announced by W. Pauli in 1924 for which, in 1945, he received the Nobel Prize in Physics. Soon after Pauli, the exclusion principle was generalized by P. Dirac and E. Fermi who – independently – integrated it into quantum mechanics. As a consequence half-integer spin particles are called Fermi–Dirac particles or fermions. PEP applies to electrons, protons, neutrons, neutrinos, quarks – and their antiparticles – as well as composite fermions such as He3 atoms. Thermodynamic properties of metals and semiconductors are largely determined by electron (fermion) behavior.

Introduction

For over 50 years the low-temperature liquid state of uncharged, spinless He4 was the only system in which a Bose–Einstein (BE) condensation was considered experimentally realized. In that case, cold (<2.19K) liquid He4 passes into an extraordinary phase of matter called a superfluid, in which the liquid's viscosity and entropy become zero.

With advances in atomic cooling (to ≈ 10–9 K) the number of Bose systems which demonstrably pass into a condensate has considerably increased. These include several isotopes of alkali gas atoms as well as fermionic atoms that pair into integerspin (boson) composites.

Although an ideal Bose gas does exhibit a low-temperature critical instability, the ideal BE gas theory is not, on its own, able to describe the BE condensate wave state. In order for a theory of bosons to account for a condensate wave state, interactions between the bosons must be included. Nevertheless, considerable interesting physics is contained in the ideal Bose gas model.