Most classical mechanics books are written by physicists for physicists. They are fluent in the way physicists think — a lever "obviously" multiplies force, a rigid body "obviously" has six degrees of freedom — and they move quickly past the steps a mathematician would want justified. Michael Spivak's Physics for Mathematicians, Mechanics I exists because he was that mathematician: someone who had learned differential geometry to the bone and still could not follow why a physics textbook's argument about a pulley was supposed to be convincing. This post is for readers wondering whether the book is for them, what they need before starting, and how it sits next to the other books a mathematician might reach for.
Who the Book Is For
Spivak states his audience in the first lines of the preface: someone "trained in modern mathematics and inculcated with its general outlook." Not a student who uses mathematics as a tool, but one for whom precise definitions and honest proofs are the normal way to think. In practice that means the book fits three kinds of reader well:
- Mathematics graduate students and faculty who want to understand mechanics as mathematics — manifolds, cotangent bundles, symplectic forms — rather than as a bag of coordinate tricks.
- Mathematically strong physics students who found the standard sequence (Taylor, Goldstein, Landau–Lifshitz) unsatisfying precisely where it was hand-waving, and want the missing step before general relativity or gauge theory.
- Self-learners who already own Spivak's other books and want the same voice — patient, historically curious, unwilling to skip the hard "obvious" parts — applied to physics.
It is not a first course in physics, and it is not a fast one. A reader who wants to compute orbital periods by Friday should use a physicist's text and come back to this later.
What You Need First
Spivak names the prerequisite himself: the first two volumes of his A Comprehensive Introduction to Differential Geometry — manifolds, tangent bundles, differential forms, Riemannian metrics, and connections. He is explicit that he chose this not because he wrote them but because, in his view, the concepts of mechanics are simply best expressed in that language. Any equivalent grounding in differential geometry will serve; what matters is comfort with manifolds and forms, not familiarity with his particular books.
Beyond that, the book asks for mathematical maturity rather than physics background. It begins with Newton's laws and builds forward, so a reader with no prior mechanics can follow — provided they are willing to read slowly.
What the Book Covers
The book runs from first principles to the modern geometric formulation of mechanics in four parts, with two additions:
- Part I — The Foundations of Mechanics. Newtonian mechanics from the ground up: central forces, conservation laws, the one- and two-body problems, rigid bodies, constraints — together with the historical and philosophical questions about how the subject was actually built. This is where Spivak spends real time on Newton's Principia, on what Newton's arguments do and do not establish, and on the elementary problems (wheels, weights, ropes, pulleys) that physicists find natural and mathematicians find mysterious.
- Part II — Building on the Foundations. Oscillations, rigid-body motion, non-inertial frames and fictitious forces, and friction — including the Painlevé paradoxes, billiards, and tippe tops.
- Part III — Lagrangian Mechanics. Variational principles, Hamilton's principle, Noether's theorem, and small oscillations.
- Part IV — Hamiltonian Mechanics. The cotangent bundle, Hamilton–Jacobi theory, canonical transformations, symplectic manifolds, Liouville integrability, and action-angle variables, ending with adiabatic invariants and the Hannay angle.
- Interlude — Light. A chapter on the history and mathematics of optics and its long entanglement with mechanics, from antiquity to Hamilton.
- Supplement — A PDE Primer. Supporting material on partial differential equations for readers who need it.
How It Compares to Arnold
The book most often mentioned in the same breath is V. I. Arnold's Mathematical Methods of Classical Mechanics, and the comparison is fair: both are mechanics for mathematicians, both arrive at symplectic geometry, and both were written by mathematicians of the first rank. They are nonetheless very different books.
Arnold is compact and fast. It assumes the reader is already comfortable with the physics and wants the geometry made precise; it is dense with exercises and famous for the amount it packs into a page. Many readers find it the better reference once they know the subject, and the harder book to learn from.
Spivak is the opposite in temperament. It is long because it refuses to assume the physics. It works through the elementary problems that Arnold treats as known, it stops to ask what Newton meant and whether his arguments hold up, and it is written in the same conversational, occasionally exasperated voice as his Calculus. Readers who want that patience love it; readers who want to get to symplectic manifolds quickly find the first part slow. Both reactions are on the record, and both are correct about what the book is.
For most mathematicians coming to mechanics for the first time, the honest recommendation is Spivak first, Arnold second: Spivak to understand what the subject is and why the geometric formulation is the right one, Arnold afterward as the sharper, more complete reference. A physicist who already knows mechanics and wants the mathematics tightened up may prefer Arnold alone.
A Note on "Mechanics I"
The title's "I" reflects Spivak's plan for a series of physics books written to the same standard. Mechanics I was the only volume completed before his death in 2020, so it stands as a complete treatment of classical mechanics rather than the first half of one. Nothing in it is left hanging for a sequel.
Editions
The book is published only by Publish or Perish and is available as a Smyth-sewn cloth hardcover, printed and bound to the same standard as the current Calculus, and as an official digital edition (a Readium LCP PDF). The digital edition is not sold anywhere else.
Did you come to Spivak's mechanics from the mathematics side or the physics side — and which part of the book earned its keep for you? Tell us in the comments.
