Mathematical Methods of Classical Mechanics (2nd Edition)
by V. I. Arnold · Springer
The book that makes Hamiltonian mechanics a branch of mathematics rather than a physics recipe. Graduate text recasting mechanics in the language of manifolds, differential forms and symplectic geometry, then developing Hamiltonian systems, canonical transformations, integrable systems, action-angle variables and KAM perturbation theory with complete proofs. Springer Graduate Texts in Mathematics 60.
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More resources on Hamiltonian Mechanics
Wolfram MathWorld
MathWorld is an online mathematics encyclopedia from Wolfram Research offering detailed, browsable articles on topics across the math spectrum, including algebra, geometry, calculus, and number theory. Each entry includes definitions, theorems, formulas, diagrams, worked examples, and links to further reading.
Classical Mechanics III (MIT 8.09)
Lagrangian and Hamiltonian mechanics, constraints, rigid body dynamics, vibrations, central forces, Hamilton-Jacobi theory, action-angle variables and perturbation theory, with introductions to fluid mechanics, turbulence and chaos. Includes about 200 pages of lecture notes and problem sets.
Classical Mechanics (3rd Edition)
The standard graduate course text. Chapters eight through ten build Hamilton's equations, canonical transformations, Poisson brackets and Hamilton-Jacobi theory, closing with the wave-mechanics analogy that leads into quantum theory, backed by extensive worked problems.
Structure and Interpretation of Classical Mechanics (2nd Edition)
Sussman and Wisdom express every mechanics derivation as executable Scheme programs, so ambiguous notation cannot hide. Covers Lagrangian mechanics, Hamilton's equations, phase space, canonical transformations, Hamilton-Jacobi theory and Lie-transform perturbation methods. MIT Press hosts the complete second edition free.
The Theoretical Minimum: Classical Mechanics
Ten recorded Stanford lectures by Leonard Susskind building from Newton's laws to phase space, the principle of least action, Hamilton's equations, Poisson brackets and Liouville's theorem, deriving conservation laws from symmetries throughout. Videos are free on YouTube. The lowest-friction on-ramp to the Hamiltonian formulation for someone without a graduate background: Susskind derives Hamilton's equations and Poisson brackets specifically to set up the jump to quantum mechanics.
David Tong: Lectures on Classical Dynamics
Cambridge Part II lecture notes, roughly 130 pages, moving from Newtonian dynamics and rigid bodies into the Lagrangian and Hamiltonian formalisms: phase space, Liouville's theorem, Poisson brackets, canonical transformations and Hamilton-Jacobi theory. Chapter PDFs and problem sheets included.