Contents -- 1 Geometric and kinematic foundations of Lagrangian mechanics -- 1.1 Curves in the plane -- 1.2 Length of a curve and natural parametrisation -- 1.3 Tangent vector, normal vector and curvature of plane curves -- 1.4 Curves in R[sup(3)] -- 1.5 Vector fields and integral curves -- 1.6 Surfaces -- 1.7 Differentiable Riemannian manifolds -- 1.8 Actions of groups and tori -- 1.9 Constrained systems and Lagrangian coordinates -- 1.10 Holonomic systems -- 1.11 Phase space -- 1.12 Accelerations of a holonomic system -- 1.13 Problems -- 1.14 Additional remarks and bibliographical notes -- 1.15 Additional solved problems -- 2 Dynamics: general laws and the dynamics of a point particle -- 2.1 Revision and comments on the axioms of classical mechanics -- 2.2 The Galilean relativity principle and interaction forces -- 2.3 Work and conservative fields -- 2.4 The dynamics of a point constrained by smooth holonomic constraints -- 2.5 Constraints with friction -- 2.6 Point particle subject to unilateral constraints -- 2.7 Additional remarks and bibliographical notes -- 2.8 Additional solved problems -- 3 One-dimensional motion -- 3.1 Introduction -- 3.2 Analysis of motion due to a positional force -- 3.3 The simple pendulum -- 3.4 Phase plane and equilibrium -- 3.5 Damped oscillations, forced oscillations. Resonance -- 3.6 Beats -- 3.7 Problems -- 3.8 Additional remarks and bibliographical notes -- 3.9 Additional solved problems -- 4 The dynamics of discrete systems. Lagrangian formalism -- 4.1 Cardinal equations -- 4.2 Holonomic systems with smooth constraints -- 4.3 Lagrange's equations -- 4.4 Determination of constraint reactions. Constraints with friction -- 4.5 Conservative systems. Lagrangian function -- 4.6 The equilibrium of holonomic systems with smooth constraints.

4.7 Generalised potentials. Lagrangian of an electric charge in an electromagnetic field -- 4.8 Motion of a charge in a constant electric or magnetic field -- 4.9 Symmetries and conservation laws. Noether's theorem -- 4.10 Equilibrium, stability and small oscillations -- 4.11 Lyapunov functions -- 4.12 Problems -- 4.13 Additional remarks and bibliographical notes -- 4.14 Additional solved problems -- 5 Motion in a central field -- 5.1 Orbits in a central field -- 5.2 Kepler's problem -- 5.3 Potentials admitting closed orbits -- 5.4 Kepler's equation -- 5.5 The Lagrange formula -- 5.6 The two-body problem -- 5.7 The n-body problem -- 5.8 Problems -- 5.9 Additional remarks and bibliographical notes -- 5.10 Additional solved problems -- 6 Rigid bodies: geometry and kinematics -- 6.1 Geometric properties. The Euler angles -- 6.2 The kinematics of rigid bodies. The fundamental formula -- 6.3 Instantaneous axis of motion -- 6.4 Phase space of precessions -- 6.5 Relative kinematics -- 6.6 Relative dynamics -- 6.7 Ruled surfaces in a rigid motion -- 6.8 Problems -- 6.9 Additional solved problems -- 7 The mechanics of rigid bodies: dynamics -- 7.1 Preliminaries: the geometry of masses -- 7.2 Ellipsoid and principal axes of inertia -- 7.3 Homography of inertia -- 7.4 Relevant quantities in the dynamics of rigid bodies -- 7.5 Dynamics of free systems -- 7.6 The dynamics of constrained rigid bodies -- 7.7 The Euler equations for precessions -- 7.8 Precessions by inertia -- 7.9 Permanent rotations -- 7.10 Integration of Euler equations -- 7.11 Gyroscopic precessions -- 7.12 Precessions of a heavy gyroscope (spinning top) -- 7.13 Rotations -- 7.14 Problems -- 7.15 Additional solved problems -- 8 Analytical mechanics: Hamiltonian formalism -- 8.1 Legendre transformations -- 8.2 The Hamiltonian -- 8.3 Hamilton's equations -- 8.4 Liouville's theorem.

8.5 Poincaré recursion theorem -- 8.6 Problems -- 8.7 Additional remarks and bibliographical notes -- 8.8 Additional solved problems -- 9 Analytical mechanics: variational principles -- 9.1 Introduction to the variational problems of mechanics -- 9.2 The Euler equations for stationary functionals -- 9.3 Hamilton's variational principle: Lagrangian form -- 9.4 Hamilton's variational principle: Hamiltonian form -- 9.5 Principle of the stationary action -- 9.6 The Jacobi metric -- 9.7 Problems -- 9.8 Additional remarks and bibliographical notes -- 9.9 Additional solved problems -- 10 Analytical mechanics: canonical formalism -- 10.1 Symplectic structure of the Hamiltonian phase space -- 10.2 Canonical and completely canonical transformations -- 10.3 The Poincaré-Cartan integral invariant. The Lie condition -- 10.4 Generating functions -- 10.5 Poisson brackets -- 10.6 Lie derivatives and commutators -- 10.7 Symplectic rectification -- 10.8 Infinitesimal and near-to-identity canonical transformations. Lie series -- 10.9 Symmetries and first integrals -- 10.10 Integral invariants -- 10.11 Symplectic manifolds and Hamiltonian dynamical systems -- 10.12 Problems -- 10.13 Additional remarks and bibliographical notes -- 10.14 Additional solved problems -- 11 Analytic mechanics: Hamilton-Jacobi theory and integrability -- 11.1 The Hamilton-Jacobi equation -- 11.2 Separation of variables for the Hamilton-Jacobi equation -- 11.3 Integrable systems with one degree of freedom: action-angle variables -- 11.4 Integrability by quadratures. Liouville's theorem -- 11.5 Invariant l-dimensional tori. The theorem of Arnol'd -- 11.6 Integrable systems with several degrees of freedom: action-angle variables -- 11.7 Quasi-periodic motions and functions -- 11.8 Action-angle variables for the Kepler problem. Canonical elements, Delaunay and Poincaré variables.

11.9 Wave interpretation of mechanics -- 11.10 Problems -- 11.11 Additional remarks and bibliographical notes -- 11.12 Additional solved problems -- 12 Analytical mechanics: canonical perturbation theory -- 12.1 Introduction to canonical perturbation theory -- 12.2 Time periodic perturbations of one-dimensional uniform motions -- 12.3 The equation D[sub(ω)]u = v. Conclusion of the previous analysis -- 12.4 Discussion of the fundamental equation of canonical perturbation theory. Theorem of Poincaré on the non-existence of first integrals of the motion -- 12.5 Birkhoff series: perturbations of harmonic oscillators -- 12.6 The Kolmogorov-Arnol'd-Moser theorem -- 12.7 Adiabatic invariants -- 12.8 Problems -- 12.9 Additional remarks and bibliographical notes -- 12.10 Additional solved problems -- 13 Analytical mechanics: an introduction to ergodic theory and to chaotic motion -- 13.1 The concept of measure -- 13.2 Measurable functions. Integrability -- 13.3 Measurable dynamical systems -- 13.4 Ergodicity and frequency of visits -- 13.5 Mixing -- 13.6 Entropy -- 13.7 Computation of the entropy. Bernoulli schemes. Isomorphism of dynamical systems -- 13.8 Dispersive billiards -- 13.9 Characteristic exponents of Lyapunov. The theorem of Oseledec -- 13.10 Characteristic exponents and entropy -- 13.11 Chaotic behaviour of the orbits of planets in the Solar System -- 13.12 Problems -- 13.13 Additional solved problems -- 13.14 Additional remarks and bibliographical notes -- 14 Statistical mechanics: kinetic theory -- 14.1 Distribution functions -- 14.2 The Boltzmann equation -- 14.3 The hard spheres model -- 14.4 The Maxwell-Boltzmann distribution -- 14.5 Absolute pressure and absolute temperature in an ideal monatomic gas -- 14.6 Mean free path -- 14.7 The 'H theorem' of Boltzmann. Entropy -- 14.8 Problems -- 14.9 Additional solved problems.

14.10 Additional remarks and bibliographical notes -- 15 Statistical mechanics: Gibbs sets -- 15.1 The concept of a statistical set -- 15.2 The ergodic hypothesis: averages and measurements of observable quantities -- 15.3 Fluctuations around the average -- 15.4 The ergodic problem and the existence of first integrals -- 15.5 Closed isolated systems (prescribed energy). Microcanonical set -- 15.6 Maxwell-Boltzmann distribution and fluctuations in the microcanonical set -- 15.7 Gibbs' paradox -- 15.8 Equipartition of the energy (prescribed total energy) -- 15.9 Closed systems with prescribed temperature. Canonical set -- 15.10 Equipartition of the energy (prescribed temperature) -- 15.11 Helmholtz free energy and orthodicity of the canonical set -- 15.12 Canonical set and energy fluctuations -- 15.13 Open systems with fixed temperature. Grand canonical set -- 15.14 Thermodynamical limit. Fluctuations in the grand canonical set -- 15.15 Phase transitions -- 15.16 Problems -- 15.17 Additional remarks and bibliographical notes -- 15.18 Additional solved problems -- 16 Lagrangian formalism in continuum mechanics -- 16.1 Brief summary of the fundamental laws of continuum mechanics -- 16.2 The passage from the discrete to the continuous model. The Lagrangian function -- 16.3 Lagrangian formulation of continuum mechanics -- 16.4 Applications of the Lagrangian formalism to continuum mechanics -- 16.5 Hamiltonian formalism -- 16.6 The equilibrium of continua as a variational problem. Suspended cables -- 16.7 Problems -- 16.8 Additional solved problems -- Appendices -- Appendix 1: Some basic results on ordinary differential equations -- A1.1 General results -- A1.2 Systems of equations with constant coeffcients -- A1.3 Dynamical systems on manifolds -- Appendix 2: Elliptic integrals and elliptic functions -- Appendix 3: Second fundamental form of a surface.

Appendix 4: Algebraic forms, differential forms, tensors.