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Document Outline

  • 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
      • A4.1 Algebraic forms
      • A4.2 Differential forms
      • A4.3 Stokes’ theorem
      • A4.4 Tensors
    • Appendix 5: Physical realisation of constraints
    • Appendix 6: Kepler’s problem, linear oscillators and geodesic
    • Appendix 7: Fourier series expansions
    • Appendix 8: Moments of the Gaussian distribution and the Euler Γ function
  • Bibliography
  • Index
    • A
    • B
    • C
    • D
    • E
    • F
    • G
    • H
    • I
    • J
    • K
    • L
    • M
    • N
    • O
    • P
    • Q
    • R
    • S
    • T
    • U
    • V
    • W
    • Y

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