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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