Mechanics notebook

This is a book about worked exercises in Theoretical Mechanics, designed to accompany a one-semester course. It was born from the Italian university tradition of splitting basic courses into a purely theoretical part, called Lezioni, and a practical part, called Esercizi e Complementi. In fact, this book is based on experience gained by the authors in teaching Esercizi e Complementi at the Universities of Firenze, Milano, Pavia, Pisa and Napoli and at the Italian Naval Academy in Livorno. The book is more then a mere collection of exercises. Each chapter contains an introductory part presenting some relevant topics of the theory, which is then applied to selected problems which are completely worked out. In this way, the student learns what is necessary and may test his understanding directly from the exercises. The authors use the language and methods of today's Mathematics, so providing a modern and concise treatment of the subject. This enables students to acquire a useful methodology, preparing them also for subsequent Physics courses. The choice of the contents has been carefully selective in order to have a balance between theory and applications. Special emphasis has been paid to "model exercises" related to topics that the student will encounter in other branches of Physics. The result is an elegant treatment with only basic, essential material. The book can be useful equally to students in Engineering, Physics and Mathematics. It is ideal for students studying by themselves.

Chapter 1. Mathematical Language.
Points, Vectors, and Tensors.
Points and vectors.
Tensors.
Diads.
Bases of L(V).
Tensor Algebra.
Transposition.
Transposition Theorem.
Trace of a tensor.
Spectral decomposition.
Spectral Theorem.
Skew Tensors.
Axial vector.
Exterior product.
Orientation of V.
Duality.
Representation Theorem of Linear Forms.
Representation Theorem of Bilinear Forms.
Determinant of a tensor.
Orthogonal group of V .

Symmetries.
Isotropic tensors.
Transversely isotropic tensors.
Curves.
General properties.
Regular curves.
Examples.
Arc-length.
Tangent unit vector.
Curvature.
Normal and binormal unit vectors.
FRENET-SERRET'S- formulae.
Torsion.
Orientation of a curve.

Special co-ordinates.
Cylindrical co-ordinates.
Spherical co-ordinates.
Examples.
Example 1.1.
Example 1.2.
Example 1.3.
Example 1.4.
Example 1.5.
Example 1.6.
Example 1.7.
Example 1.8.
Example 1.9.

Chapter 2. Inertia.
Inertia of Rigid Bodies.
BOSCOVICH'S hypothesis.
Tensor of inertia.
Moments of inertia.
Principal axes of inertia.
Ellipsoid of inertia.
HUYGENS-STEINER'S Theorem.
STEINER's formula.
Composition Theorem.
Material symmetries.
Symmetry group.
Symmetry requirement.
Templates.
Planar systems.
Mirror symmetries.
Reflections.
Plane symmetry.
Uniaxial symmetry.
Spherical symmetry.
Inertia properties of simple homogeneous bodies.
Examples.
Example 2.1.
Example 2.2.
Example 2.3.
Example 2.4.
Example 2.5.
Example 2.6.
Example 2.7.
Example 2.8.
Example 2.9.
Example 2.10.
Example 2.11.
Example 2.12.

Chapter 3. Motion.
WEIERSTRASS' Theory.
One-dimensional systems.
Barriers.
Asymptotic limits and inversion points.
Periodic and non-periodic orbits.
Equilibrium stability.
Multi-dimensional systems.
Effective kinetic energy and effective potential Merostatic orbits.
Templates.
Central motion.
Degenerate central orbits.
Degenerate KEPLER 's problem.
Non-degenerate central orbits.
Orbit equation.
Orbit features.
Non-degenerate KEPLER'5 problem.
Examples.
Example 3.1.
Example 3.2.
Example 3.3.
Example 3.4.
Example 3.5.
Example 3.6.
Example 3.7.
Example 3.8.
Example 3.9.
Example 3.10.
Example 3.11.
Example 3.12.
Example 3.13.

Chapter 4. Equilibrium.
The Principle of Virtual Work.
Configurations.
Equilibrium configurations.
Virtual velocities.

Virtual power.
Constitutive restriction.
Principle of Virtual Work.
Rigid bodies.
Holonomic systems.
LIAPUNOV's Stability.
LlAPUNOV's definition.
DIRICHLET-LAGRANGE's Stability Theorem.

Instability Theorems.
LIAPUNOV 's First Instability Theorem.
LlAPUNOV 's Second Instability Theorem.
CHETAEV's Instability Theorem.
HAGEDORN-TALIAfERRO's Instability Theorem.
Normal modes.
Algebraic Lemma.
Simultaneous diagonalization.
Motion near equilibrium.
Quadratic Lagrangian.
Approximate motion.
Periodic approximate motions.
Stabilising constraints.
Templates.
Wake-up of latent modes.
Gyroscopic stabilization.
Examples.
Example 4.1.
Example 4.2.
Example 4.3.
Example 4.4.
Example 4.5.
Example 4.6.
Example 4.7.
Example 4.8.
Example 4.9.
Example 4.1O.
Example 4.11.
Example 4.12.
Example 4.13.
Example 4.14.
Example 4.15.
Example 4.16.

Chapter 5. Impact.
Impact Dynamics.
Impact.
Instantaneous impulse.
NEWTON's second law.
Balance laws.
Rigid bodies.
Unconstrained rigid body.
Kinetic energy.
Templates.
Free rigid bodies.
Restitution coefficient.
Direct impact.
Central impact.
Energy loss.
Rebound against a wall.
Incidence and reflection angles.
Impact between constrained rigid bodies.
Impulsive force on a fixed point.
Planar impact.
Impact mass.
Breakdown of an axis of rotation.
Motion after breakdown.
Examples.
Example 5.1.
Example 5.2.
Example 5.3.
Example 5.4.

Cited Authors.
Index.
Conversion Table: Italian-English.