The simplest version of the Lagrange's equations (valid only for holonomic systems whose motion is described through the time derivative of coordinates) is presented as an analytically systematized version of the method of virtual power. They provide the equations of motion of the system from the derivatives of its mechanical energy and the generalized forces associated with the nonconservative interactions. Two methods to calculate the constraint unknowns are given. The first one is based on that simple version of the Lagrange's equations, while the second one leads to the Lagrange's equations with multipliers. Hamilton's principle is presented as the gateway to analytical dynamics. Finally, the equilibrium configurations of an n degree of freedom system are considered.

The work–energy theorem in a general reference frame is presented. It relates the change of the mechanical energy of the system in a finite time interval to the impulse work of the interaction forces (and inertia forces, if the frame is a non-Galilean one). Different expressions for the calculation of the kinetic energy of a rigid body are provided. The concept of conservative system and conservative force are introduced, and from them, that of potential energy. The particular case of gravitational energy and energy associated with linear springs is studied. A section is devoted to the study of equilibrium configurations, and their stability, of systems of one degree of freedom. Finally, the impossibility of perpetual motion is proved. The rotation stability of a free rigid body is analyzed through Poinsot’s ellipsoid in an appendix.

All the previous theorems and methods are adapted to percussive problems (problems with sudden changes of velocities). The main idea is to integrate them over the percussion interval. The behavior of unilateral constraints is analyzed in detail. The energy balance in percussive situations is particularly challenging. The formulation of energy dissipation through coefficients of restitution (used extensively in the scientific literature but not always in a consistent way) is discussed with a rigor unusual in many texts on classical mechanics. Multiple-point and rough impacts (impacts with friction) are introduced at the end of the chapter.

Description of the interactions between rigid bodies with a particular emphasis on constraint forces. The resultant torsor (resultant force-moment) associated to a system of forces is defined. Interaction torsors associated with the usual interactions in mechanical systems (gravitational attraction, tosion springs, and dampers) are formulated. The chapter proposes an original and rigorous treatment of constraint links with total and tangent redundancy, yielding indeterminacy and ill-conditioning in the mechanical constraints design. The analytical characterization of constraint torsors is proved and applied to different examples. Finally, the limit conditions leading to constraint partial or total loss (contact loss, overturning, sliding, rolling, pivoting) are explored.

Formulation and application of the vector theorems (linear momentum theorem and the angular momentum theorem) for the case of systems with constant matter. Those theorems relate the rate of change of a vector associated with the system (the linear momentum and the angular momentum about a point, respectively) to the net interactions (forces and moments) exerted on the system. Three different versions of the angular momentum theorem are presented: about a point fixed to a Galilean frame, about the system's center of mass, and about a point moving relative to a Galilean frame. Rotational dynamics, where the behavior of rigid bodies is often counterintuitive, is analyzed in a general and rigorous way. The dynamic role of the principal axes of inertia is discussed through several didactic examples. The formulation of the vector theorems in non-Galilean frames is also included. An appendix is devoted to the static and dynamic balancing of a rotor.

The method of virtual power is a powerful tool to obtain the equations of motion and the constriant forces and moments in multibody systems (especially in planar motion). The d'Alembert torsor of inertial forces is defined, and then the concept of virtual motion is introduced. Numerous examples illustrate the application of that method to different mechanical systems.

Introduction of all concepts related to the mass distribution of the system (center of mass and inertia tensor about a point) needed to apply the vector theorems that solve the dynamics of a rigid body. A few theorems that help calculate those elements (Pappus–Guldon theorems, Steiner's theorem) are presented. The qualitative assessment of the inertia tensor from the mass distribution geometry is discussed and illustrated through several examples. Principal directions of inertia (or of rotation) are introduced, and symmetrical and spherical rotors are defined. The inertia ellipsoid (a tool to visualize the inertia tensor) is presented in the last section.