The derivative in sense of distributions and distributional model of external collision. The generalized Euler-Lagrange equations. Internal collision and impact theories of the Hertz type - approximative models. Energy dissipation during impacts. The Kelvin-Zener model of viscoelastic body. Fractional derivative and the fractional Kelvin-Zener model of viscoelastic body. Restrictions on the rheological model that follow from the Clausius-Duhem inequality. The Mittag-Leffler function and the Laplace transform of the left Riemann-Liouville fractional derivative. The Post inversion formula. Simple deformation pattern and parameter identification based on rheological experiments. The Post-Newton method. Dry friction force models. Multifunctions (set-valued functions) and the Coulomb dry friction model. Dual nature of friction force in mechanics. Dual nature of friction force in mathematics. Differential inclusions. The Filippov theorem. Motion of a rigid body with fractional viscoelastic layer, the Cauchy problems given in form of integro-differential inclusion as models of motion: impact and forced oscillations of mechanical systems. Generalized derivative and generalized differential. The Hertz-Signorini-Moreau law of normal contact. Linear complementarity problems. Non-smooth potentials and the Frémon approach in collision mechanics. Numerical integration for dynamical systems with unilateral constraints: slack variable, application of the Laplace transform to differential equations with discontinuous right hand side. Examples: prismatic jount between two segments of robot manipulator, collision of two balls in the presence of adhesion, combinatorial analysis for impact of two bodies, reconstruction of three phases of motion in vehicle accidents, ABS braking system. The augmented Lagrangian method. Basic stability theory. The Lyapunov function. The Lyapunov theorems. First approximation and the Routh-Hurwitz criterion.