Ordinary Differential Equations in Mechanics (FSI-SRM-A)

Aim of the course: The aim of the course is to acquaint the students with the fundamentals of the qualitative theory of ordinary differential equation, dynamical systems, and analytical mechanics. The task is also to show a possible use of the theoretical results in analysis of ordinary differential equations appearing in mathematical models in mechanics.

Acquired knowledge and skills: Students will acquire the skills to apply theoretical mathematical apparatus in analysis of differential equations appearing in selected mathematical models in mechanics. In particular, they will be able to derive equations of motion of simpler mechanical systems and to determine stability and type of the equilibria of the obtained non-linear autonomous systems of ordinary differential equations. Students will also be familiarized with ordinary differential equations as mathematical models in mechanics and other disciplines.

In the field of mathematics: Linear algebra, differential calculus of functions of one and several variables, integral calculus of functions of one variable, solving of linear ordinary differential equations and their systems.

In the field of mechanics: Vectorial representation of forces and moments. Free body diagrams. Potential and kinetic energy.

The course provides an introduction to the qualitative theory of ordinary differential equations, in particular, with the questions of existence, uniqueness and extension of a solution to the initial value problems for non-linear non-autonomous systems of the first-order ordinary differential equations. Stability of solutions to the non-autonomous differential systems (and their particular cases) and fundamentals of the theory of dynamical systems will also be discussed. Finally, the basics of classical mechanics (kinematics and dynamics of a point, rigid body and systems of rigid bodies, Lagrange equations) will also be recalled, which are needed to derive the equations of motion of mechanical systems. The obtained mathematical apparatus will be used in the analysis of the ordinary differential equations appearing in selected mathematical models in mechanics, such as models of vibration of linear and nonlinear mechanical systems with one or more degrees of freedom.

Attendance at lectures and in seminars is obligatory and checked. Absence may be compensated for based on an agreement with the teacher.

Course-unit credit is awarded on the following conditions: Active participation at seminars, passing a wtritten test (at least half of possible points in the test).

Examination: The exam will be done orally, it tests the knowledge of definitions and theorems (especially the ability of their application to the given problems). Detailed information will be announced at the end of the semester.

Stability of solutions to linear systems of ordinary differential equations (ODEs), Lyapunov's exponents. Stability of solutions to quasi-linear systems.
Initial value problem for systems of non-linear ODEs: Existence, uniqueness, and extension of solutions.
Structure of the set of solutions to initial value problem for systems of non-linear ODEs.
Stability of solutions to systems of non-linear ODEs, direct Lyapunov's method.
Autonomous systems of first-order ODEs: Trajectory, phase portrait, equilibrium and its stability.
Planar non-linear systems of ODEs: Stability and classification of equilibria, linearization.
Hamiltonian systems, non-linear autonomous second-order equations.
Kinematics and dynamics of motion of a point and systems of points.
Kinematics and dynamics of simple motions of rigid bodies.
Introduction to analytical mechanics, Lagrange equations.
Linear vibrations with 1 degree of freedom, different types of damping.
Non-linear vibrations with 1 degree of freedom, mathematical and physical pendulum.
Linear vibrations with n degree of freedom.
Mathematical modelling of motions of dislocations in crystals.

Solving of selected types of non-linear first-order ODEs.
Stability of linear systems of ODEs with constant coefficients, Hurwitz criterion.
Implicit differential equations: Lagrange and Clairot equations.
Geometric problems leading to the closed-form solutions of ODEs.
Euler differential equation in stress-analysis of thick-walled cylindrical vessels and analysis of deformation of shells.
Kinematics and dynamics of motion of a point and systems of points.
Kinematics and dynamics of simple motions of rigid bodies.
Equation of catenary.
Planar non-linear autonomous systems of ODEs: Stability and classification of equilibria, phase portrait.
Autonomous non-linear second-order equations: Stability and classification of equilibria.