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.

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 fundamentals of the classical mechanics (kinematics and dynamics of point masses and their systems, 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.

Attendance at lectures is recommended, attandance 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.

Examination: The exam tests the knowledge of definitions, theorems a selected proofs in the field of mathematics, basic notions and principles in the field of mechanics and the ability of application of theoretical aparatus to given problems. Detailed information will be announced at the end of the semester.

Linear systems of ordinary differential equations (ODEs), stabilita, fundamentals of Floquet theory.
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 quasi-linear systems. Direct Lyapunov's method.
Autonomous systems of first-order ODEs: Trajectory, phase portrait, equilibrium and its stability, stability of periodic solutions.
Planar non-linear systems of ODEs: Stability and classification of equilibria, linearization.
Hamiltonian and gradient systems.
Non-linear autonomous second-order equations.
Basic notions and principles of kinematics and dynamics of motion of a point mass and systems of point masses.
Fundamentals of analytical mechanics, Lagrange equations.
Variational principles of classical mechanics, heuristric foundations of Hamiltonian mechanics.
Dynamical stabilization of inverted pendulum.
Mathematical modelling of motions of dislocations in crystals.

Geometric problems leading to the closed-form solutions of ODEs.
Qualitative analysis of the solutions of some differential and integral equations.
Stability and classification of the equilibria of non-linear autonomous systems of ODEs and of the second-order differential equations.
Constructing equations of motion of selected mechanical systems with 1 or more degrees of freedom and their qualitative analysis.