Ordinary Differential Equations in Mechanics (FSI-SRM)

Academic year 2026/2027
Supervisor: doc. Ing. Jiří Šremr, Ph.D.  
Supervising institute: ÚM all courses guaranted by this institute
Teaching language: Czech
Course type: departmental course
Aims of the course unit:

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 the analysis of differential equations appearing in selected mathematical models in mechanics. In particular, they will be able to derive the equations of motion of simpler mechanical systems and determine the stability of equilibria and periodic solutions 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.
Learning outcomes and competences:
 
Prerequisites:
 
Course contents:

The course provides an introduction to the qualitative theory of ordinary differential equations, particularly focusing on the questions of existence, uniqueness, and extension of a solution to initial value problems for non-linear, non-autonomous systems of first-order ordinary differential equations. The stability of solutions to non-autonomous differential systems and their particular cases (e.g., autonomous systems, Hamiltonian systems, and second-order differential equations) will also be discussed. Finally, the fundamentals of 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 utilised in the analysis of ordinary differential equations that appear in selected mathematical models in mechanics, such as models of vibration of linear and nonlinear mechanical systems.
Teaching methods and criteria:
 
Assesment methods and criteria linked to learning outcomes:
 
Controlled participation in lessons:
 
Type of course unit:
    Lecture  13 × 3 hrs. optionally                  
    Exercise  13 × 1 hrs. compulsory                  
Course curriculum:
    Lecture

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.
Stability of equilibria and periodic solutions to autonomous systems.
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.
    Exercise

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, Hamiltonian systems, and the second-order differential equations.
Constructing equations of motion of selected mechanical systems with 1 or more degrees of freedom and their qualitative analysis.
Literature - fundamental:
1. ANDRONOV, A. A.; LEONTOVICH, E. A.; GORDON, I. I. a MAIER, A. G. Qualitative Theory of Second-order Dynamic Systems. New York: John Wiley, 1973. ISBN 0470031956.
2. CODDINGTON, E. A. a LEVINSON, N. Theory of ordinary differential equations. Malabar: Krieger Publishing Company, 1984. ISBN 0-89874-755-4.
3. LEVI M.Classical Mechanics With Calculus of Variations and Optimal Control: An Intuitive Introduction.Student Mathematical Library 69, American Mathematical Society, 2014.ISBN 978-0-8218-9138-4.
4. DEMIDOVICH B. P. Lectures on the mathematical theory of stability. Izdat. "Nauka'', Moscow 1967.
5. HARTMAN, P. Ordinary differential equations. Philadelphia: SIAM, 2002. ISBN 0-89871-510-5.
6. PERKO, L. Differential equations and dynamical systems. New York: Springer Science+Business Media, 2001. ISBN 0-387-95116-4.ew York, 2001, ISBN 0-387-95116-4.
7. KRATOCHVÍL C. a SLAVÍK, J. Mechanika těles: dynamika. Brno: Akademické nakladatelství CERM, 2007. ISBN 978-80-214-3446-2.
Literature - recommended:
1. CODDINGTON, E. A. a LEVINSON, N. Theory of ordinary differential equations. Malabar: Krieger Publishing Company, 1984. ISBN 0-89874-755-4.
2. HARTMAN, P. Ordinary differential equations. Philadelphia: SIAM, 2002. ISBN 0-89871-510-5.
3. KALAS J. a RÁB M. Obyčejné diferenciální rovnice. Masarykova univerzita, Brno, 1995. ISBN 80-210-1130-0.
4. PERKO, L. Differential equations and dynamical systems. New York: Springer Science+Business Media, 2001. ISBN 0-387-95116-4.ew York, 2001, ISBN 0-387-95116-4.
5. KRATOCHVÍL C. a SLAVÍK, J. Mechanika těles: dynamika. Brno: Akademické nakladatelství CERM, 2007. ISBN 978-80-214-3446-2.
The study programmes with the given course:
Programme Study form Branch Spec. Final classification   Course-unit credits     Obligation     Level     Year     Semester  
N-MAI-P full-time study --- no specialisation -- Cr,Ex 6 Compulsory 2 2 W
C-AKR-P full-time study CZS -- Cr,Ex 6 Elective 1 1 W