Dynamical modelling

Practical Mathematics
3 year
first or second
Hours per week – 1. or 2. semester:
Content (Syllabus outline)

Gravitation and central forces.
Motion in gravitational and central force fields. Kepler's laws. Integrability of motion in central force field.
Seminar: Differential equation of the orbit and energy equation of motion.
Non-inertial systems.
Accelerated coordinate systems and inertial forces. Influence of Earth rotation. Foucault's pendulum.
Seminar: Projectile motion including air resistance and Earth rotation. Numerical integration and graphical representation of trajectory of motion.
Systems with many degrees of freedom. Eigenfrequencies and eigenmodes. Linearization of motion in the vicinity of the equilibrium state. Small oscillations in the vicinity of equlibrium state.
Seminar: Double pendulum. Eigenmodes and eigenfrequencies. Normal modes of harmonic motion.
Rigid body kinematics and dynamics.
Motion about a fixed point. Coordinate transformations and Euler's angles. Euler's dynamic equations. Free gyro or top.
Seminar: General motion of a symmetric gyro or top. Steady precession of a vector of angular velocity. Uniform rotation and stability of motion.
Systems with variable mass.
General equations of motion for systems with variable mass. Motion of one-stage space rocket.
Seminar: Motion optimization for multistage rockets. Optimization of fuel consumption and trajectory of flight.


M. W. McCall: Classical Mechanics, a modern introduction, John Wiley, Chichester, 2001.
W. Greiner: Classical Mechanics: Point Particles and Relativity, Springer, 2004.
W. T. Thomson: Introduction to Space Dynamics, Dover, Publ. Inc., 1986.

Objectives and competences

Students will acquire knowledge about more comprehensive facts and ingredients of rational mechanics with emphasis on strict mathematical formulation based on previously mastered mathematical knowledge.

Intended learning outcomes

Knowledge and understanding: To establish knowledge and understanding of fundamental principles of rational mechanics at more advanced level.
Application: Mastered coursework represents a foundation for specialized research in the field of mechanics.
Reflection: Connecting acquired mathematical knowledge within the course with implementation of that knowledge in a general field of mechanics.
Transferable skills: A global overview of mathematical methods of modelling within the framework of rational mechanics. Solving problems from related areas of applied mathematics, natural sciences and engineering.

Learning and teaching methods

Theoretical lectures and seminar work. During the course student gets acquainted with application of symbolic languages (like Mathematica etc.) in real-problem solving. Student becomes familiar with a project problem solving approach which consist of different phases: the phase of planning, the phase of mathematical conceptualization, the choice of appropriate method of solution together with implementation of available computational tools.


Active participation at the theoretical part of lectures is required. Completion of seminar assignments or projects by individual choice.
Oral defense of seminar assignments.
Written defense of theoretical part
Grade is combination of the above.
Grading: 1-5 (fail), 6-10 (pass) (according to the Statute of UL)

Lecturer's references

KOC, Pino. Sea-wave dynamic loading of sailing yacht`s retractable keel. Strojniški vestnik, ISSN 0039-2480, Mar. 2014, vol. 60, no. 3, str. 203-209, ilustr. [COBISS-SI-ID 13401627]
KOC, Pino, HALILOVIČ, Miroslav, ŠTOK, Boris. Impact of restrained thermal expansion on NPP Krško primary loop piping. Tehnički vjesnik, ISSN 1330-3651, 2013, god. 20, br. 5, str. 897-904, ilustr. [COBISS-SI-ID 13212955]
KOC, Pino, ŠTOK, Boris. Computer-aided identification of the yield curve of a sheet metal after onset of necking. Computational materials science, ISSN 0927-0256. [Print ed.], 2004, letn. 31, št. 1/2, str. 155-168. [COBISS-SI-ID 7467803]