Dynamics of Complex Systems 390-FG1-2DUZ

Study profile: general academic

Form of studies: full-time

Module: fundamentals of physics

Field and discipline of science: natural sciences, physical sciences

Level of education: first-cycle studies

Year of study / semester: 2nd year / 2nd semester

Prerequisites: differential and integral calculus, mechanics, numerical methods and algorithms, programming

Number of teaching hours: lecture (15 hours), tutorials (30 hours), laboratory (15 hours)

Teaching methods: lecture, solving calculation problems, presentation and analysis of program code, independent coding, homework assignments, problem-oriented discussions, consultations, independent study of literature

ECTS credits: 5

Student workload balance: participation in classes (60 hours, 2 ECTS credits), preparation for classes, assessment, and examination (75 hours, 3 ECTS credits); additionally, students are offered the opportunity to participate in consultations (15 hours per semester)

Quantitative indicators: student workload requiring direct teacher contact (60 hours, 2 ECTS credits), student workload not requiring direct teacher contact (75 hours, 3 ECTS credits)

Rules for the use of artificial intelligence (AI):

1. The use of AI is permitted for: verifying the correctness of symbolic and numerical calculations, visualizing results, supporting the analysis of models (differential equations, nonlinear systems), and providing suggestions regarding methods for solving equations.
   
2. AI may propose solutions that lack physical meaning. The responsibility for verification lies with the person using AI.
   
3. The use of AI in coursework, reports, or analyses should be clearly indicated, e.g.: “The linear differential equation was symbolically transformed with the help of AI (which one), and then solved numerically in the environment (which one).”
   
4. The use of AI is prohibited for: copying ready-made solutions to homework assignments or projects, creating coursework without one’s own contribution, using answers without understanding, generating conclusions, and during tests and examinations.

Course program (lecture):

1. Dynamics of a material point (Fundamentals of kinematics and dynamics: position, velocity, acceleration, momentum, mass, and force. Newton’s laws. Force models: elastic, gravitational, Lorentz, non-conservative forces. Variable-mass motion, Meshchersky equation. Conservation of momentum, energy, and angular momentum. Work, power, kinetic and potential energy. Conservative forces, potentials, central forces.)
   
2. Constrained material point and Lagrangian formalism (Mechanical constraints. D’Alembert’s principle. Generalized coordinates and forces. Lagrange equations of the second kind.)
   
3. Hamiltonian formalism (Phase space, canonical variables. Legendre transformation and Hamiltonian. Hamilton’s equations. Geometric interpretation and symmetries.)
   
4. Dynamics of systems of material points (Holonomic and non-holonomic systems. Configuration space. Virtual work principle.)
   
5. Rigid body dynamics (Kinematics and dynamics of rigid bodies. Inertia tensor. Equations of motion. Contact and interaction of rigid bodies.)

Course program (tutorials):

Analytical solution of example problems closely related to lecture content.

Course program (laboratory):

1. Numerical solution of systems of differential equations using the RK4 method; comparison with other numerical methods.
   
2. Solution of simple physical models: a parachutist; projectile motion with air resistance (cannon shot).
   
3. Newton’s equations of motion; bodies moving in a gravitational potential; the three-body problem.
   
4. Introduction to Lagrangian mechanics – the simple pendulum and the Kapitza pendulum.
   
5. Systems of oscillators coupled by springs; normal modes of vibration: the Fermi–Pasta–Ulam–Tsingou problem.
   
6. Midterm test.
   
7. Integrals of motion illustrated by the Kepler problem; the Hamiltonian formalism.