|
Lecturer(s)
|
-
Richterek Lukáš, Mgr. Ph.D.
|
|
Course content
|
I. Introduction to the Study of Theoretical Physics II. Mechanics of a Particle and Systems of Particles 1. Basic concepts in particle kinematics. Position vector, trajectory, velocity, acceleration, surface velocity. Natural components of velocity and acceleration; velocity and acceleration in curvilinear coordinates. 2. Particle dynamics. Newton's laws. The two fundamental problems of dynamics. Specific problems in particle dynamics. 3. Systems of particles. D'Alembert's principle and the equations of motion for a system of particles. The centre of mass of the system. Classical integrals of motion. Motion of a particle with variable mass. III. Lagrangian Formulation of Mechanics 1. Systems subject to constraints. Classification of constraints, virtual displacements. 2. The principle of virtual work and its application to certain problems of system equilibrium. D'Alembert-Lagrange principle. 3. Lagrange's equations of the first and second kind and their solutions for certain specific problems; small oscillations of mechanical systems; the two-body problem. 4. Generalized potential for the Lorentz force. Levi-Civita symbol; symbolic representation of the vector product. IV. Rigid-Body Mechanics 1. Basic concepts of rigid-body kinematics. Translation and rotation of a rigid body. Inertia tensor and moments of inertia. 2. Motion of a rigid body with a fixed point, Euler's equations, Euler angles. Motion of flywheels, motion in rotating frames of reference. V. Hamilton's Formulation of Mechanics 1. Hamilton's Principle. Hamilton's Canonical Equations. 2. Phase Space and Phase Trajectories, Canonical Transformations, the Hamilton-Jacobi Equation. 3. Canonical Transformations and Their Invariants. Poisson Brackets, Conservation Laws. VI. Introduction to Continuum Mechanics 1. Stress tensor. Volume and surface forces. Stress vector. Equations of equilibrium for a continuum. Equations of motion for a continuum. 2. Displacement vector and strain tensor. Translational, rotational, and deformational motion of a continuum. 3. Fundamentals of the mechanics of elastic bodies, generalized Hooke's law. Equations of equilibrium for an isotropic elastic body. Some applications. Equations of motion for an isotropic elastic body. 4. Waves in an elastic body, vibrations of elastic bodies. The wave equation and its solution for string vibrations. VII. Fundamentals of Fluid Mechanics 1. Fluid statics; Pascal's and Archimedes' laws. 2. Equations of motion for an ideal fluid, their integrals, and irrotational flow. 3. Motion of a viscous fluid. The Navier-Stokes equations and similarity theory; turbulence; the Reynolds number; Hagen-Poiseuille's law.
|
|
Learning activities and teaching methods
|
Lecture, Work with Text (with Book, Textbook), Activating (Simulations, Games, Dramatization)
- Attendace
- 13 hours per semester
- Homework for Teaching
- 20 hours per semester
|
|
Learning outcomes
|
The aim of the course is to deepen students' knowledge of classical theoretical mechanics and to develop their ability to formulate, mathematically solve, and physically interpret mechanical problems within the frameworks of Newtonian, Lagrangian, and Hamiltonian formulations, rigid-body mechanics, and the fundamentals of continuum mechanics. The course guides students in working with physical models, evaluating the assumptions and limitations of solutions, and linking formal calculations with physical understanding. For students in teacher education programs, it also contributes to the development of the ability to teach difficult concepts and procedures in classical mechanics.
The course is primarily focused on gaining a deeper understanding of classical theoretical mechanics and on developing the ability to use mathematical tools in modelling mechanical systems. At the same time, it develops selected KRAAU competencies, particularly in the areas of subject-specific understanding, teaching the subject to students, and evaluating and reflecting on professional development. Upon completion of the course, the student will be able to: a) explain the basic principles of the Newtonian, Lagrangian, and Hamiltonian formulations of mechanics and clarify their interrelationships, advantages, and limitations of application; use appropriate physical concepts, quantities, and mathematical tools; KRAAU: 1.1.1; 1.1.2; 1.2.2; b) solve problems in particle mechanics, systems of particles, rigid bodies, small oscillations, continua, elasticity, and fluids; construct an appropriate model, write down the equations of motion, perform calculations, and interpret the physical significance of the results; KRAAU: 1.1.1; 1.1.2; 1.2.2; 4.1.1; c) distinguishes between physical reality, an idealized model, and a mathematical formulation; assesses the assumptions and limitations of models, such as those of a material point, a rigid body, an ideal fluid, a continuum, or small oscillations; KRAAU: 1.2.3; 1.2.4; 1.2.5; d) critically analyzes technical texts, textbooks, problem sets, numerical results, and digital resources; verifies the correctness of solutions and recognizes formal calculations lacking physical understanding; KRAAU: 1.2.3; 1.2.4; 6.2.1; 6.2.3; e) identifies typical challenging concepts in mechanics, such as the significance of constraints, generalized coordinates, virtual displacements, conservation laws, phase space, inertial forces, stability, the inertia tensor, or the stress tensor, and proposes an appropriate method for teaching them; KRAAU: 1.2.1; 1.2.6; 2.2.1; 4.2.1; f) reflects on their own problem-solving approach, mistakes, and difficulties in teaching theoretical mechanics, and uses feedback for further professional development; KRAAU: 4.3.1; 6.1.1; 6.2.1.
|
|
Prerequisites
|
Students are expected to have a basic understanding of mechanics, mathematical analysis, and linear algebra, particularly differential and integral calculus, vector and matrix calculus, as well as the ability to solve basic problems in Newtonian mechanics and to work with physical quantities, units, and the mathematical notation of physical models.
|
|
Assessment methods and criteria
|
Mark, Student performance
Credit is awarded based on ongoing work in the seminars, solving problems at the blackboard, homework, verification of understanding of selected homework solutions, and a written credit assignment. Students can earn a maximum of 55 points in the course assessment: - attendance and active participation in seminars: 5 points; - commented solution of a problem at the blackboard during a seminar: 5 points; - homework assignments: 15 points; - assessment of understanding of a selected homework solution: 10 points; - written exam: 20 points. To pass the course, students must earn at least 40 points, attend at least 50% of the seminars, submit the required minimum number of homework assignments (at least 9), and demonstrate understanding of at least one selected solution. The exam component is worth a maximum of 70 points and consists of a midterm exam and a final exam. The midterm exam is worth a maximum of 30 points, and the final exam is worth a maximum of 40 points. The overall grade for the course is based on the sum of points from the credit and exam components, up to a maximum of 125 points. Grading scale: 107-125 points: A; 99-106 points: B; 91-97 points: C; 83-90 points: D; 75-82 points: E; 74 points or fewer: F. Homework assignments serve primarily as a means of ongoing preparation and practice of computational procedures. Their evaluation is therefore not based solely on the submitted result, but also on the ability to explain the solution, justify the assumptions used, interpret the result, and correct any errors. An oral explanation may be requested, particularly in cases where there are doubts about the student's independent work or understanding of the submitted solution. A maximum of three attempts is allowed to retake the credit exam and each test; only the score from the best attempt is counted. If the required grade is not achieved after the credit exam and both tests, students may request an oral exam to improve their grade by one level.
|
|
Recommended literature
|
-
Bajer. (2007). Mechanika 1-3. chlup.cz.
-
Brdička, M., Hladík, A. (1987). Teoretická mechanika. Praha.
-
Brdička, M., Samek L., Sopko B. (2000). Mechanika kontinua. Praha.
-
Elsgolc, L. E. (1965). Variační počet. Praha.
-
Goldstein, H. (1980). Classical Mechanics. Addison-Wesley Publishing Company.
-
Greiner W., & Bromley D. A. (2004). Classical mechanics: point particles and relativity. New York.
-
Greiner W. et al. (2003). Classical mechanics. System of particles and Hamiltonoan mechanics. New York.
-
Horský J., Novotný J., Štefaník M. (2001). Mechanika ve fyzice. Praha.
-
Jex I., Štoll I., Tolar J. (2017). Klasická teoretická fyzika. Praha.
-
Podolský J. (2024). Teoretická mechanika ve třech knihách. Praha.
-
Tillich J., Richterek L. (2007). Klasická mechanika.
|