Analytical mechanics is important in systems and control. Newton’s equations of motion can take complicated form in many-body systems or in systems with constraints. In analytical dynamics, simple and general description of the system dynamics is developed, which is invariant under coordinate transformations. Relations between the symmetries of the system and conserved quantities such as the energy and angular momentum are clarified. In the Hamiltonian description, the system dynamics is described as trajectories in the phase space spanned by the position and momentum of the system, which is further generalized to the dynamical systems theory. The following topics will be covered in the course: Lagrange’s equations, generalized coordinates, symmetries and conservation laws, variational methods, Hamilton’s equations, phase space and Liouville’s theorem, oscillations, rotation of rigid bodies.
The aim of this course is to understand the Lagrangian and Hamiltonian formalisms of the laws of motion, which are generalizations of Newton’s equations of motion, to learn the related mathematical methods such as coordinate transformations and variational methods, and to apply the formalisms of analytical dynamics to actual problems.
Lagrange’s equations, Hamilton’s equations, phase space, generalized coordinates, symmetries, conservation laws
|✔ Specialist skills||Intercultural skills||Communication skills||Critical thinking skills||Practical and/or problem-solving skills|
lectures, exercises, homework
|Course schedule||Required learning|
|Class 1||Introduction||To understand the background and objectives of analytical mechanics|
|Class 2||Lagrange’s equation of motion||To understand Lagrange's equation of motion|
|Class 3||Euler-Lagrange equation of motion||To understand Euler-Lagrange equation of motion|
|Class 4||Symmetries and conservation laws||To understand relations between symmetries and conservation laws|
|Class 5||Small oscillations||To understand how to treat small oscillations|
|Class 6||Small oscillations||To understand how to treat small oscillations|
|Class 7||Motion of rigid bodies||To understand how to describe the motion of rigid bodies|
|Class 8||Motion of rigid bodies||To understand how to describe the motion of rigid bodies|
|Class 9||Variational principle||To understand the notion of the variational principle|
|Class 10||Variational principle||To understand the notion of the variational principle|
|Class 11||Hamilton’s equation of motion||To understand Hamilton’s equation of motion|
|Class 12||Canonical transformations||To understand the notion of canonical transformations|
|Class 13||Canonical transformation / symplectic form||To understand the notion of canonical transformations|
|Class 14||Phase space and Liouville’s theorem / other issues||To understand the notion of phase space and Liouville's theorem|
To enhance effective learning, students are encouraged to spend approximately 100 minutes preparing for class and another 100 minutes reviewing class content afterwards (including assignments) for each class by referring to textbooks and other course material.
H. Goldstein, Classical Mechanics, Pearson Education
L. D. Landau and E. M. Lifshitz, Mechanics, Elsevier
Grading will be based on the final examination and homework scores.
Fundamentals of Mechanics 1, 2
Fundamental Kinematics and Kinetics for Mechanical Systems