Analytical dynamics is important in mechanical, control, and systems engineering. 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 objectives of analytical dynamics |
Class 2 | Lagrange’s equations of motion | To understand Lagrange's equations of motion |
Class 3 | Generalized coordinates | To understand the notion of generalized coordinates |
Class 4 | Symmetries and conservation laws | To understand relations between symmetries and conservation laws |
Class 5 | Applications | To apply the formalisms of analytical dynamics to practical examples |
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 | Variational methods | To understand the variational methods and their use |
Class 9 | Summary and exercises | Summary and exercises |
Class 10 | Hamilton’s equations of motion | To understand Hamilton's equations of motion |
Class 11 | Canonical transformations | To understand canonical transformations and their use |
Class 12 | Applications | To apply the formalisms of analytical dynamics to practical examples |
Class 13 | Phase space and Liouville’s theorem | To understand the notion of phase space and Liouville's theorem |
Class 14 | Introduction to dynamical systems theory | To glance at general dynamical systems theory |
Class 15 | Examination | Examination |
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.
They should do so by referring to textbooks and other course material.
L. D. Landau and E. M. Lifshitz, Mechanics, Elsevier
H. Goldstein, Classical Mechanics, Pearson Education
Grading will be based on the final examination and homework scores.
Fundamentals of Mechanics 1, 2
Fundamental Kinematics and Kinetics for Mechanical Systems