### 2022　Fundamentals of Analytical Dynamics (Systems and Control)

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Undergraduate major in Systems and Control Engineering
Instructor(s)
Nakao Hiroya
Class Format
Lecture    (Face-to-face)
Media-enhanced courses
Day/Period(Room No.)
Tue1-2(W934)  Fri1-2(W934)
Group
-
Course number
SCE.S205
Credits
2
2022
Offered quarter
4Q
Syllabus updated
2022/4/20
Lecture notes updated
-
Language used
Japanese
Access Index ### Course description and aims

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.

### Student learning outcomes

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.

### Keywords

Lagrange’s equations, Hamilton’s equations, phase space, generalized coordinates, symmetries, conservation laws

### Competencies that will be developed

 ✔ Specialist skills Intercultural skills Communication skills Critical thinking skills Practical and/or problem-solving skills

### Class flow

lectures, exercises, homework

### Course schedule/Required learning

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 Phase space and Liouville’s theorem To understand the notion of phase space and Liouville's theorem
Class 14 Introduction to more general dynamical systems / other issues To have a look at more general dynamical systems / other issues
Class 15 Examination Examination

### Out-of-Class Study Time (Preparation and Review)

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.

Not specified

### Reference books, course materials, etc.

H. Goldstein, Classical Mechanics, Pearson Education
L. D. Landau and E. M. Lifshitz, Mechanics, Elsevier

### Assessment criteria and methods

Grading will be based on the final examination and homework scores.

### Related courses

• LAS.P101 ： Fundamentals of Mechanics 1
• LAS.P102 ： Fundamentals of Mechanics 2
• SCE.M201 ： Fundamental Kinematics and Kinetics for Mechanical Systems

### Prerequisites (i.e., required knowledge, skills, courses, etc.)

Fundamentals of Mechanics 1, 2
Fundamental Kinematics and Kinetics for Mechanical Systems 