2016 Fundamentals of Analytical Dynamics (Systems and Control)

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Academic unit or major
Undergraduate major in Systems and Control Engineering
Nakao Hiroya 
Class Format
Media-enhanced courses
Day/Period(Room No.)
Tue7-8(W936)  Fri7-8(W936)  
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Syllabus updated
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Course description and aims

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.

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.


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 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 Small oscillations To understand how to treat small oscillations
Class 6 Motion of rigid bodies To understand how to describe the motion of rigid bodies
Class 7 Variational methods To understand the variational methods and their use
Class 8 Summary and exercises Summary and exercises
Class 9 Hamilton’s equations of motion To understand Hamilton's equations of motion
Class 10 Canonical transformations To understand canonical transformations and their use
Class 11 Hamilton-Jacobi equation To understand the Hamilton-Jacobi equation and its 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 Summary and exercises Summary and exercises


L. D. Landau and E. M. Lifshitz, Mechanics, Elsevier

Reference books, course materials, etc.

H. Goldstein, Classical Mechanics, Pearson Education

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

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