Analytical Dynamics

Junichiro INOUE Professor

Department: School of Engineering / Graduate School of Engineering

Class Time: 2003 Fall Friday
Recommended for: Quantum Science and Energy Engineering students

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Course Aims

A general course of physics contains classical mechanics, electromagnetic dynamics, thermo dynamics, statistical mechanics and quantum mechanics. Analytical mechanics is re-formulation of classical mechanics, which involve advanced mathematical approaches. The basics of classical mechanics are Newton's laws of motion, and they provide relationships between the motion of the body and the force acting on the body. However, Lagrangian mechanics and Hamiltonian mechanics can also provide the equivalent relationships. In this lecture we will focus on the application of Lagrangian mechanics and Hamiltonian mechanics for the complex system. We will also focus on the fundamental idea of classical mechanics and its relations to Lagrangian mechanics and Hamiltonian mechanics.

Key Features

Lecture is given based on the lecture notes distributed in the beginning of each class. This lecture involves with some basics of calculus 3 by 3 matrix operations. It is very important for students to derive the equation introduced in this class. Therefore exercise session will be given just after the lecture. Two teaching assistants will be organizing the exercise session, and there will be about hundred students. They will be ready to answer questions about the classes. Midterm exams are also given to make sure students understand the basics of Analytical Physics.

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Course Aims

We will be reviewing the Newton's second law and its application for the case of mass system for the first part of this course. We will then focus on the vibration of multi-degree-of-freedom system and rigid-body dynamics. Furthermore we will learn about variational principles and canonical equations of motion, which are deeply related to the field of quantum mechanics.

Requirements and recommended courses

Mechanics 1, Mechanics 2, Calculus, Linear Algebra


Lecture notes are provided.


Practice exercises on the last part of chapter are sometimes given for homework.

Course Schedule

Lecture notes consist of 11 chapters. The amount of material covered in this course is much considering that we only have 2.5 hour of class every week. Therefore, in order to enforce your ability for analytical physics, practice exercises on the last part of each chapter are highly recommended for the review.

Session Chapter Contents
1 1st chapter Classical Newtonian mechanics -Review of Newton's laws of motion and its application for the one dimensional motion
2 1st chapter Classical Newtonian mechanics continued -Equation of motion in polar coordinates and its application for the two dimensional motion
3 2nd chapter Central force motion -General solution for the central force motion
-Example problems of the motion of planets
4 3rd chapter Scattering problem -Scattering problem as one of the example of central force motion and its example problems
5 4th chapter Lagrangian and Lagrange's equation -Introduction of Lagrangian and application of the Lagrange's equation.
-Advantages of Lagrange's equation
6 5th chapter Microscopic vibration -Example problem on vibration motion and typical usage of Lagrangian.
-Microscopic vibration in multiple degree of freedom system.
7 5th chapter Microscopic vibration continued -Example problems on vibration in multiple degree of freedom system and its characteristics
8 6th chapter Motion of rigid body -Introduction of inertia tensor for the case of motion of rigid body
-Conservation of energy equation for the case of rigid body motion
9 6th chapter Motion of rigid body continued - Method of calculating inertia tensor and inertia moment
10 7th chapter Lagrangian of the rigid body and its Equation of motion -Equation of motion of the rigid body and its example problems.
11 7th chapter Lagrangian of the rigid body and its Equation of motion continued. -More difficult example problems.
12 8th chapter Euler angle and motion of the spinning top -Euler angle as the representation of rotating motion -Example problems about spinning top
13 9th chapter Variational principles -Variational principles, or the fundamental idea of deriving equation of state, and its practice problems
-The relation between variational principles and principle of least action
14 10th chapter Principle of virtual work -Principle of virtual work
-Derivation of Lagrange's equation
15 11th chapter Hamilton's principle -Derivation of Hamilton's principle from principle of least action
-Its relation to quantum physics

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Lecture Handouts

Lecture Notes (PDF, 973KB)

Lecture notes are provided but only in Japanese. Lecture notes consist of 7 chapters containing example problems at the end of each chapter. I would like you to use these problems for your review.

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Page last updated February 11, 2009

The class contents were most recently updated on the date indicated. Please be aware that there may be some changes between the most recent year and the current page.

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