# Dynamics and Stability – Readings

Course week(s)
1. Newton’s Laws and Fictitious Forces
2. Work and Energy
3. Lagrange Equations
4. Forces of Constraint and Stability of Steady Motion
5. Stability of Conservative Systems and Euler Equations
6. Euler Angles and Variational Calculus
7. Hamilton’s Principle and Ritz Mehtod

The readings for this course consist of one book and some “recommended literature”. The first book presented is the one that is most important. The recommended literature is additional material that can be useful for extra exercises.

The book: **Analytical Mechanics: With an Introduction to Dynamical System – by Joseph S. Torok**. Published in November 1999 by Wiley. ISBN 978-0-471-33207-7. You can view a number of pages for free under this google books link.

## Summary of Analytical Mechanics

The textbook consists of 6 chapters. Below, a short overview of the contents of each chapter is given below. You can use this to find out which chapters deal with which concepts, or use it as a checklist: do you know what all the definitions mean, and how to apply all the methods?

Below the chapter summaries, topics are named that are not required for this course. This includes topics that are either assumed to be already known or material that goes beyond the scope of this course.

## Chapter 1: Principles of Dynamics

This first chapter is an introduction to dynamics. General principles are elaborated in here:

- Mechanics
- Kinematics
- (Virtual) Work and energy
- Conservative systems

For this course on dynamics and stability, it is assumed that the student already understands the basics of mechanics. Therefore, the subjects *systems of particles* (p. 29) *and planar motion of rigid bodies* (p.41) is excluded from the lecture material of this course.

## Chapter 2: Lagrangian Dynamics

As the chapter title suggests, this part of the book introduces the Lagrangian approach to dynamics. Previously, a Newtonian approach was shown. Now, the Lagrange equations of motions are explained, as well as the Lagrange function. It is shown how to analyze systems by looking at its energy. Therefore, the Lagrangian method is also referred to as variational or energy method. Generalized coordinates are introduced to simplify systems. It is also shown how to derive constraint equations. Also, the important concept of steady motion is explained in this chapter.

Excluded topics in Chapter 2 are: *Lagrange equations for impulsive forces* (p. 131) and*electromechanical analogies* (p. 133).

## Chapter 3: Calculus of Variations

This chapter brings another method to analyze the dynamics of systems. Another quantity is examined: action. This method deals with functionals, as opposed to ordinary calculus which deals with functions. Functionals are real-valued expressions of functions defined within an interval. In a variational problem, one is interested in extremizing the functional to find a maximum or minimum value.

From this chapter, it is assumed that you already understand the topic on *extrema of functions* (p. 164), since this is first year’s mathematics. Furthermore, *variational problems with constraints* (p. 176) is not discussed in this course.

## Chapter 4: Dynamics of Rotating Bodies

In the previous chapters, different methods where explained to establish the dynamics of a system of particles. In this chapter, that knowledge is extended to analyze systems containing rigid bodies. Since a rigid body can be considered as a system of continuously distributed particles, all of the results derived in the previous chapters remain valid. A rigid body differs from a particle in the fact that it can rotate with respect to its center of mass. This brings along an extra degree of freedom.

It is shown how the equations of motion of a rotating body is established. In addition, the Euler equations and Euler angles are derived. The concept of steady precession is introduced.

Four topics from chapter 4 are excluded for this course: *Kinematics of rotating bodies* (p. 193),*Translation theorem for angular momentum* (p. 205), *Moment-free motion* (p. 209) and *General solution* (p. 224).

## Chapter 5: Hamiltonian Systems

This chapter is completely* excluded* for this course.

## Chapter 6: Stability Theory

This chapter deals with analyzing the global behavior of mechanical systems: stability. Analyzing stability gives information whether a system is stable or not, without calculating the (mostly) complex solution of a (nonlinear) system. It is first shown how to analyze the stability of linear systems. Thereafter, nonlinear systems are analyzed by first linearizing the equations of motion.

Corresponding to this chapter, *variational equations* (p. 296), *higher-order systems* (p. 306) and*Lyapunov’s direct method* (p. 308) are excluded topics.

## Recommended literature

Your first year’s Engineering Mechanics texts (the fourth edition is also o.k.). In particular the exercises on virtual work (Statics volume) and three-dimensional dynamics of rigid bodies (Dynamics volume) can be useful:

- Engineering Mechanics: Statics – by J. L. Meriam, L. G. Kraige
- Engineering Mechanics: Dynamics – by J. L. Meriam, L. G. Kraige

Your second year’s text on differential equations. It is recommended read the chapter on non-linear differential equations and stability at least one more time:

- Elementary Differential Equations and Boundary Value Problems – by W.E. Boyce

Dynamics and Stability by TU Delft OpenCourseWare is licensed under a Creative Commons Attribution-ShareAlike 4.0 International License.

Based on a work at https://ocw.tudelft.nl/courses/dynamics-and-stability/.