Classical Mechanics

A Computational Approach with Examples Using Mathematica and Python

Christopher W. Kulp, Vasilis Pagonis

• Éditeur: CRC Press
• Année: 2020
• ISBN:
  9781351024365
• Édition: 1
• Sujet: Physical Sciences
• Sous-sujet: Mathematical & Computational Physics

Classical Mechanics: A Computational Approach with Examples using Python and Mathematica provides a unique, contemporary introduction to classical mechanics, with a focus on computational methods. In addition to providing clear and thorough coverage of key topics, this textbook includes integrated instructions and treatments of computation.

Full of pedagogy, it contains both analytical and computational example problems within the body of each chapter. The example problems teach readers both analytical methods and how to use computer algebra systems and computer programming to solve problems in classical mechanics. End-of-chapter problems allow students to hone their skills in problem solving with and without the use of a computer. The methods presented in this book can then be used by students when solving problems in other fields both within and outside of physics.

It is an ideal textbook for undergraduate students in physics, mathematics, and engineering studying classical mechanics.

Features:

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• Gives readers the "big picture" of classical mechanics and the importance of computation in the solution of problems in physics

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• Numerous example problems using both analytical and computational methods, as well as explanations as to how and why specific techniques were used

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• Online resources containing specific example codes to help students learn computational methods and write their own algorithms

A solutions manual is available via the Routledge Instructor Hub and extra code is available via the Support Material tab

CHAPTER 1

The Foundations of Motion and Computation

So, you have decided to—or are required to—learn the subject of classical mechanics. But, what is classical mechanics? Is it fixing old cars? No, but a knowledge of classical mechanics will help you understand how your car works! To understand the term, classical mechanics let us first understand the term classical physics. Classical physics are the fields of physics that don’t involve either quantum theory or the theory of relativity. Mechanics is the branch of physics that deals with the actions of forces on an object that involve motion. So, classical mechanics involves the study of forces on objects that are well-described without using quantum theory and whose motion is nonrelativistic (i.e., cases where relativity is not needed to correctly model the motion). In other words, classical mechanics is the physics of your day-to-day world! An understanding of classical mechanics will help you understand how to build roller coasters, merry-go-rounds, and airplanes. Our knowledge of classical mechanics also allows us to make predictions on the motion of objects like comets and punted footballs. The world you interact with on a daily basis is generally the world of classical mechanics.

At first it might seem that classical mechanics is a dusty old subject that you need to learn in order to get to the “interesting stuff” like quantum mechanics. While the field of classical mechanics is one of the older subjects in physics, it is certainly not dusty! In fact, there is a lot of intriguing current research done in classical systems. For example, in Chapter 13 we will explore the field of nonlinear systems, which occur in in many of the natural and social sciences, not only in physics, and they display interesting types of behaviors including chaos.

1.1The world of physics

It is helpful to break up the field of physics into different branches. It should be noted that such divisions are largely arbitrary but useful human constructs that help us understand the world around us. As we sometimes tell students: Mother Nature doesn’t care if a particular system is a thermodynamics problem or a classical mechanics problem; humans make those distinctions in order to better understand how to describe and model the system. In fact, most real-world problems involve multiple fields of physics. So with that in mind, we can loosely break up physics into the following branches:

•Classical Mechanics deals with how forces cause motion in classical systems.

•Thermodynamics deals with relationships between all forms of energy; often the focus is on heat and its relationship with other forms of energy.

•Electromagnetism deals with the interaction of electrically-charged particles using the concepts of electric and magnetic fields.

•Statistical Mechanics deals with understanding how macroscopic properties, such as temperature and pressure, emerge from a large number of particles that make up the system.

•Relativity deals with the dependence of physical phenomenon on the relative motion between the observer and the observed. Physicists often consider three cases of relativity: Galilean relativity (which falls under the category of classical physics), Einstein’s theory of special relativity, and Einstein’s theory of general relativity. Special relativity focuses on the dependencies between inertial frames of reference while general relativity, a generalization of special relativity, takes into account noninertial frames.

•Quantum Mechanics deals with the interactions between subatomic particles and between subatomic particles and radiation.

Classical mechanics is one of the first topics learned by a physics student because it deals with the more intuitive concepts of force and motion. It provides tools for not just describing motion but also predicting motion. One of the central ideas of classical mechanics is that if one knows the position and velocity of a particle at time, t, then one can find the particle’s position and velocity at any point in the past or future (except for the case of chaotic systems, which we will deal with later). The ability to make predictions is critical in science, hence, classical mechanics provides a powerful set of tools, which are widely applicable to many different types of systems.

As mentioned earlier, almost all real-world problems involve more than one branch of physics, and classical mechanics is often an important element to those problems. For example, one might want...