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Unformatted text preview: lecture 1 Topics: What is classical mechanics? Degrees of freedom The Art of Theoretical Physics Motion, trajectories and F = ma F = ma implies two initial conditions per degree of freedom Two initial conditions per degree of freedom implies F = ma So what? Finding trajectories numerically Forces of the form F ( t ) What is classical mechanics? Classical mechanics is a category defined by what it is not! It is not quantum mechanics. A classi cal mechanical system is any collection of objects that we can describe to a good approximation without worrying about quantum mechanics. This includes most of the systems you are used to in everyday life (if you don’t worry too much about what goes on inside your computer or CD player or TV set). In this course, we will discuss these mechanical systems, and we will push beyond your everyday experience to discuss what happens when objects move at speeds close to the speed of light. We won’t study quantum mechanics explicitly, but we will talk about it at times. The underlying laws of the Universe seem to be quantum mechanical, and it is fun, instructive, and not at all trivial to think about how classical physics emerges as an approximation to the quantum mechanical world. Degrees of freedom We will start slowly, with some useful definitions. Coordinates — The coordinates of a physical system are the numbers (possibly dimensional) that describe the system at a given fixed time. Configuration — The configuration q ( t ) of a mechanical system is a number or vector consist ing of values of a set of independent coordinates that complete describe the system. We will call the configuration q , without specifying (at least for another few seconds) whether q is a single variable, or some kind of vector describing several coordinates at once. The word “inde pendent” in the definition means that none of the numbers in our set of coordinates is redundent or dependent of the others. That is we asume that the coordinates in the configuration are indepen dent in the sense that each can in independently changed, and each different set of values describes something physically different. Given a set of values of the coordinates at some particular time, we can figure out what the configuration is and thus what the system looks like at that time. Thus a configuration is just a mathematical snapshot of the system at a given time. One way of describing the underlying problem of classical mechanics is that we want to under stand how the configuration of the system evolves with time. That is we want to put the snapshots together into a mathematical movie to describe how the system moves. 1 Degrees of freedom — The number of independent components of the configuration q is called the number of degrees of freedom of the mechanical system....
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 Spring '09
 JohnRobertson
 Calculus, Derivative, order differential equation

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