What is a degree of freedom? Configuration spaces and their topology.

Understanding degrees of freedom in classical mechanics is fundamental to characterizing physical systems. Counting them is usually easy, especially if we can assign them a clear meaning. However, the precise definition of a degree of freedom is not usually presented in first-year physics courses since it requires mathematical knowledge only learned in more advanced courses. In this paper, we use a pedagogical approach motivated by simple but non-trivial mechanical examples to define degrees of freedom and configuration spaces. We highlight the role that topology plays in understanding these ideas. Editor's Note Landau and Lifshitz defined a degree of freedom as "the number of independent quantities which must be specified in order to define uniquely the position of any system." Undergraduate physics students may find this concept obvious when a system is simple, such as a particle free to move in three dimensions, but struggle with the concept when faced with a more complex system, especially one with constraints. This paper shows how to present the concept of "degrees of freedom" in a mathematically rigorous but accessible way using examples like the coupled double pendulum. [ABSTRACT FROM AUTHOR]