Section 5.1 Goals of the new Techniques of Theoretical Mechanics course
The primary goals were to deepen students' ability to envision the physical systems described by more advanced mathematics than is used in the introductory physics sequence and to emphasize sense-making strategies helpful in setting up, solving, and evaluating answers obtained for more complex problems. The course proposal described these goals as follows:
We propose a course to explicitly focus on physics majors' mathematical fluency and sense making skills. Introductory courses begin to develop these aspects, but the primary goals of those courses tend to be for students to apply simple concepts (Newton's Laws, energy conservation, etc) to relatively simple and very idealized problems. This course will be taken just after the introductory physics sequence. It will leverage students' understandings of simple mechanical systems from introductory physics to develop their mathematical fluency in the context of less idealized, more sophisticated systems. Each physics topic (velocity-dependent forces, Lagrangian mechanics, Hamiltonian mechanics, special relativity) will focus on specific math methods (e.g. differential equations, calculus of variations, generalized coordinates, Legendre transformations, linear transformations, hyperbola geometry) and will highlight strategies to make physical sense of mathematics (symbolic problem solving, reading equations like sentences, idealization, coordinating representations, limiting case analysis, dimensional checking).
This new course was intended to create a bridge for majors from the introductory physics sequence to the upper level physics courses where the problem-solving required was lengthy and complex.