Covered the general heuristics for the setup of mechanics problems: generalized coordinates, counting degrees of freedom, proper sets of coordinates, and using overcomplete sets of coordinates with Lagrange multipliers to get constraints.

1. For system with N particles, D dimensions
2. specifying a particle position takes D numbers (projections on each coordinate axis), so we naively need DN generalized coordinates to describe the instantaneous position of the system.
3. However, there may be equations of constraint. Assume we have m 1-dimensional constraint equations (that is, count each equation for each component as one equation). These make some of our coordinates dependent on each other.
4. Such a system has s = DN - m independent coordinates, called "degrees of freedom". Each degree of freedom corresponds to an independent direction in which a system element can move, without violating any constraint.
5. To describe the system's motion (to build its Lagrangian), we can use ANY s independent quantities that we want (e.g ). They need not be physical lengths -- they can be dimensionless, have dimensions of , whatever. We have this freedom because the Lagrangian is coordinate-invariant. We showed that regardless of what coordinate transformation we need apply to Cartesian coordinates to obtain our chosen coordinate system, the Euler-Lagrange equations always do all our chain rule coordinate and axis differentiations correctly for us, automatically, to get the right Newton's law . This is because the master equation responsible for the E-L equations, , does not change when we change coordinates.
6. We call "generalized coordinates" any set of quantities specifying the system. We usually label them q, so we have . Note that -- we must have enough to completely specify the system, and we can have up to the full number to specify N particle locations independently in D-space.
7. We have a "proper" set of generalized coordinates when r=s -- we have just enough coordinates to fully specify the system, and all are independent (no constraints relate them).

This leads to the following heuristic to solve mechanics problems:

1. To solve only for the motion, you can
   1. choose a proper set of generalized coordinates, build L, and get 1 E-L equation for each coordinate
   2. keep an overcomplete set of r generalized coordinates, include Lagrange multiplier terms for each constraint, and solve all r E-L equations for the coordinates (in addition to imposing the constraints).
2. To find a constraint force, you must reintroduce an additional coordinate for each constraint, then find the constraint force by using the Lagrange multiplier method. It's most economical to introduce your new coordinate along the direction in which the constraint force acts. For example, to find a tension that is holding r constant, reintroduce r as a coordinate, with the constraint g = r-c, and find the tension from the Lagrange multiplier method.

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