Made some final comments on constraints. Warned

1. that our methods apply only for holonomic constraints -- that is, constraints that depend only on coordinates. We can handle velocity-dependent constraints only to the extent that they are integrable. The rolling without slipping constraint is an example of an integrable constraint: implies that , a holonomic constraint.
2. that our methods also apply only to ``workless'' constraints -- if our constraint force does work during the classical motion, then we cannot describe the system as containing only conservative forces, allowing us to write L = T-V where V is a conservative potential. The constraint does more than limit the possible physical motions; it also dissipates energy and prevents V from being a valid potential energy for the system. We discussed why rolling without slipping is a workless constraint, since the instantaneous velocity vanishes at the contact point where the frictional constraint force is applied.

We began to discuss symmetries and conservation laws. In this context we see the third major advantage of Lagrangian formalism over Newtonian (the first two were coordinate independence and the ability to handle constraints).

We reviewed how the three major classical conservation laws (momentum, energy, and angular momentum) can be obtained via manipulations of Newton II. The method of finding them was unsystematized and relied on "noticing" the conserved quantities hiding in the differential equations.

We then turned to the Lagrangian formalism. We noted that the E-L equations, which govern the system's motion, do not uniquely specify the Lagrangian. Instead Lagrangians differing by a total time derivative give the same E-L equations and are thus physically indistinguishable.

We then noted that we have already encountered some conserved quantities in the Lagrangian formalism: 1) when the Lagrangian had no explicit time dependence, we found that the Jacobi integral was conserved; and 2) when the Lagrangian had no explicit dependence on a generalized coordinate, E-L implied that its conjugate momenta was conserved. Such conjugate momenta are components of linear momentum, when the gen. coord. is a distance; and of angular momentum, when the gen. coord. is an angle.

We noted that both these type of conservation laws were related to invariances of the Lagrangian: the first because , and the second because . So we turned our attention to looking for symmetries. We defined a symmetry as a transformation of the coordinates which leaves the system's E-L equations unchanged, then detailed possible examples of such symmetries: space translations and time translations. We'll continue with more candidate symmetries next time.

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