Discussed why certain problems -- involving constraints or time-dependent coordinates -- are particularly complicated in the Newtonian approach.
Considered transforming Newton II explicitly from Cartesian
coordinates 
to some other choice of coordinates 
. The new
choice could be motivated by symmetries of the physical problem, or by
a desire to eliminate some constrained variable, by choosing one of
our to be some function kept constant by the constraint.
Before now we have always rederived Newton II from first principles for each new coordinate system, calculating the acceleration in our new coordinates by laborious application of the chain rule. Now we are trying to derive a more systematic rule for how Newton II must transform. We argued that it is easiest to understand how vector quantities like the components of Newton II transform by examining a coordinate-independent quantity, like the work performed in a particle motion between fixed endpoints.
We thus dotted Newton II with an infinitesimal path element
, giving on the left hand side a coordinate-independent
infinitesimal contribution to the work. We related our equation,
which must be true for independent coordinate variations 
, to
the Cartesian components of Newton II -- obtaining the i-th
component of Newton II by forcing coefficients of the independent
variations to be equal.
We then related our equation to variations 
of our new
coordinates by using the chain rule (since we know how our old
coordinates depend on the new coordinates ). Setting
coefficients of the independent variations equal gives the
transformed, -th component of Newton II.
While this gives the transformed version of Newton II, the physical content is not very transparent. We will work on getting simplified general expressions for both sides of the transformed equation we have obtained.
We simplified the left hand side, showing (for conservative forces)
that it equals to the generalized force 
. Since we allowed to be arbitrary independent functions of
, we may not always be able to interpret 
as a component of
the physical force along some unit vector. However, we showed that for
a linear transformation, where we may interpret our new coordinates as
distances along a transformed set of axes 
, that we do have
exactly this physical interpretation. That is, is simply the
component of the force, since 
, due to the connection between
directional derivatives and the gradient.
We then adjourned for more mathematica intuition-building on multivariable functions and the gradient. Next time we will simplify the right hand side, which has to do with accelerations.
-- KB