We continued the derivation from last time, transforming Newton's Laws
into a new coordinate basis, 
. We focused on the acceleration
side of the transformed equation, using a roundabout trick to express
the acceleration side in terms of partial derivatives of the kinetic
energy T:
Here T is the coordinate-transformed kinetic energy, which depends
generally both on the new coordinates and their time-derivatives
. The partial derivatives were taken ignoring the
relationship between a coordinate and its time-derivative
; that is, varying one while holding the other fixed (along
with all the other 
and 
for 
, which are held
fixed in taking a or partial derivative).
Recombining this result with the generalized force side of our transformed Newton II gives, for the tranformed Newton II,
By defining the Lagrangian L = T- V, we rewrite this as
the Euler-Lagrange equations.
Using the Euler-Langrange equations -- that is, the Lagrangian approach --
is easier than using Newton II directly
because you only need to know positions and velocities as a function
of your generalized coordinates . You simply calculate T, calculate V,
and plug L = T - V into the Euler-Lagrange equations. These
automatically generate the right expressions for the acceleration
components needed to use Newton II.
-- KB