Lagrange Equations in Generalized Coordinates

This section provides a quick introduction to Lagrange Equations in Generalized Coordinates.

It is amazing to see that Lagrange Equations stay in the same form as in Cartesian Coordinates:

```d(∂L/∂q')/dt = ∂L/∂q               (C.3)

or:
d(∂L/∂q'1)/dt = ∂L/∂q1
d(∂L/∂q'2)/dt = ∂L/∂q2
d(∂L/∂q'3)/dt = ∂L/∂q3
```

Here is the proof using the first component of the vector only as provided in David Morin's book.

```d(∂L/∂q'1)/dt = ∂L/∂q1             (C.4)
# To be approved
```

Start from left side term:

```d(∂L/∂q'1)/dt = d(∂L/∂r'∙∂r'/∂q'1)/dt
# Chain rule applied

d(∂L/∂q'1)/dt = d(∂L/∂r'∙∂r/∂q1)/dt
# Since ∂r'/∂q' = ∂r/∂q from G.29

d(∂L/∂q'1)/dt = d(∂L/∂r')/dt∙∂r/∂q1 + ∂L/∂r'∙d(∂r/∂q1)/dt
# Chain rule applied again

d(∂L/∂q'1)/dt = d(∂L/∂r')/dt∙∂r/∂q1 + ∂L/∂r'∙∂r'/∂q1
# Since d(∂r/∂q1)/dt = ∂r'/∂q1 by derivative switching

d(∂L/∂q'1)/dt = ∂L/∂r∙∂r/∂q1 + ∂L/∂r'∙∂r'/∂q1
# Since d(∂L/∂r')/dt = ∂L/∂r by Lagrange Equations in Cartesian coordinates

d(∂L/∂q'1)/dt = ∂L/∂q1             (C.4)
# Reverse chain rule applied
```

Cool. We have proved Lagrange Equations in Generalized Coordinates based Lagrange Equations in Cartesian Coordinates.