Generalized Coordinates and Generalized Velocity

This section provides a quick introduction to the Generalized Coordinates and Generalized Velocity.

What are Generalized Coordinates? Generalized Coordinates are independent functions qi(t), that can be used to represent positions in Cartesian coordinates through a set of transformation functions.

```q(t) = (q1(t), q2(t), q3(t))
# Generalized coordinates

r(t) = (rx(t), ry(t), rz(t))
# Cartesian coordinates

r(t) = (r1(q1(t), q2(t), q3(t)),
(r2(q1(t), q2(t), q3(t)),
(r3(q1(t), q2(t), q3(t)))
# r1(), r2(), and r3() are transformation functions

or:
r(t) = r(q(t))                  (C.1)
```

The diagram below shows the spherical coordinates as an example of generalized coordinates (source: quora.com)

What Is Generalized Velocity? Generalized Velocity is a vector of time derivatives of generalized coordinates.

```q'(t) = (dq1/dt, dq2/dt, dq3/dt)
```

The velocity in Cartesian coordinates can be expressed in generalized velocity through transformation functions:

```r'(t) = d(r(q(t))) / dt

r'(t) = (∑ ∂r1/∂qi * dqi/dt,
∑ ∂r2/∂qi * dqi/dt,
∑ ∂r3/∂qi * dqi/dt)
# The chain rule for derivatives applied

or:
r'(t) = (∂r1/∂q ∙ dq/dt,
∂r2/∂q ∙ dq/dt,
∂r3/∂q ∙ dq/dt)
# Dot product operation applied

or:
r'(t) = ∂r/∂q ∙ dq/dt

or:
r' = ∂r/∂q ∙ q'                 (C.2)
```