Exercise 1.38: Velocity transformation

Use the procedure Gamma-bar to construct a procedure that transforms velocities given a coordinate transformation. Apply this procedure to the procedure p->r to deduce (again) equation (1.67) on page 42.

$$ \begin{align*} v_x &= \dot{r} \cos{\varphi} − r \dot{\varphi} \sin{\varphi} \\ v_y &= \dot{r} \sin{\varphi} + r \dot{\varphi} \cos{\varphi}\tag{1.67} \end{align*} $$

We need to define a function F->C_v that takes a local tuple coordinate transformation function $F$ and returns a new local tuple function $C_v$. $C_v$ when evaluated on a local tuple, returns the expression for velocity in terms of the new coordinates.

If the fucntion $F$ is defined as $x = F(t, x’)$, then $v = C_v(t, x’, v’)$.

We start with a path-dependent function $\bar{C_v}$ with path $q’$ as its input, defined as:

$$ \bar{C_v}[q'] := \dot{Q} \circ \Gamma \left[ F \circ \Gamma[q'] \right] $$

where $\dot{Q}$ is a selector function that extracts the velocity component of a local tuple. We get $C_v$ by applying the Gamma-bar function on $\bar{C_v}$

$$ C_v = \bar{\Gamma}[ \bar{C_v} ] $$
(defn F->C_v
    (fn C_v [local]
        (let [f-bar (fn [q-prime]
                        (let [q (compose F (Gamma q-prime))] ;; q = F . Gamma[q']
                        (compose velocity (Gamma q))         ;; 
        ((Gamma-bar f-bar) local))

(rendertex ((F->C_v p->r)
           (up 't (up 'r 'varphi) (up 'rdot 'varphidot)))
\begin{pmatrix}\displaystyle{- r\,\dot {\varphi}\,\sin\left(\varphi\right) + \dot r\,\cos\left(\varphi\right)} \cr \cr \displaystyle{r\,\dot {\varphi}\,\cos\left(\varphi\right) + \dot r\,\sin\left(\varphi\right)}\end{pmatrix}

^ The above answer matches Eq. 1.67

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