# Exercise 1.17: Bead on a helical wire

```clojurescript id=be81f546-e959-44af-bfd0-e2971df222ea
(require '[sicmutils.env :refer :all])
```

This, and the next three exercises, are here to give you practice in the real art, of difficulty, of any dynamics problem. It's easy to change coordinates. So what coordinates do you use?

> A bead of mass $m$ is constrained to move on a frictionless helical wire. The helix is oriented so that its axis is horizontal. The diameter of the helix is $d$ and its pitch (turns per unit length) is $h$. The system is in a uniform gravitational field with vertical acceleration $g$. Formulate a Lagrangian that describes the system and find the Lagrange equations of motion.

I'll replace this with a better picture later, but this is the setup:

![2020-06-25_11-03-55_screenshot.png][nextjournal#file#6214db2f-53e5-4c69-9865-9bb8a1746f16]

```clojurescript id=9a7f0142-c0f6-4005-9ce3-883aaf5757cc
(defn turns->rect [d h]
  (fn [local]
    (let [turns (coordinate local)
          theta (* turns 2 'pi)]
      (up (/ turns h)
          (* (/ d 2) (cos theta))
          (* (/ d 2) (sin theta))))))
```

Or you could do this. Remember, these transformations need to be functions of a local tuple, so if you're going to compose them, remember to put `coordinate` at the beginning of the composition.

```clojurescript id=80a4528a-2393-4cc6-ac84-ad05b251e1dc
(defn turns->x-theta [h]
  (fn [q]
    (up (/ q h)
        (* q 2 'pi))))

(defn x-theta->rect [d]
  (fn [[x theta]]
    (up x
        (* (/ d 2) (cos theta))
        (* (/ d 2) (sin theta)))))

(defn turns->rect* [d h]
  (compose (x-theta->rect d)
           (turns->x-theta h)
           coordinate))
```

The transformations are identical:

```clojurescript id=2379cfa0-55a6-405f-b680-cf9691ab1ebc
((- (turns->rect 'd 'h)
    (turns->rect* 'd 'h))
 (up 't 'n 'ndot))
```

Define the Lagrangian:

```clojurescript id=bdfb517c-395a-44f2-b588-e0647e478b8d
(defn L-rectangular [m U]
  (fn [[_ q v]]
    (- (* (/ 1 2) m (square v))
       (U q))))

(defn L-turns [m d h U]
  (compose (L-rectangular m U)
           (F->C (turns->rect d h))))
```

The potential is a uniform gravitational acceleration:

```clojurescript id=64df3975-4a41-4fa6-8a33-ffc58d0d46c2
(defn U-grav [m g]
  (fn [q]
    (* m g (ref q 2))))
```

Final Lagrangian:

```clojurescript id=4f2c3ff0-3eb1-477a-aeef-a31f5453f0ec
((L-turns 'm 'd 'h (U-grav 'm 'g))
 (up 't 'n 'ndot))
```

Lagrange equations of motion:

```clojurescript id=039c5626-b00e-4690-ab0c-8b464dc23538
(let [L (L-turns 'm 'd 'h (U-grav 'm 'g))
      n (literal-function 'n)]
  (((Lagrange-equations L) n) 't))
```

[nextjournal#file#6214db2f-53e5-4c69-9865-9bb8a1746f16]:
<https://nextjournal.com/data/QmQnpXZH3d3CRfupGQpu4jzZR6Wkyq31g86Bcn6vzL4Dma?content-type=image/png&node-id=6214db2f-53e5-4c69-9865-9bb8a1746f16&filename=2020-06-25_11-03-55_screenshot.png&node-kind=file>

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