巴黎先賢祠傅科擺

北極，角速度為零

北緯 30° 一周爾天

之前曾經談過『張衡』的『候風地動儀』為何失傳，它與漢代『讖緯之學』的關係，於此摘引『前因後果』的一小段

既然叫做『候風地動儀』，它的命名必然有些來歷。西漢末年隨著社會的衝突加劇，『讖緯之學』開始廣泛大流行。《後漢書‧光武帝紀》光武帝於中元元年宣布『圖讖』於天下，把圖讖國教化。生於之後的張衡自當深知讖緯之術。就像『物候曆』的傳統，比如說《禮記‧月令》更是其來有自。然後發展成用『占候』來『預測』人事的『吉凶禍福』。因此命名裡那個『候』字應是指『徵候』，藉著此徵候來『預測』之義。而『風』字當是『風角』之術的觀『八方』風的用法，藉以表達『八個方位』的意思。如此看來這個候風地動儀的名義就是『測知八方地動之器 』。
……
自公元一三二年張衡發明候風地動儀以來，接連發生了幾次地震，到公元一三四年的隴西地震，張衡名氣大造，候風地動儀也聲名遠播。然而因著『天象』結合了『政爭』，頻起的『地震』究竟是『誰的過錯』？懺緯之說如是說︰地震起於『用人不當』，此上天之所以『罰罪』。縱使張衡有『天才之能』亦『無力分說』那個『地震之是非』。因此公元一三四年有『高官免職』後，張衡的『官運』也就步上了『黯淡之途』。由於沒有人希望能夠再『測出地震』，這時那個候風地動儀已經成為了『不祥之器』！短短幾年後，到了公元一三九年張衡抑鬱而逝。東漢末年，公元一九零年，董卓一把大火燒毀了『洛陽城』，一切終歸於『灰飛煙滅』！！

由此觀之，持守『科學精神』的『理性』實屬不易，『科技文明』的『發達』，也很難度杜絕『無根之言』，也許應該說面對『大自然』的『神奇奧妙』，人類其實『所知甚少』。而且一些雖然說是人們『已知之事』，但由於是『抽象』的，在缺乏了『直接經驗』下，總是顯得有些『難明難了』的吧！舉個例子來說，我們都知道『地球自轉』產生了太陽的『東升西落』，也學過牛頓力學所講的『慣性系統』，可是我們並不感覺地球在自轉的啊！一八五一年二月法國物理學家『萊昂‧傅科』Jean Bernard Léon Foucault 首度次在『巴黎天文台』的子午儀室公開展示了一的『單擺』。幾星期之後，傅科他又在『巴黎先賢祠』的拱頂下，用一根長六十七公尺的鋼索，其下懸掛了一顆重二十八公斤的鉛錘，然後使之擺動。這個單擺的『擺動平面』它每小時順時針方向旋轉 11° 度，經三十點七小時後環繞一圈。這就是大名鼎鼎的『傅科擺』 Foucault pendulum ，它的旋轉角速度 $\omega$ 與『緯度』 $\varphi$ 成正比，可以表示為 $\omega=360\sin\varphi\ ^\circ/\mathrm{day}$

，此處，『北緯』角度為『正』，表示『順時針方向旋轉』。據聞一八五五年，這個單擺被移到了國立巴黎工藝技術學院之國立工藝博物館。然後在二零一零年四月六日，國立工藝博物館內懸掛鉛錘的鋼索不知何故斷裂，使得單擺和博物館的大理石地板都受到無法修補的損壞。或許自傅科擺第一次以簡單的實驗證明『地球自轉』以來，這個擺已經善盡了『告知大眾』的『義務』的吧！！

─── 《水的生命！！下》

即使人工神經網路尚須深度學習訓練！既然探索豈能不盡興乎？

若是試問

Nonminimal Coordinates Pendulum

In this example we demonstrate the use of the functionality provided in mechanics for deriving the equations of motion (EOM) for a pendulum with a nonminimal set of coordinates. As the pendulum is a one degree of freedom system, it can be described using one coordinate and one speed (the pendulum angle, and the angular velocity respectively). Choosing instead to describe the system using the $x$ and $y$ coordinates of the mass results in a need for constraints. The system is shown below:

The system will be modeled using both Kane’s and Lagrange’s methods, and the resulting EOM linearized. While this is a simple problem, it should illustrate the use of the linearization methods in the presence of constraints.

定義參考系 $A$ 的目的是什麼呢？

那麼答之以方便

Kane’s Method

First we need to create the dynamicsymbols needed to describe the system as shown in the above diagram. In this case, the generalized coordinates $q_{1}$ and $q_{2}$ represent the mass $x$ and $y$ coordinates in the inertial $N$ frame. Likewise, the generalized speeds $u_{1}$ and $u_{2}$ represent the velocities in these directions. We also create some symbols to represent the length and mass of the pendulum, as well as gravity and time.

……

設定『束縛條件』的也！

As this system has more coordinates than degrees of freedom, constraints are needed. The configuration constraints relate the coordinates to each other. In this case the constraint is that the distance from the origin to the mass is always the length $L$ (the pendulum doesn’t get longer). Likewise, the velocity constraint is that the mass velocity in the A.x direction is always 0 (no radial velocity).

>>> f_c = Matrix([P.pos_from(pN).magnitude() - L])
>>> f_v = Matrix([P.vel(N).express(A).dot(A.x)])
>>> f_v.simplify()

不知讀者以為然耶？？

說來『擺』

Pendulum

A pendulum is a weight suspended from a pivot so that it can swing freely.^[1] When a pendulum is displaced sideways from its resting, equilibrium position, it is subject to a restoring force due to gravity that will accelerate it back toward the equilibrium position. When released, the restoring force acting on the pendulum’s mass causes it to oscillate about the equilibrium position, swinging back and forth. The time for one complete cycle, a left swing and a right swing, is called the period. The period depends on the length of the pendulum and also to a slight degree on the amplitude, the width of the pendulum’s swing.

From the first scientific investigations of the pendulum around 1602 by Galileo Galilei, the regular motion of pendulums was used for timekeeping, and was the world’s most accurate timekeeping technology until the 1930s.^[2] The pendulum clock invented by Christian Huygens in 1658 became the world’s standard timekeeper, used in homes and offices for 270 years, and achieved accuracy of about one second per year before it was superseded as a time standard by the quartz clock in the 1930s. Pendulums are also used in scientific instruments such as accelerometers and seismometers. Historically they were used as gravimeters to measure the acceleration of gravity in geophysical surveys, and even as a standard of length. The word “pendulum” is new Latin, from the Latin pendulus, meaning ‘hanging’.^[3]