光的『本性』到底是什麼？一個光的『折射問題』就曾經在科學史上引發過大『論戰』！追根溯源西元二世紀時古希臘的托勒密 Claudius Ptolemy 在所著之《光學》Optics 第五卷裡，提出了他的折射實驗與定律。也許在那個時代，並不清楚『正弦』Sin 的概念，所以他結論並不正確。其後於九八四年伊朗學者伊本‧沙爾 Ibn Sahl 在《論點火鏡子與透鏡》On Burning Mirrors and Lenses 裡最早正確地描述了『折射定律』，並且將之應用於找出能夠讓『光聚焦』而又不會產生『幾何像差』之『透鏡』的形狀。然而只可惜他的研究結果並未為其它的學者所注意到。因此往後的許多年，人們又再次的從托勒密的『錯誤理論』開始『研究折射』。到了十一世紀初阿拉伯的學者海什木 Al Hazen 『重新再做』托勒密的實驗，雖然在著作的《光學書》Book of Optics 中總結出了一些法則，卻也沒能夠得出折射的『正弦定律』。如此又過了五百年，一六零二年英國天文學家托馬斯‧哈里奧特 Thomas Harriot 重新發現了『折射定律』，不過他並沒有發表這個結果，只是在與德國天文學家約翰內斯‧克卜勒 Johannes Kepler 通信中曾提及過這件事。其後於一六二一年荷蘭天文學家威理博‧司乃耳 Willebrord Snellius 推導出了一個數學上的『等同形式』，然而在其有生之年，人們並不知道他的成就。作者雖然不知這些偉大的『天文學家們』為何當時『人不知』他們的『研究結論』？然而設想從事『竹藤工藝』者，假使不知道『如何彎曲』那個『竹片』與『藤條』的話，大概想做『什麼家具』都可能是困難的吧！那麼如果不知道如何『屈折光線』，一個天文學家又怎麽能夠製造『好的透鏡』，用以『觀測天象』的呢？？

一六三七年法國的大哲學家勒內‧笛卡兒 René Descartes 在其專著《屈光學》Dioptrics 裡，推導出了這個『折射定律』，並且用自己的理論解析了一系列的『光學問題』。在這推導裡，他做了兩個『假設』︰一、『光』的『傳播速度』與周遭的『介質密度』成『正比』；二、『光的速度』沿著『交界面』方向的『速度分量守恆』。一六六二年法國律師和業餘數學家皮埃爾‧德‧費馬 Pierre de Fermat 發表了『最短時間原理』：光線傳播的路徑是需時最少的路徑。藉此推導出了『折射定律』，但是該原理假設了與笛卡兒相反之『光的傳播速度與介質密度成反比』，為此費馬強烈的反駁笛卡兒的導引，認為笛卡爾的假設是錯誤的。根據歷史學者以撒‧福雪斯 Issac Vossius 一六六二年在著作《De natura lucis et proprietate》裏的敘述，笛卡兒事先閱讀了司乃耳的論文，然後調製出自己的導引。有些歷史學者覺得這指控太過誇張，令人難以置信；也有很多歷史學者都存疑過曾經發生了這回事，然而費馬與惠更斯卻分別多次重複地譴責笛卡兒之行為缺失。在此不論歷史上的『是非對錯』，這場光的『粒子說』與『波動說』之大戰正方興未艾！一六六四年英國博物學家羅伯特‧虎克 Robert Hooke 開始提倡光的『波動說』。但是一六六九年被授予劍橋大學三一學院盧卡斯數學教授席位的牛頓卻是笛卡兒的『光粒子說』之發揚者。一六七零年到一六七二年期間，牛頓負責在校講授光學。他研究了光的折射，發表『三稜鏡』可以將白光發散成彩色光譜，而且藉著透鏡和另一個三稜鏡可以將彩色光譜合組為白光。雖然虎克本人曾經公開批評牛頓的光微粒說。但是因為牛頓在多門物理領域的成就，使得他被公認是這場『光本性爭論』的贏家。

一六七八年荷蘭物理、天文和數學家克里斯蒂安‧惠更斯 Christiaan Huygens 依據虎克的提議，在其著作《光論》（Traité de la Lumiere）裡應用他創造的『子波原理』 ── 今天的惠更斯原理──，從光的波動性質，成功的推導出並且解釋了司乃耳定律。之後於一七零三年惠更斯在其著作《Dioptrica》中又談到了這定律，並且正式的將這定律的發現歸功於司乃耳。一八零二年英國的科學家與醫生托馬斯‧楊 Thomas Young ── 被譽為『世界上最後一個什麼都知道的人』 ── 做實驗發現，當光波從較低密度介質傳播到較高密度介質時，光波的波長會變短，他因此歸結出光波的傳播速度會降低。楊氏之所以大名鼎鼎在於他所提出的『雙縫實驗』 double-slit experiment 就是這一場『古典光本性大戰』勝負之最終『判定性實驗』。之後這個『光本性問題』又發生了量子力學史上的『愛因斯坦‧波耳』大戰，以及波耳所提出的『波、粒互補性原理』。那麼光到底是什麼呢？？請看

這段動畫影片道盡了現今科學所了解的『量子本性』，妙哉！沒有『觀察者』時它是『波』，想確定『它』而『作觀察』時，它又是『粒子』，果真是 Oh! My God !!

─── 《【SONIC Π】聲波之傳播原理︰原理篇《一》》

一場光之『折射定律』大論戰，一篇費馬的『最短時間原理』

Fermat

In the 1600s, Pierre de Fermat postulated that “light travels between two given points along the path of shortest time,” which is known as the principle of least time or Fermat’s principle.^[21]

文本，誰知開啟了物理學『最小作用量原理』之思維門戶︰

Principle of least action

The principle of least action – or, more accurately, the principle of stationary action – is a variational principle that, when applied to the action of a mechanical system, can be used to obtain the equations of motion for that system. In relativity, a different action must be minimized or maximized. The principle can be used to derive Newtonian, Lagrangian and Hamiltonian equations of motion, and even general relativity (see Einstein–Hilbert action). The physicist Paul Dirac^[1], and after him Julian Schwinger and Richard Feynman demonstrated how this principle can also be used in quantum calculations.^[2]^[3] It was historically called “least” because its solution requires finding the path that has the least value.^[4] Its classical mechanics and electromagnetic expressions are a consequence of quantum mechanics, but the stationary action method helped in the development of quantum mechanics.^[5]

The principle remains central in modern physics and mathematics, being applied in thermodynamics,^[6] fluid mechanics,^[7] the theory of relativity, quantum mechanics,^[8] particle physics, and string theory^[9] and is a focus of modern mathematical investigation in Morse theory. Maupertuis’ principle and Hamilton’s principle exemplify the principle of stationary action.

The action principle is preceded by earlier ideas in optics. In ancient Greece, Euclid wrote in his Catoptrica that, for the path of light reflecting from a mirror, the angle of incidence equals the angle of reflection.^{[citation needed]} Hero of Alexandria later showed that this path was the shortest length and least time.^[10]

Scholars often credit Pierre Louis Maupertuis for formulating the principle of least action because he wrote about it in 1744^[11] and 1746.^[12] However, Leonhard Euler discussed the principle in 1744,^[13] and evidence shows that Gottfried Leibniz preceded both by 39 years.^[14]^[15]^[16]

In 1933, Paul Dirac discerned the quantum mechanical underpinning of the principle in the quantum interference of amplitudes.^[17]

General statement

As the system evolves, q traces a path through configuration space (only some are shown). The path taken by the system (red) has a stationary action (δS = 0) under small changes in the configuration of the system (δq).^[18]

The starting point is the action, denoted $\displaystyle {\mathcal {S}}$ (calligraphic S), of a physical system. It is defined as the integral of the Lagrangian L between two instants of time t₁ and t₂ – technically a functional of the N generalized coordinates q = (q₁, q₂, … , q_N) which define the configuration of the system:

$\displaystyle {\mathcal {S}}[\mathbf {q_{1}} (t_{1}),\mathbf {q_{2}} (t_{2}),\mathbf {q} (t)]=\int _{t_{1}}^{t_{2}}L(\mathbf {q} (t),\mathbf {\dot {q}} (t),t)dt$

where the dot denotes the time derivative, and t is time.

Mathematically the principle is^[19]^[20]^[21]

$\displaystyle \delta {\mathcal {S}}=0$

where δ (Greek lowercase delta) means a small change. In words this reads:^[18]

The path taken by the system between times t₁ and t₂ and configurations q₁ and q₂ is the one for which the action is stationary (no change) to first order.

In applications the statement and definition of action are taken together:^[22]

$\displaystyle \delta \int _{t_{1}}^{t_{2}}L(\mathbf {q} ,\mathbf {\dot {q}} ,t)dt=0$

The action and Lagrangian both contain the dynamics of the system for all times. The term “path” simply refers to a curve traced out by the system in terms of the coordinates in the configuration space, i.e. the curve q(t), parameterized by time (see also parametric equation for this concept).

後於古典力學時期，哈密頓歸結了此原理︰

哈密頓原理

在物理學裏，哈密頓原理（英語：Hamilton’s principle）是愛爾蘭物理學家威廉·哈密頓於1833年發表的關於平穩作用量原理的表述。哈密頓原理闡明，一個物理系統的拉格朗日函數，所構成的泛函的變分問題解答，可以表達這物理系統的動力行為。拉格朗日函數又稱為拉格朗日量，包含了這物理系統所有的物理內涵。這泛函稱為作用量。哈密頓原理提供了一種新的方法來表述物理系統的運動。不同於牛頓運動定律的微分方程式方法，這方法以積分方程式來設定系統的作用量，在作用量平穩的要求下，使用變分法來計算整個系統的運動方程式。

雖然哈密頓原理本來是用來表述古典力學，這原理也可以應用於古典場，像電磁場或重力場，甚至可以延伸至量子場論等等。

概念

微分方程式時常被用來表述物理定律。微分方程式指定出，隨著極小的時間、位置、或其他變數的變化，一個物理變數如何改變。總合這些極小的改變，又加上已知這變數在某一點的數值或導數值，就能求得物理變數在任何點的數值。

哈密頓原理用積分方程式來表述物理系統的運動。我們只需要設定系統在兩個點的狀態，叫做最初狀態與最終狀態。然後，經過求解系統作用量的平穩值，我們可以得到系統在，兩個點之間，其他點的狀態。不但是關於古典力學中的一個單獨粒子，而且也關於古典場像電磁場與萬有引力場，這表述都是正確的。更值得一提的是，現今，哈密頓原理已經延伸至量子力學與量子場論了。

用變分法數學語言來表述，求解一個物理系統作用量的平穩值（通常是最小值），可以得到這系統隨時間的演變（就是說，系統怎樣從一個狀態演變到另外一個狀態）。更廣義地，系統的正確演變對於任何微擾必須是平穩的。這要求導致出描述正確演變的微分方程式。

……

Hamilton’s principle

In physics, Hamilton’s principle is William Rowan Hamilton‘s formulation of the principle of stationary action (see that article for historical formulations). It states that the dynamics of a physical system is determined by avariational problem for a functional based on a single function, the Lagrangian, which contains all physical information concerning the system and the forces acting on it. The variational problem is equivalent to and allows for the derivation of the differential equations of motion of the physical system. Although formulated originally for classical mechanics, Hamilton’s principle also applies to classical fields such as the electromagnetic andgravitational fields, and plays an important role in quantum mechanics, quantum field theory and criticality theories.

Mathematical formulation

Hamilton’s principle states that the true evolution q(t) of a system described by N generalized coordinates q = (q₁, q₂, …, q_N) between two specified states q₁ = q(t₁) and q₂ = q(t₂) at two specified times t₁ and t₂ is astationary point (a point where the variation is zero), of the action functional

$\displaystyle {\mathcal {S}}[\mathbf {q} ]\ {\stackrel {\mathrm {def} }{=}}\ \int _{t_{1}}^{t_{2}}L(\mathbf {q} (t),{\dot {\mathbf {q} }}(t),t)\,dt$

where $\displaystyle L(\mathbf {q} ,{\dot {\mathbf {q} }},t)$ is the Lagrangian function for the system. In other words, any first-order perturbation of the true evolution results in (at most) second-order changes in $\displaystyle {\mathcal {S}}$ . The action $\displaystyle {\mathcal {S}}$ is a functional, i.e., something that takes as its input a function and returns a single number, a scalar. In terms of functional analysis, Hamilton’s principle states that the true evolution of a physical system is a solution of the functional equation

Hamilton’s principle

$\displaystyle {\frac {\delta {\mathcal {S}}}{\delta \mathbf {q} (t)}}=0$

Euler–Lagrange equations derived from the action integral

Requiring that the true trajectory q(t) be a stationary point of the action functional $\displaystyle {\mathcal {S}}$ is equivalent to a set of differential equations for q(t) (the Euler–Lagrange equations), which may be derived as follows.

Let q(t) represent the true evolution of the system between two specified states q₁ = q(t₁) and q₂ = q(t₂) at two specified times t₁ and t₂, and let ε(t) be a small perturbation that is zero at the endpoints of the trajectory

$\displaystyle {\boldsymbol {\varepsilon }}(t_{1})={\boldsymbol {\varepsilon }}(t_{2})\ {\stackrel {\mathrm {def} }{=}}\ 0$

To first order in the perturbation ε(t), the change in the action functional $\displaystyle \delta {\mathcal {S}}$ would be

$\displaystyle \delta {\mathcal {S}}=\int _{t_{1}}^{t_{2}}\;\left[L(\mathbf {q} +{\boldsymbol {\varepsilon }},{\dot {\mathbf {q} }}+{\dot {\boldsymbol {\varepsilon }}})-L(\mathbf {q} ,{\dot {\mathbf {q} }})\right]dt=\int _{t_{1}}^{t_{2}}\;\left({\boldsymbol {\varepsilon }}\cdot {\frac {\partial L}{\partial \mathbf {q} }}+{\dot {\boldsymbol {\varepsilon }}}\cdot {\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\right)\,dt$

where we have expanded the Lagrangian L to first order in the perturbation ε(t).

Applying integration by parts to the last term results in

$\displaystyle \delta {\mathcal {S}}=\left[{\boldsymbol {\varepsilon }}\cdot {\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\right]_{t_{1}}^{t_{2}}+\int _{t_{1}}^{t_{2}}\;\left({\boldsymbol {\varepsilon }}\cdot {\frac {\partial L}{\partial \mathbf {q} }}-{\boldsymbol {\varepsilon }}\cdot {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\right)\,dt$

The boundary conditions $\displaystyle {\boldsymbol {\varepsilon }}(t_{1})={\boldsymbol {\varepsilon }}(t_{2})\ {\stackrel {\mathrm {def} }{=}}\ 0$ causes the first term to vanish

$\displaystyle \delta {\mathcal {S}}=\int _{t_{1}}^{t_{2}}\;{\boldsymbol {\varepsilon }}\cdot \left({\frac {\partial L}{\partial \mathbf {q} }}-{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}\right)\,dt$

Hamilton’s principle requires that this first-order change $\displaystyle \delta {\mathcal {S}}$ is zero for all possible perturbations ε(t), i.e., the true path is a stationary point of the action functional $\displaystyle {\mathcal {S}}$ (either a minimum, maximum or saddle point). This requirement can be satisfied if and only if

Euler–Lagrange equations

$\displaystyle {\frac {\partial L}{\partial \mathbf {q} }}-{\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {\mathbf {q} }}}}=0$

These equations are called the Euler–Lagrange equations for the variational problem.

Canonical momenta and constants of motion

The conjugate momentum p_k for a generalized coordinate q_k is defined by the equation

$\displaystyle p_{k}\ {\stackrel {\mathrm {def} }{=}}\ {\frac {\partial L}{\partial {\dot {q}}_{k}}}$ .

An important special case of the Euler–Lagrange equation occurs when L does not contain a generalized coordinate q_k explicitly,

$\displaystyle {\frac {\partial L}{\partial q_{k}}}=0\quad \Rightarrow \quad {\frac {d}{dt}}{\frac {\partial L}{\partial {\dot {q}}_{k}}}=0\quad \Rightarrow \quad {\frac {dp_{k}}{dt}}=0\,,$

that is, the conjugate momentum is a constant of the motion.

In such cases, the coordinate q_k is called a cyclic coordinate. For example, if we use polar coordinates t, r, θ to describe the planar motion of a particle, and if L does not depend on θ, the conjugate momentum is the conserved angular momentum.

至今仍然光芒萬丈也☆

何不就借 SymPy

Calculus-related methods.

This module implements a method to find Euler-Lagrange Equations for given Lagrangian.

※ 連接兩點間最短距離之路徑是直線。

好好探究一番耶◎

FreeSandal

每日彙整: 2018-04-13

STEM 隨筆︰古典力學︰運動學【三‧II 】