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戴榕菁
前文《量子論與古老的力學》中提到了兩個不適用拉格朗日力學的保守係統的例子。因為很多人對那個兩個例子都不熟悉，而那又涉及到對物理學基礎尤其是對量子力學理論框架的基礎的一種挑戰，我覺得有必要對為什麽那兩個例子不適用拉格朗日力學稍作進一步的說明。
（1）ETDPMS係統
第一個例子是ETDPMS係統，這是我自己設計來表明宏觀物理過程不一定總是滿足能量守恒的一個例子。我最早是在中文博客給出這個係統的設計【[1]】，並在該文的評論區給該係統取名為“等邊三角無平衡態永動架”，不久後又寫了英文文章介紹該係統【[2]】，並按照最初起的中文名起了個英文名為“Equilateral Triangle Deequilibrium & Perpetual Motion System”，縮寫為ETDPMS。後來發現有網絡學校將該文收為參考教材【[3]】。我曾將該文給各知名雜誌投稿，但均被拒絕。盡管他們沒說，我相信原因很簡單：他們根本讀不懂！
該運動係統有一個廣義變量θ，另外就是時間變量t。該係統的勢能為
V = -24ρgl2 F(θ) （1）
F(θ) = sin(4θ)/ U(θ) （1a）,
U(θ) 是θ的一個三角函數。該係統的角動能為：
T = Io(θ)ω2/2 （2）
轉動慣量Io(θ) = (2640-29sin22θ)ρl3/3U(θ) （2a）
ω是角速度，它與dθ/dt的關係是：
ω =(dθ/dt)/2 （3）
這是因為θ所在平麵與係統轉動平麵之間有一個夾角。
所以係統的拉格朗日函數L=T -V= Io(θ)ω2/2 + 24ρgl2F(θ) （4）
係統的歐拉-拉格朗日方程為：
∂L/∂θ – (d(∂L/∂ω)/dt)/2 = 0 （5）
而∂L/∂θ = (ω2/2)dIo(θ)/dθ + 24ρgl2dF(θ)/dθ （6）
d(∂L/∂ω)/dt = d(Io(θ)ω)/dt = Io(θ)dω/dt + ω(dIo(θ)/dθ) (dθ/dt)
= Io(θ)dω/dt + 2ω2 dIo(θ)/dθ （7）
我們可求出dω/dt = 42√3g G(θ)/29l （8）
G(θ) = 1-2640/(2640-29sin22θ) （9）
把(6), (7), (8)代入(5)：
5ω2 (dIo(θ)/dθ)/2 = X(θ) （10）
X(θ) = -[24ρgl2dF(θ)/dθ + 42√3g Io(θ)G(θ)/29l] （10a）
很顯然。（10）給出的ω是θ的一個周期函數。但實際上，如果我們用牛頓定律來分析該係統。我們會得到如鏈接的參考文章中給出那種發散的解。由於整個上述討論中我們沒有考慮摩擦力，所以是一個無耗散的係統。所以，即便在無耗散的條件下，ETDPMS也無法用歐拉-拉格朗日來正確求解；或者說，ETDPMS係統給出了一個拉格朗日力學與牛頓力學不一致的宏觀無耗散係統的例子。
2. DDWFTTW
與ETDPMS相比，大家會更熟悉DDWFTTW係統，因為網上有很多實際運行的DDWFTTW係統的視頻。如果說有人會因為我還沒有能實際做出一個ETDPMS係統而不相信該解的真實性的話，DDWFTTW是已經在世界上運行了十多年的係統，因此它作為自然界的一個真實的解是沒有疑問的。
不過，DDWFTTW因為涉及到固體與氣體的相互作用，用拉格朗日力學求解還是比較複雜的，我本人並沒有實際去那樣求解，但這並不妨礙我們來分析為什麽它無法用拉格朗日力學求解。乍看起來似乎我們可以把坐標係設在運動的DDWFTTW的剛體車身上，但這樣一來我們必須把整個宇宙的動能考慮進去了------這也反映出用拉格朗日力學的一個局限性。
不過不要緊，因為諾伊德已經在更一般的條件下用最小作用原理證明了任意多點的體係滿足能量守恒，但是如我在去年和前年已反複論證指出的，DDWFTTW顯然是打破了能量守恒的。也就是說，DDWFTTW是不符合諾伊德用最小作用原理得出的結論的！
討論
ETDPMS和DDWFTTW給出了兩個在宏觀世界不符合最小作用原理的例子。另外，盡管我還沒有具體深入去分析，憑直覺我感到那個著名的Chain Fountain【[4]】也不符合最小作用原理。
雖然我這裏討論的不滿足最小作用原理的都是宏觀世界的例子，而量子力學研究的是微觀世界。但有兩點需要引起我們的注意：1）本文給出的例子表明，最小作用原理並非天生合理地一定成立，而人們至今沒有一個理論論述為什麽它應該成立；2）量子場論有一個對於運用最小作用原理來說很危險的假設，那就是基本作用力的介子都是從真空（或凝聚態）借來的能量，而這過程是短暫地打破能量守恒的------這對最小作用原理是大大地不好不好地。。。。ETDPMS和DDWFTTW這兩個例子已表明，隻要不符合能量守恒，就可能不符合最小作用原理。
但問題是，假如（僅僅是假如）最小作用原理不能用在量子力學的話，那樣對量子論來說的後果比推翻狹義相對論要嚴重的不知多少倍了。因為整個量子力學的理論框架都是建立在最小作用原理之上的！
或許又一個暴風雨快要來了。。。。。。
【[1]】 戴榕菁. 杠杆原理與永動機
【[2]】R. Dai (2022) “De-equilibrium State & Perpetual Motion Machine”. Retrieved from: https://www.researchgate.net/publication/358575784_De-equilibrium_State_Perpetual_Motion_Machine_Rongqing_Dai
【[3]】R. Dai (2022) “De-equilibrium State & Perpetual Motion Machine”. Retrieved from: https://www.coursehero.com/file/130822719/DeequilibriumPerpetualMotionMachinedocx/
【[4]】R. Dai (2022) “The Dynamics of the Chain Fountain”. Retrieved from: https://globaljournals.org/GJSFR_Volume22/1-The-Dynamics-of-the-Chain.pdf
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