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戴榕菁
前文《量子论与古老的力学》中提到了两个不适用拉格朗日力学的保守系统的例子。因为很多人对那个两个例子都不熟悉，而那又涉及到对物理学基础尤其是对量子力学理论框架的基础的一种挑战，我觉得有必要对为什么那两个例子不适用拉格朗日力学稍作进一步的说明。
（1）ETDPMS系统
第一个例子是ETDPMS系统，这是我自己设计来表明宏观物理过程不一定总是满足能量守恒的一个例子。我最早是在中文博客给出这个系统的设计【[1]】，并在该文的评论区给该系统取名为“等边三角无平衡态永动架”，不久后又写了英文文章介绍该系统【[2]】，并按照最初起的中文名起了个英文名为“Equilateral Triangle Deequilibrium & Perpetual Motion System”，缩写为ETDPMS。后来发现有网络学校将该文收为参考教材【[3]】。我曾将该文给各知名杂志投稿，但均被拒绝。尽管他们没说，我相信原因很简单：他们根本读不懂！
该运动系统有一个广义变量θ，另外就是时间变量t。该系统的势能为
V = -24ρgl2 F(θ) （1）
F(θ) = sin(4θ)/ U(θ) （1a）,
U(θ) 是θ的一个三角函数。该系统的角动能为：
T = Io(θ)ω2/2 （2）
转动惯量Io(θ) = (2640-29sin22θ)ρl3/3U(θ) （2a）
ω是角速度，它与dθ/dt的关系是：
ω =(dθ/dt)/2 （3）
这是因为θ所在平面与系统转动平面之间有一个夹角。
所以系统的拉格朗日函数L=T -V= Io(θ)ω2/2 + 24ρgl2F(θ) （4）
系统的欧拉-拉格朗日方程为：
∂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）
我们可求出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）
很显然。（10）给出的ω是θ的一个周期函数。但实际上，如果我们用牛顿定律来分析该系统。我们会得到如链接的参考文章中给出那种发散的解。由于整个上述讨论中我们没有考虑摩擦力，所以是一个无耗散的系统。所以，即便在无耗散的条件下，ETDPMS也无法用欧拉-拉格朗日来正确求解；或者说，ETDPMS系统给出了一个拉格朗日力学与牛顿力学不一致的宏观无耗散系统的例子。
2. DDWFTTW
与ETDPMS相比，大家会更熟悉DDWFTTW系统，因为网上有很多实际运行的DDWFTTW系统的视频。如果说有人会因为我还没有能实际做出一个ETDPMS系统而不相信该解的真实性的话，DDWFTTW是已经在世界上运行了十多年的系统，因此它作为自然界的一个真实的解是没有疑问的。
不过，DDWFTTW因为涉及到固体与气体的相互作用，用拉格朗日力学求解还是比较复杂的，我本人并没有实际去那样求解，但这并不妨碍我们来分析为什么它无法用拉格朗日力学求解。乍看起来似乎我们可以把坐标系设在运动的DDWFTTW的刚体车身上，但这样一来我们必须把整个宇宙的动能考虑进去了------这也反映出用拉格朗日力学的一个局限性。
不过不要紧，因为诺伊德已经在更一般的条件下用最小作用原理证明了任意多点的体系满足能量守恒，但是如我在去年和前年已反复论证指出的，DDWFTTW显然是打破了能量守恒的。也就是说，DDWFTTW是不符合诺伊德用最小作用原理得出的结论的！
讨论
ETDPMS和DDWFTTW给出了两个在宏观世界不符合最小作用原理的例子。另外，尽管我还没有具体深入去分析，凭直觉我感到那个著名的Chain Fountain【[4]】也不符合最小作用原理。
虽然我这里讨论的不满足最小作用原理的都是宏观世界的例子，而量子力学研究的是微观世界。但有两点需要引起我们的注意：1）本文给出的例子表明，最小作用原理并非天生合理地一定成立，而人们至今没有一个理论论述为什么它应该成立；2）量子场论有一个对于运用最小作用原理来说很危险的假设，那就是基本作用力的介子都是从真空（或凝聚态）借来的能量，而这过程是短暂地打破能量守恒的------这对最小作用原理是大大地不好不好地。。。。ETDPMS和DDWFTTW这两个例子已表明，只要不符合能量守恒，就可能不符合最小作用原理。
但问题是，假如（仅仅是假如）最小作用原理不能用在量子力学的话，那样对量子论来说的后果比推翻狭义相对论要严重的不知多少倍了。因为整个量子力学的理论框架都是建立在最小作用原理之上的！
或许又一个暴风雨快要来了。。。。。。
【[1]】 戴榕菁. 杠杆原理与永动机
【[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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