# JK 也能听懂的理论物理（一）

09-9-2020

### 时空

1887年，抱着证明以太存在的想法，迈克尔逊(Michelson)和莫雷(Morley)尝试使用实验测量以太对光速的影响，但没有获得预期结果。在传统力学环境下，不同参考系的光速可变似乎是合理的，而光速可变势必证明了传播光乃至电磁活动的介质存在。他们的实验数据启发了洛伦兹(Lorentz)在1904年使用洛伦兹变换解释该结果，使用以下公式：

$L = L_0\sqrt{1 - \frac{v^2}{c^2}}$

1. 光速不变性原理：在所有参考系中，光速是一个确定的常数 $$c$$。
2. 相对性原理：所有参考系都有相同的物理定理形式，这也是伽利略(Galileo)在经典力学得出的结论的推广。

Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.
– Hermann Minkowski

$(c\Delta t_1)^2 - (\Delta x_1)^2 - (\Delta y_1)^2 - (\Delta z_1)^2 = (c\Delta t_2)^2 - (\Delta x_2)^2 - (\Delta y_2)^2 - (\Delta z_2)^2$

$ds^2 = \begin{pmatrix} dx_0 & dx_1 & dx_2 & dx_3 \end{pmatrix} \begin{pmatrix} 1 & 0 & 0 & 0 \\ 0 & -1 & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{pmatrix} \begin{pmatrix} dx_0 \\ dx_1 \\ dx_2 \\ dx_3 \end{pmatrix} = dx_{\mu}\eta^{\mu\nu}dx_{\nu}$

$dx_{\mu} \to dx'_{\mu} = \Lambda_{\mu}^{\sigma}dx_{\sigma}$

$ds'^2 = dx'_{\mu}\eta^{\mu\nu}dx'_{\nu} = \Lambda_{\mu}^{\sigma}dx_{\mu}\eta^{\mu\nu}\Lambda_{\nu}^{\gamma}dx_{\gamma} = dx_{\mu}\eta^{\mu\nu}dx_{\nu}$

$\Lambda_{\sigma}^{\mu}\eta^{\sigma\gamma}\Lambda_{\gamma}^{\nu} = \eta^{\mu\nu}$

$\Lambda^T\eta\Lambda = \eta$

$\begin{pmatrix} \gamma & -\beta\gamma & 0 & 0 \\ -\beta\gamma & \gamma & 0 & 0 \\ 0 & 0 & 1 & 0 \\ 0 & 0 & 0 & 1 \end{pmatrix}$

### 拉格朗日量

$\frac{\partial \mathcal{L}}{\partial x} - \frac{d}{dt}(\frac{\partial \mathcal{L}}{\partial \dot{x}}) = 0$

$-\frac{\partial V}{\partial x} = \frac{d(m\dot{x})}{dt} = m\ddot{x}$

### 附录：欧拉-拉格朗日方程的推导

$S' - S = \int_{t_1}^{t_2}(\mathcal{L}(x + \epsilon, \dot{x} + \dot{\epsilon}, t) - \mathcal{L}(x, \dot{x}, t))dt = 0$

$\mathcal{L}(x + \epsilon, \dot{x} + \dot{\epsilon}, t) = \mathcal{L}(x) + (x + \epsilon - x)\frac{\partial \mathcal{L}}{\partial x} + (\dot{x} + \dot{\epsilon} - \dot{x})\frac{\partial \mathcal{L}}{\partial \dot{x}} + ...$

$\int_{t_1}^{t_2}(\epsilon\frac{\partial \mathcal{L}}{\partial x} + \dot{\epsilon}\frac{\partial \mathcal{L}}{\partial \dot{x}})dt = 0$

$\int_{t_1}^{t_2}dt\dot{\epsilon}\frac{\partial \mathcal{L}}{\partial \dot{x}} = \epsilon\frac{\partial \mathcal{L}}{\partial \dot{x}}\biggr\rvert_{t_1}^{t_2} - \int_{t_1}^{t_2}dt\epsilon\frac{d}{dt}(\frac{\partial \mathcal{L}}{\partial \dot{x}})$

$\int_{t_1}^{t_2}dt\epsilon(\frac{\partial \mathcal{L}}{\partial x} - \frac{d}{dt}(\frac{\partial \mathcal{L}}{\partial \dot{x}})) = 0$

$\frac{\partial \mathcal{L}}{\partial x} - \frac{d}{dt}(\frac{\partial \mathcal{L}}{\partial \dot{x}}) = 0$

### 参考资料

1. Physics from Symmetry(2nd Edition), J. Schwichtenberg, Springer, 2018.
2. Prof Kenneth Young on A Special Lecture: Principle of Least Action, The Chinese University of Hong Kong.
3. 最小作用量原理, 林琦焜, 數學傳播 35卷1期, 2009.
4. Many entries on Wikipedia.