Noether定理
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Noether定理
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对于任何一种在坐标连续变化下系统的哈密顿作用量的不变性都存在相应的运动积分
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位形空间的运动积分
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设拉格朗日函数$L(q,\dot{q},t)$描述一封闭系统，设无限小变化为
$$
t\to\overline{t}=t+\epsilon\xi(q(t),t)\\
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q_i \to \overline{q}_i=q_i+\epsilon\eta_{i}(q(t),t)$$
如果系统的哈密顿作用量在变换下保持不变，或拉格朗日函数具有形式协变性，即
$$\overline{S}=\int_{\overline{t}_1}^{\overline{t}_2}L(\overline{q}(\overline{t}),\frac{d\overline{q}(\overline{t})}{d\overline{t}},\overline{t})d\overline{t}=\int_{t_1}^{t_2}L(q(t),\dot{q}(t),t)dt=S\\
\quad\\
L(\overline{q},\frac{d\overline{q}}{d\overline{t}},\overline{t})\frac{d\overline{t}}{dt}=L(q,\dot{q},t)+\epsilon\frac{d}{dt}\Lambda(q,t)
$$
则系统的运动积分为
$$I=\sum\limits_{i=1}^{n}\eta_{i}p_i-\xi H-\Lambda$$
$\nabla$证
$$\small首先\frac{d\overline{q}_i}{d\overline{t}}=\frac{d(q_i+\epsilon\eta_i)}{dt}=\frac{1}{\frac{d(t+\eta\xi)}{dt}}=[\dot{q_i}+\epsilon\dot{\eta_i}][1-\epsilon\dot{\xi}]=\dot{q_i}+\epsilon(\dot{\eta_i}-\dot{q_i}\dot{\xi})\\\\ L(\overline{q},\frac{d\overline{q}}{d\overline{t}},\overline{t})\frac{d\overline{t}}{dt}=L(q,\dot{q},t)+\epsilon\frac{d}{dt}\Lambda(q,t)\\\\ \quad\\\\ \small一阶展开\\\\ \quad\\\\ \scriptsize[L(q,\dot{q},t)+\sum\limits_{i=1}^{n}\frac{\partial L(q,\dot{q},t)}{\partial q_i}(\overline{q}_i-q_i)+\sum\limits_{i=1}^{n}\frac{\partial L(q,\dot{q},t)}{\partial \dot{q_i}}(\frac{d\overline{q}_i}{d\overline{t}}-\dot{q_i})+\frac{\partial L(q,\dot{q},t)}{\partial t}(\overline{t}-t)] [1+\epsilon\frac{d\xi}{dt}]\\\\ \quad\\\\=L(q,\dot{q},t)+\epsilon\frac{d}{dt}\Lambda(q,t)\\\\ \quad\\\\ \Downarrow\\\\ \quad\\\\ \scriptsize[L(q,\dot{q},t)+\sum\limits_{i=1}^{n}\frac{\partial L(q,\dot{q},t)}{\partial q_i}\epsilon\eta_i+\sum\limits_{i=1}^{n}\frac{\partial L(q,\dot{q},t)}{\partial \dot{q_i}}\epsilon(\dot{\eta_i}-\dot{q_i}\dot{\xi})+\frac{\partial L(q,\dot{q},t)}{\partial t}\epsilon\xi] [1+\epsilon\frac{d\xi}{dt}]\\\\ \quad\\\\=L(q,\dot{q},t)+\epsilon\frac{d}{dt}\Lambda(q,t)\\\\ \quad\\\\ \Downarrow\\\\ \quad\\\\ L\frac{d\xi}{dt}+\sum\limits_{i=1}^{n}\frac{\partial L}{\partial q_i}\eta_i+\sum\limits_{i=1}^{n}\frac{\partial L}{\partial \dot{q_i}}(\dot{\eta_i}-\dot{q_i}\dot{\xi})+\frac{\partial L}{\partial t}\xi-\frac{d\Lambda}{dt}=0\\\\ \quad\\\\ \Downarrow\\\\ \quad\\\\ \frac{d[L\xi]}{dt}+\sum\limits_{i=1}^{n}\frac{\partial L}{\partial q_i}(\eta_i-\dot{q_i}\xi)+\sum\limits_{i=1}^{n}\frac{\partial L}{\partial \dot{q_i}}(\dot{\eta_i}-\dot{q_i}\dot{\xi}-\ddot{q_i}\xi)-\frac{d\Lambda}{dt}=0\\\\ \quad\\\\ \Downarrow\\\\ \quad\\\\ \sum\limits_{i=1}^{n}(\eta_i-\dot{q_i}\xi)[\frac{\partial L}{\partial q_i}-\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q_i}}\right)] +\\\\ \quad\\\\ \sum\limits_{i=1}^{n}\frac{d}{dt}\left[(\eta_i-\dot{q_i}\xi)\frac{\partial L}{\partial \dot{q_i}}\right]+\frac{d[L\xi]}{dt}-\frac{d\Lambda}{dt}=0\\\\ \quad\\\\ \small故运动积分为\\\\ \quad\\\\ \sum\limits_{i=1}^{n}\left[(\eta_i-\dot{q_i}\xi)\frac{\partial L}{\partial \dot{q_i}}\right]+L\xi-\Lambda =\sum\limits_{i=1}^{n}\eta_ip_i-\xi H-\Lambda $$
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参考与引用来源
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