查看“等效位能”的源代码

等效位能是將許多效應綜合成單一位能的數學表達式。在[[古典力學]]中，等效位能即為離心力位能與位能的和。等效位能常被用來計算[[行星]]的軌道及半古典的原子計算，可用來降低問題的維度。

==定義==
等效位能 <math>U_\text{eff}</math> 可定義如下：

<math> U_\text{eff}(\mathbf{r}) = \frac{L^2}{2mr^2} + U(\mathbf{r}) </math>
:''L'' 為[[角動量]]。
:''r'' 為兩物體之距離。
:''m'' 為繞行物體之質量。
:''U(r)'' 為位能。

等效力，也就是等效位能的梯度取負號則為：

<math> \begin{align}
\mathbf{F}_\text{eff} &= -\nabla U_\text{eff}(\mathbf{r}) \\
                      &= \frac{L^2}{mr^3}\hat{\mathbf{r}} - \nabla U(\mathbf{r})
\end{align}
</math>

其中 <math>\hat{\mathbf{r}}</math> 為r方向的單位向量。

==特性==
等效位能有許多有用的特性：

:<math> U_\text{eff} \leq E </math>

要找到一個圓形軌道的半徑，我們只要將等效位能對r取最小值，或是找<math>r_0</math>使總力為0：

:<math> \frac{d U_\text{eff}}{dr} = 0 </math>
在解出<math>r_0</math>後，代回<math>U_\text{eff}</math>以求等效位能之最大值<math>U_\text{eff}^\text{max}</math>。

也可求得微小振盪之頻率：

:<math> \omega = \sqrt{\frac{U_\text{eff}''}{m}} </math>
其中[[角分符號]]代表對r的微分。

==例子：重力位能==
以質量為m的小天體繞行質量為M的大天體為例，並假設M遠大於m。在牛頓力學中，大天體的運動可被忽略，且[[能量]]E及[[角動量]]L守恆：

:<math>E = \frac{1}{2}m\left(\dot{r}^2 + r^2\dot{\phi}^2\right) - \frac{GmM}{r},</math>
:<math>L = mr^2\dot{\phi} \, </math>

其中

:<math>\dot{r}</math> 為r對時間的微分，
:<math>\dot{\phi}</math> 為小天體之[[角速度]]，
:''G'' 為重力常數。

因為整個運動發生在一平面，所以我們只需要兩個變數r及<math>\phi</math>。將第二個式子代回第一個式子，整理之後可得

:<math>m\dot{r}^2 = 2E - \frac{L^2}{mr^2} + \frac{2GmM}{r} = 2E - \frac{1}{r^2}\left(\frac{L^2}{m} - 2GmMr\right),</math>
:<math>\frac{1}{2}m\dot{r}^2 = E - U_\text{eff}(r),</math>

其中

:<math>U_\text{eff}(r) = \frac{L^2}{2mr^2} - \frac{GmM}{r} </math>

即為等效位能。<ref group="Note">A similar derivation may be found in Jos&eacute; & Saletan, ''Classical Dynamics: A Contemporary Approach'', pgs. 31–33</ref> 如同上式所述，原問題的兩個變數被化簡成單變數的問題。在許多應用中，等效位能可視為一維系統的位能，譬如說用等效位能來決定轉折點及[[力學平衡|穩定平衡區]]。類似的方法可用來決定[[廣義相對論]]中[[史瓦西度規]]的軌道。


==注释==
<references group="Note"/>

==参考资料==
{{Reflist}}

* {{Cite book|last1=Jos&eacute;|first1=JV|last2=Saletan|first2=EJ|year=1998|title=Classical Dynamics: A Contemporary Approach|edition=1st|publisher=Cambridge University Press|isbn=0-521-63636-1|postscript=<!-- Bot inserted parameter. Either remove it; or change its value to "." for the cite to end in a ".", as necessary. -->{{inconsistent citations}}}}.

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    |title = Gaussian effective interaction between flexible dendrimers of fourth generation: a theoretical and experimental study 
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{{DEFAULTSORT:Effective Potential}}
[[Category:Mechanics]]