# 保守力

## 保守力的性質

1、$\mathbf{F}$旋度是零：
$\nabla \times\mathbf{F} = 0$
2、假設粒子從某閉合路徑 $\mathbb{C}$ 的某一位置，經過這閉合路徑 $\mathbb{C}$ ，又回到原先位置，則力向量場 $\mathbf{F}$ 所做的機械功 $W$ 等於零：
$W = \oint_\mathbb{C} \mathbf{F}\cdot \mathrm{d}\mathbf{r}= 0$
3、 作用力 $\mathbf{F}$ 是某位勢 $\Phi$梯度
$\mathbf{F} = - \nabla \Phi$

### 數學證明

1⇒2：

$\int_\mathbb{S} (\nabla \times \mathbf{F}) \cdot \mathrm{d}\mathbf{a} = \oint_\mathbb{C} \mathbf{F} \cdot \mathrm{d}\mathbf{r}$

2⇒3：

$\Phi(\mathbf{x}) = - \int_\mathbf{O}^\mathbf{x} \mathbf{F} \cdot \mathrm{d}\mathbf{r}$

$\mathbf{F}(\mathbf{x}) = - \nabla \Phi(\mathbf{x})$

3⇒1：

\begin{align}\nabla\times\mathbf{F} & = - \nabla \times \nabla \Phi \\ & = - \left( \frac{\partial^2 \Phi}{\partial y \partial z} - \frac{\partial^2 \Phi}{\partial z \partial y} \right) \hat{x} - \left( \frac{\partial^2 \Phi}{\partial z \partial x} - \frac{\partial^2 \Phi}{\partial x \partial z} \right)\hat{y} - \left( \frac{\partial^2 \Phi}{\partial x \partial y} - \frac{\partial^2 \Phi}{\partial y \partial x} \right)\hat{z} \\ & =\boldsymbol{0}\ \ _{\circ} \\ \end{align}

## 參考文獻

1. ^ David Halliday，《Fundamentals of Physics Extended》，第9版，173：「This result is called the principle of conservation of mechanical energy. (Now you can see where conservative forces got their name.)」，即「遵守力學能『守恆』的力」稱為「守恆力」。
2. ^ HyperPhysics - Conservative force
3. ^ Louis N. Hand, Janet D. Finch. Analytical Mechanics. Cambridge University Press. 1998: 41. ISBN 0521575729.
4. ^ For example, Mechanics, P.K. Srivastava, 2004, page 94: "In general, a force which depends explicitly upon the velocity of the particle is not conservative. (However, the magnetic force (qv×B) can be included among conservative forces in the sense that it acts perpendicular to velocity and hence work done is always zero".
5. ^ For example, The Magnetic Universe: Geophysical and Astrophysical Dynamo Theory, Rüdiger and Hollerbach, page 178.