# 保守力

（重定向自守恒力

## 保守力的性質

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

### 數學證明

1⇒2：

${\displaystyle \int _{\mathbb {S} }(\nabla \times \mathbf {F} )\cdot \mathrm {d} \mathbf {a} =\oint _{\mathbb {C} }\mathbf {F} \cdot \mathrm {d} \mathbf {r} }$

2⇒3：

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

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

3⇒1：

{\displaystyle {\begin{aligned}\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{aligned}}}

## 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. [2012-01-20]. （原始内容存档于2012-01-04）.
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.