Conservative force

A Force $F$ is called conservative iff $\exists V:F=-\nabla V$, where

• $V$ is the associated Potential energy
• $\nabla V$ is the Del or Nabla of $V$

Tip

For a Force $F(x):{\mathbb{R}}^n\to {\mathbb{R}}^n$ with no dependency on the velocity $v=\dot{x}$, Total Energy is constant only if the Force is conservative

Properties

A force is conservative iff the partial derivatives commute

$F$ is a conservative force iff $\frac{\delta F_j}{\delta x_k}=\frac{\delta F_k}{\delta x_j}$ for the domain $U$ of $F$, where

• $U$ is a simply connected domain in ${\mathbb{R}}^n$
• $F:U\to {\mathbb{R}}^n$ is a smooth function
• $V$ is the Potential energy associated with the Force $F$.

Proof

$\Rightarrow$

If $F$ is conservative, $\exists V:F=-\nabla V$ . Hence $\frac{\partial F_j}{\partial x_k}=\frac{{\partial }^{2}V}{\partial x_k\partial x_j}=\frac{{\partial }^{2}V}{\partial x_j\partial x_k}=\frac{\partial F_k}{\partial x_j}$

$\Leftarrow$

Conversely, If $F$ satisfies $\frac{\partial F_j}{\partial x_k}=\frac{\partial F_k}{\partial x_j}$ we can use path integrals to define $V$, and use Stokes theorem to show that the integral is independent of the choice of paths.

• Rename the result above and move it to it’s own note.