Lecture Notes | A blue-sky wanderer

1 Central Force and Central Potential

The central force on an object is a force directed along the line joining the object and the origin. We often describe such forces as

$\mathbf{F}(\mathbf{r})=F(\mathbf{r})\hat{\mathbf{r}},$ …(1)

where $\mathbf{r}$ is the position, or called radius vector, with $\hat{\mathbf{r}}$ being its unit vector (or called unit radial vector) and $r$ being its magnitude. $\mathbf{F}$ is the force as a vector-valued function of position. The (vector) field derived from a central force is called a central field.

Here we restrict our attention to conservative systems (i.e. without energy dissipation due to non-conservative forces, such as friction), then the magnitude $F(r)$ of a central force can always be expressed as the derivative of a time-independent potential energy function $U(r)$ :

$F(r)=-\dfrac{\text{d}U(r)}{\text{d}r}.$ …(2)

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Kepler Problem

1 Central Force and Central Potential