General formulae for perturbations

In the following we will use the same notations as before: vectors are bold, time derivative is dot, $\mu = GM$. Position vector is ${\bf r}$, velocity vector is ${\bf v}$, i.e.,

$$\dot{{\bf r}}= {\bf v}.$$

scalar product is ${\bf r}\cdot {\bf v}$ for two vectors, ${\bf r}$ and ${\bf v}$. The length of ${\bf r}$ is $\Vert {\bf r}\Vert$, also denoted by $r$, the length of ${\bf v}$ is $\Vert {\bf v}\Vert = v$. We will need some basic facts from vector analysis.

$$r = \sqrt{{\bf r}\cdot {\bf r}}, \;\;\;\;\ v = \sqrt{{\bf v}\cdot {\bf v}}$$

From

$$\dot{(r^2)} = \frac{d}{\rd t}({{\bf r}\cdot {\bf r}}) = 2 {\bf r}\cdot \dot{{\bf r}} = 2 {\bf r}\cdot {\bf v}$$

and

$$\dot{(r^2)} = 2 r \dot{r}$$

we obtain

$$r \dot{r} = {\bf r}\cdot {\bf v}$$

The following general vector identity will be particularly useful::

$${\bf a} \times ({\bf b} \times {\bf c}) = ({\bf a} \cdot {\bf c}) {\bf b} - ({\bf a} \cdot {\bf b}) {\bf c}$$

Consider the perturbed Kepler problem

$$\ddot{{\bf r}} = -\frac{\mu}{r^3}{\bf r} + {\bf F}$$

This is the differential equation for the Kepler problem which we encountered (and solved) before, however, it contains an addtional term ${\bf F}$ which is the perturbation. For now we leave its form unspecified, but we note that we will assume that this force is small, when compared to the central gravitational force $-\frac{\mu}{r^3}{\bf r}$. The mass of the particle whose motion is under consideration does not enter our equations.