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[parent] motion in central-force field (Definition)

Let us consider a body with mass $m$ in a gravitational force field exerted by the origin and directed always from the body towards the origin.  Set the plane through the origin and the velocity vector $\vec{v}$ of the body.  Apparently, the body is forced to move constantly in this plane, i.e. there is a question of a planar motion.  We want to derive the trajectory of the body.

Equip the plane of the motion with a polar coordinate system $r,\,\varphi$ and denote the position vector of the body by $\vec{r}$.  Then the velocity vector is

$\displaystyle \vec{v} \;=\; \frac{d\vec{r}}{dt} \;=\; \frac{d}{dt}(r\vec{r}^{\,0}) \;=\; \frac{dr}{dt}\vec{r}^{\,0}+r\frac{d\varphi}{dt}\vec{s}^{\,0},$ (1)
where $\vec{r}^{\,0}$ and $\vec{s}^{\,0}$ are the unit vectors in the direction of $\vec{r}$ and of $\vec{r}$ rotated 90 degrees anticlockwise ( $\vec{r}^{\,0} = \vec{i}\cos\varphi+\vec{j}\sin\varphi$,  whence  $\frac{\vec{r}^{\,0}}{dt} = (-\vec{i}\sin\varphi+\vec{j}\cos\varphi)\frac{d\varphi}{dt} = \frac{d\varphi}{dt}\vec{s}^{\,0}$).  Thus the kinetic energy of the body is

$\displaystyle E_k \;=\; \frac{1}{2}m\left\vert\frac{d\vec{r}}{dt}\right\vert^2 ... ...!\left(\frac{dr}{dt}\right)^2\!+\!\left(r\frac{d\varphi}{dt}\right)^2\right)\!.$
Because the gravitational force on the body is exerted along the position vector, its moment is 0 and therefore the angular momentum

$\displaystyle \vec{L} \;=\; \vec{r}\!\times\!m\frac{d\vec{r}}{dt} \;=\; mr^2\frac{d\varphi}{dt}\vec{r}^{\,0}\!\times\!\vec{s}^{\,0}$
of the body is constant; thus its magnitude is a constant,

$\displaystyle mr^2\frac{d\varphi}{dt} \;=\; G,$
whence
$\displaystyle \frac{d\varphi}{dt} \;=\; \frac{G}{mr^2}.$ (2)
The central force  $\displaystyle\vec{F} := -\frac{k}{r^2}\vec{r}^{\,0}$  (where $k$ is a constant) has the scalar potential   $U(r) = -\frac{k}{r}$.  Thus the total energy  $E = E_k\!+\!U(r)$ of the body, which is constant, may be written

$\displaystyle E \;=\; \frac{1}{2}m\!\left(\frac{dr}{dt}\right)^2\!+\frac{1}{2}m... ... \frac{m}{2}\!\left(\frac{dr}{dt}\right)^2\!+\frac{G^2}{2mr^2}\!-\!\frac{k}{r}.$
This equation may be revised to

$\displaystyle \left(\frac{dr}{dt}\right)^2\!+\frac{G^2}{m^2r^2}-\frac{2k}{mr}+\frac{k^2}{G^2} \;=\; \frac{2E}{m}+\frac{k^2}{G^2},$
i.e.

$\displaystyle \left(\frac{dr}{dt}\right)^2\!+\left(\frac{k}{G}-\frac{G}{mr}\right)^2 \;=\; q^2$
where

$\displaystyle q \;:=\; \sqrt{\frac{2}{m}\left(\!E\!+\!\frac{mk^2}{2G^2}\right)}$
is a constant.  We introduce still an auxiliary angle $\psi$ such that
$\displaystyle \frac{k}{G}-\frac{G}{mr} \;=\; q\cos\psi, \quad \frac{dr}{dt} \;=\; q\sin\psi.$ (3)
Differentiation of the first of these equations implies

$\displaystyle \frac{G}{mr^2}\cdot\frac{dr}{dt} \;=\; -q\sin\psi\frac{d\psi}{dt} \;=\; -\frac{dr}{dt}\cdot\frac{d\psi}{dt},$
whence, by (2),

$\displaystyle \frac{d\psi}{dt} \;=\; -\frac{G}{mr^2} \;=\; -\frac{d\varphi}{dt}.$
This means that  $\psi = C\!-\!\varphi$, where the constant $C$ is determined by the initial conditions.  We can then solve $r$ from the first of the equations (3), obtaining
$\displaystyle r \;=\; \frac{G^2}{km\left(1-\frac{Gq}{k}\cos(C-\varphi)\right)} \;=\; \frac{p}{1-\varepsilon\cos(\varphi-C)},$ (4)
where

$\displaystyle p \;:=\; \frac{G^2}{km}, \quad \varepsilon \;:=\; \frac{Gq}{k}.$

The result (4) shows that the trajectory of the body in the gravitational field of one point-like sink is always a conic section whose focus contains the sink causing the field.

As for the type of the conic, the most interesting one is an ellipse.  It occurs when  $\varepsilon < 1$.  This condition is easily seen to be equivalent with a negative total energy $E$ of the body.

One can say that any planet revolves around the Sun along an ellipse having the Sun in one of its foci — this is Kepler's first law.



"motion in central-force field" is owned by pahio.

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See Also: Kepler's first two laws of planetary motion, Kepler's third law of planetary motion, Kepler's three laws of planetary motion summarized

Other names:  Kepler's first law

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Cross-references: field, section, energy, scalar, magnitude, angular momentum, kinetic energy, unit vectors, position vector, system, motion, vector, velocity
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This is version 6 of motion in central-force field, born on 2009-03-31, modified 2009-04-04.
Object id is 615, canonical name is MotionInCentralForceField.
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Physics Classification: 45.50.Pk (Celestial mechanics )
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