The differential equation for the path of a planet around the sun (or any object in an inverse square force field) is, in polar coordinates, 1 r 2 d d θ 1 r 2 d r d θ − 1 r 3 = − k r 2 . Make the substitution u = 1 / r and solve the equation to show that the path is a conic section .
The differential equation for the path of a planet around the sun (or any object in an inverse square force field) is, in polar coordinates, 1 r 2 d d θ 1 r 2 d r d θ − 1 r 3 = − k r 2 . Make the substitution u = 1 / r and solve the equation to show that the path is a conic section .
The differential equation for the path of a planet around the sun (or any object in an inverse square force field) is, in polar coordinates,
1
r
2
d
d
θ
1
r
2
d
r
d
θ
−
1
r
3
=
−
k
r
2
.
Make the substitution
u
=
1
/
r
and solve the equation to show that the path is a conic section.
Curve that is obtained by the intersection of the surface of a cone with a plane. The three types of conic sections are parabolas, ellipses, and hyperbolas. The main features of conic sections are focus, eccentricity, and directrix. The other parameters are principal axis, linear eccentricity, latus rectum, focal parameter, and major and minor axis.
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