Page 156 - Intro to Tensor Calculus

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Figure 1.5-2. Parabolic and elliptic conic sections

The equation of motion of mass m with respect to mass M is obtained from Newton’s second law. Let
~ ρ = ρ b e ρ denote the position vector of mass m with respect to the origin. Newton’s second law can then be
written in any of the forms

~
2
−GmM d ~ρ dV −GmM
~
F = b e ρ = m = m = ~ ρ (1.5.128)
ρ 2 dt 2 dt ρ 3
and from this equation we can show that the motion of the mass m can be described as a conic section.
Recall that a conic section is defined as a locus of points p(x, y) such that the distance of p from a fixed
point (or points), called a focus (foci), is proportional to the distance of the point p from a fixed line, called
a directrix, that does not contain the fixed point. The constant of proportionality is called the eccentricity
and is denoted by the symbol .For = 1 a parabola results; for 0 ≤ ≤ 1 an ellipse results; for > 1a
hyperbola results; and if = 0 the conic section is a circle.
With reference to figure 1.5-2, a conic section is defined in terms of the ratio FP = where FP = ρ and
PD
PD =2q − ρ cos φ. From the ratio we solve for ρ and obtain the polar representation for the conic section
p
ρ = (1.5.129)
1+ cos φ