Section 4.2 & Figure 2 • Planetary Orbits (Central Forces)
dL/dt = 0 (L = |r||p|sinθ Conserved)
As the planet orbits the star (Figure 2), it accelerates as it nears periapsis (|r| small, |p| large) and decelerates toward apoapsis (|r| large, |p| small), while the angle θ between ext. r and p is only 90° at the apsides. All three variables—|r|, |p|, and sin(θ)—continuously compensate to keep L = |r||p|sin(θ) strictly constant, while rotational kinetic energy Krot = L²/(2mr²) surges near periapsis.
Interactive: Drag the planet directly on the canvas to scrub its position
Visual Overlays:
Dynamic Balancing of Variables (Figure 2): L = |r| · |p| · sin(θ) & Krot = L² / (2mr²)
Bars show each factor normalized between its orbital min and max. Notice how |r| and |p| move in opposite directions while sin(θ) dips between apsides, and Krot varies as 1/|r|².
Radius |r|
mid-orbit
1.500
min
max
×
Momentum |p|
mid-speed
1.200
min
max
×
sin(θ)
θ = 90.0°
1.000
min
max 1.00 (90°)
Instantaneous Product:
|r|(1.500) × |p|(1.200) × sin(90.0°)(1.000)
L =
1.800
CONSTANT
Rot. KE (Krot = ½Iω² = L²/2mr²):
0.360
∝ 1/|r|²
min (apo)
Ktot = 0.720
max (peri)
Pos. |r|
VARYING
1.500
Mom. |p|
VARYING
1.200
Angle θ
sinθ=1.00
90.0°
Lever |r|sinθ
b = r⊥
1.500
p⊥ = |p|sinθ = 1.200
Krot
VARYING
0.360
I=2.25 • ω=0.53
L = |r×p|
CONSTANT
1.800
|r||p|sinθ Conserved
dA/dt
CONSTANT
0.900
|L| / 2m (Eq. 8)
Interactive System Parameters
Modify geometry & dynamics in real time
1.0×
Presets:
0.55
0.00 (Circular, θ=90°)
0.75 (Large |r|, |p|, K_rot swings)
1.75 AU
1.00 kg
45°
Section 2 Proof • Equation (2) Live Evaluator
dL/dt = (dr/dt × p) + (r × dp/dt)
1. (dr/dt × p) = m(v × v)
Eq. (3): Cross product of collinear vectors is identically zero
2. (r × dp/dt) = r × F = τ
Eq. (4): Central force F ∥ −r ⟹ r × F = 0
Eq. (5): dL/dt = 0 + τ
dL/dt = 0.000 (L Conserved)
Real-Time Strip Chart: |r|, |p|, sin(θ), Krot vs. L
Gold line L stays flat while |r|, |p|, sin(θ), and Krot fluctuate
|r(t)|
|p(t)|
sin θ
Krot(t)
L