Kepler’s Laws from Comoment Maps
1
Introduction
2
Foundations
▶
2.1
Phase Space
2.2
Energy and Conserved Vectors
3
Dynamics
▶
3.1
The Vector Field
4
Conserved Quantities
▶
4.1
Constancy Along Solutions
4.2
Algebraic Relations
5
Poisson Brackets and Comoment Maps
▶
5.1
Canonical Bracket
5.2
Rotation Observables
6
Hidden \(\mathfrak {so}(4)\) Symmetry
▶
6.1
Fixed Negative-Energy Surface
7
Orbit Geometry and Kepler’s First Law
▶
7.1
Geometric Parameters
7.2
Orbit Equation
8
Kepler’s Second Law
▼
8.1
Areal Quantities
9
Periods and Kepler’s Third Law
▶
9.1
Periods
10
Kepler’s Laws: Combined Statement
Dependency graph
8 Kepler’s Second Law
We define areal velocity and prove the equal-areas statement.
8.1
Areal Quantities