# The Two-Body World

## Contents

# 1 The Two-Body World@

- Info
**The Two-Body World:**The equations of motion for the general 2-body problem are derived using the Newtonian, Lagrangian, and Hamiltonian approaches, showing that each method provides the same system using different methods.- Description
Derive the governing 2-body equations using 3 approaches (Newtonian, Lagrangian, Hamiltonian).

## 1.1 The simplest scenario@

The **Kepler** (also called the **2-body**) **problem** is fundamental to the study of astrodynamics,
especially so for analyzing the more general 3-body problem. It consists of
determining the positions of 2 bodies under mutual Newtonian gravitational
attraction.

Here, \(m_1\) and \(m_2\) are the body masses and a general position vector \(\mathbf{r}\) is represented

with basis elements provided by the inertial coordinate system.

## 1.2 Equations of motion@

### 1.2.1 using Newton’s laws@

Applying Newton’s laws to each body under mutual Newtonian gravitational attraction directly provides the equations of motion

### 1.2.2 using the Lagrangian@

We can write the system’s Lagrangian as simply the difference between total kinetic energy and total potential energy.

Since gravity is conservative and the system is unconstrained, the corresponding Lagrange’s equations are written

Writing the above result of Lagrange’s equations in their most compact form provides

Applying \((i,j) = (1,2), (2,1)\) produces the equations of motion for each of the two bodies.
You can verify that these equations are the **same** as (1.2.1).

### 1.2.3 using the Hamiltonian@

Using the Lagrangian (1.2.2), the Hamiltonian is defined

where \(\mathbf{q}_i = \mathbf{r}_i\) are the generalized coordinates and \(\mathbf{p}_i = m_i\dot{\mathbf{r}}_i\) are the generalized momenta for each body (\(i = 1,2\)). Hamilton’s canonical equations then say

for \((i,j) = (1,2), (2,1)\). Despite having twice as many equations, you can verify that they *are* in fact the **same** as (1.2.1).