# 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).

Author

Matt Werner

## 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.

Fig. 1.1.1 The most general setup of the Kepler (2-body) problem. The problem consists of two masses, $$m_1$$ and $$m_2$$, gravitationally attracted to one another whose positions, $$\mathbf{r}_1$$ and $$\mathbf{r}_2$$, are tracked in an inertial coordinate system.@

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

$\begin{split}\mathbf{r} := \begin{pmatrix}X \\ Y \\ Z\end{pmatrix}\end{split}$

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

(1.2.1)@$\begin{split}\ddot{\mathbf{r}}_1 &= -\frac{G m_2}{|\mathbf{r}_1 - \mathbf{r}_2|^3}(\mathbf{r}_1 - \mathbf{r}_2) \\ \ddot{\mathbf{r}}_2 &= -\frac{G m_1}{|\mathbf{r}_2 - \mathbf{r}_1|^3}(\mathbf{r}_2 - \mathbf{r}_1).\end{split}$

### 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.

(1.2.2)@$L = \underbrace{ \vphantom{\frac{G m_1 m_2}{|\mathbf{r}_2 - \mathbf{r_1}|}} \frac{1}{2}\left(m_1 |\dot{\mathbf{r}}_1|^2 + m_2|\dot{\mathbf{r}}_2|^2\right)}_{\text{kinetic}} + \underbrace{\frac{G m_1 m_2}{|\mathbf{r}_2 - \mathbf{r_1}|}}_{-\text{potential}}$

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

$\begin{split}\mathbf{0} &= \frac{d}{dt}\frac{\partial L}{\partial\dot{\mathbf{r}}_i} - \frac{\partial L}{\partial\mathbf{r}_i} && (i = 1,2) \\ &= \frac{d}{dt} (m_i \dot{\mathbf{r}}_i) - \frac{G m_1 m_2}{|\mathbf{r}_j - \mathbf{r}_i|^3} (\mathbf{r}_j - \mathbf{r}_i) \qquad\quad && (i+j=3) \\ &= m_i \ddot{\mathbf{r}}_i + \frac{G m_1 m_2}{|\mathbf{r}_i - \mathbf{r}_j|^3} (\mathbf{r}_i - \mathbf{r}_j).\end{split}$

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

(1.2.3)@$\ddot{\mathbf{r}}_i = -\frac{G m_j}{|\mathbf{r}_i - \mathbf{r}_j|^3}(\mathbf{r}_i - \mathbf{r}_j).$

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

$\begin{split}H &:= \sum_{i = 1}^2 (\mathbf{p}_i \cdot \dot{\mathbf{q}}_i) - L \\ &= \underbrace{\frac{1}{2}\left(\frac{|\mathbf{p}_1|^2}{m_1} + \frac{|\mathbf{p}_2|^2}{m_2}\right)}_{\text{kinetic}} - \underbrace{\frac{G m_1 m_2}{|\mathbf{q}_2 - \mathbf{q}_1|}}_{-\text{potential}},\end{split}$

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

(1.2.4)@$\begin{split}\dot{\mathbf{q}}_i &= \frac{\mathbf{p}_i}{m_i} \\ \dot{\mathbf{p}}_i &= -\frac{G m_1 m_2}{|\mathbf{q}_i - \mathbf{q}_j|^3} (\mathbf{q}_i - \mathbf{q}_j),\end{split}$

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).