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.

Diagram of the 2-body problem with respect to a general, inertial coordinate system

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).