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.

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 (6 of them)

(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@

Recall

Lagrange’s (conservative) equations are

\[\frac{d}{dt}\frac{\partial L}{\partial \dot{\mathbf{q}}} - \frac{\partial L}{\partial \mathbf{q}} = \mathbf{0}\]

for generalized coordinates \(\mathbf{q} \in \mathbb{R}^n\), where \(L \in \mathbb{R}\) is the Lagrangian and \(n\) is the number of degrees of freedom in the system.

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 (6 of them) 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@

Recall

Hamilton’s canonical equations are

\[\dot{\mathbf{q}} = +\frac{\partial H}{\partial \mathbf{p}} \qquad \text{and} \qquad \dot{\mathbf{p}} = -\frac{\partial H}{\partial \mathbf{q}}\]

for generalized coordinates and momenta \(\mathbf{q},\mathbf{p} \in \mathbb{R}^n\), where \(H \in \mathbb{R}\) is the Hamiltonian and \(n\) is the number of degrees of freedom in the system.

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 (12 of them) 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).