Jürgen Vollmer, Universität Leipzig
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One of the most important objectives of physics is the description of the motion of interacting particles. As a first step in this direction we discuss how to employ constants of motion to determine the motion of two point particles that interact with a conservative force depending only on the scalar distance between the particles, the interaction most commonly encountered in physical systems. The impact of spatial extension will be the topic of Chapter 5.
Definition 4.6 Point Particles A point particle is an idealization of a physical object where its mass is considered to be concentrated in a single point in space $\mathbf x$. Point particles can not collide. However, their motion can be subjected to forces that depend on their position $\mathbf x$.
Example 4.10 Kepler Problem The Kepler problem addresses the motion of a planet of mass $m$ that orbits around a sun of mass $M$. The sun and the planet are so far apart that it is justified to consider their masses as concentrated in the positions $\mathbf q_P$ and $\mathbf q_S$, and to approximate their interaction as arising from the potential \begin{align*} \Phi( R ) = -\frac{ mMG }{ R } \end{align*} where $G = 6.67259 \times 10^{-11}\, \, m^3 kg^{-1} s^{-2}$ is the constant of gravitation and $R = \lvert \mathbf q_P - \mathbf q_S \rvert$ is the distance between planet and sun. Planet and sun are considered as point particles.
Remark 4.12. The approximation of point particles has been introduced by Newton upon providing the first mathematical model for the Kepler problem. Subsequently, it has extremely successfully been applied in celestial mechanics. Celestial Mechanics addresses the problem of discussing the motion of all planets and their moons based on pair interactions deriving from the potential provided in Example 4.10. How to the tiny interactions between the planets impact their motion over long times? Is our solar system stable, or will–at some time in the far future–some planet or moon borrow energy from the other bodies and escape into outer space?
Remark 4.13. A straightforward application of the Kepler problem is the discussion of the motion of the Moon around Earth where the predictions have been tested extremely accurately based on satellite data and the return time of light signals send to Moon and reflected by mirrors on its surface that have been left there by space missions. The measurements clearly reveal the limitations of the model: Most noticeably, the Moon gives rise to tidal forces on Earth that induce a tiny amount of dissipation. Even in celestial mechanics there are small dissipative corrections to conservative interaction.
In Section 3.4 we learned that conservation laws impose constraints on the motion of bodies that can be used to simplify the description of their motion. We consider the motion of $N$ particles of masses $m_i$, $i=1,\dots,N$ at the positions $ \mathbf q_i$, $i=1, \dots , N $ that are subjected to forces $\mathbf F_{ij}$ acting between every pair $(i,j)$ of particles. There is not self-interaction $\mathbf F_{ii} = \mathbf 0$, and the forces obey Newtons 3rd law, $\mathbf F_{ij} = - \mathbf F_{ji}$. Moreover, they are conservative, and depend only on the distance of the particles, $\mathbf F_{ij} = \nabla \Phi_{ij}\bigl( \lvert \mathbf q_i - \mathbf q_j \rvert \bigl)$. Here the indices $ij$ indicate that the force may depend on additional scalar parameters such as the mass or charge of the particles.
We first determine the evolution of the position of the center of mass $\mathbf Q$ of the system \begin{align} \mathbf Q = \frac{1}{M} \sum_i m_i \: \mathbf q_i \quad\text{with total mass}\quad M = \sum_i m_i \tag{4.6.1} \end{align} Its evolution is not subjected to external forces \begin{align} \ddot{\mathbf Q} = \frac{1}{M} \sum_i m_i \: \ddot{\mathbf q_i} = \frac{1}{M} \sum_i \sum_j \mathbf F_{ij} = \mathbf 0 \tag{4.6.2} \end{align} due to Newtons 3rd law. Hence, we find for an initial position $\mathbf Q_0$ and initial velocity $\mathbf V_0$ at an initial time $t_0$ that
\begin{align} \mathbf Q(t) = \mathbf Q_0 + \mathbf V_0 \: ( t - t_0 ) \tag{4.6.3} \end{align} Now we introduce the coordinates relative to the center of mass $\mathbf r_i = \mathbf q_i - \mathbf Q$ and we observe that \begin{align} m_i \, \ddot{\mathbf r}_i &= m_i \, \ddot{\mathbf q}_i - m_i \, \ddot{\mathbf Q} = m_i \, \ddot{\mathbf q}_i \nonumber \\ &= \sum_j \mathbf F_{ij} = - \nabla_{\mathbf q_i} \Phi_{ij}\bigl( \lvert \mathbf q_i - \mathbf q_j \rvert \bigr) = -\frac{ \mathbf q_i - \mathbf q_j }{ \lvert \mathbf q_i - \mathbf q_j \rvert } \; \Phi_{ij}'\bigl( \lvert \mathbf q_i - \mathbf q_j \rvert \bigr) \nonumber \\ &= -\frac{ \mathbf r_i - \mathbf r_j }{ \lvert \mathbf r_i - \mathbf r_j \rvert } \; \Phi_{ij}'\bigl( \lvert \mathbf r_i - \mathbf r_j \rvert \bigr) \tag{4.6.4} \end{align} where $\Phi_{ij}'(x)$ denotes the derivative of $\Phi_{ij}(x)$ with respect to its scalar argument $x$. Hence, the EOMs for $\mathbf Q$ and for the positions $\mathbf r_i$ relative to the center of mass can be solved separately of each other, and the EOM for the CM has a trivial solution, Equation 4.6.3. We may therefore always address the motion of the particles in a setting where their center of mass is fixed at the origin of the coordinate system.
