calculus & modeling

summer camp @ constructor school, together with Shrajesh Thapa

the two, three body problem

by Herman Fuxen

This work explores how gravitational attraction governs the motion of celestial bodies, using systems of differential equations to describe and analyze their dynamics. The focus is on deriving, solving, and explaining the resulting equations of motion. Clear communication, correct modeling, and insight into solution structure are prioritized.

deliverables

1. Modeling Planetary Motion

Develop and explain a model describing the motion of a small planet orbiting the sun. You may assume the sun doesn’t move.

Requirements

The differential equation for the motion of the planet is described by

\[\frac{d^2 \vec{x}}{dt^2} = -\frac{GM}{\|\vec{x}\|^3} \cdot \vec{x}\]

where \(\vec{x}(t)\) is the position vector of the planet at time \(t\), \(G\) is the gravitational constant, and \(M\) is the mass of the sun.

Figure illustrates the Sun at the center and an orbiting planet
  • Introduce a velocity vector \(\vec{v}(t) = \frac{d\vec{x}}{dt}\), and rewrite the equation as two first-order equations.1
  • What information would you need to completely determine the motion of the planet, assuming the differential equation is solved?
  • Using Newton’s law of universal gravitation, explain why the force gets stronger when the planet gets closer to the sun.

2. The Three-Body Problem

Extend your model to a system of three bodies (e.g., Sun, Earth, and Jupiter) where all bodies exert gravitational forces on each other. No approximations—everybody attracts everybody.

Requirements

The equations of motion for three bodies are given by a system of second-order ODEs. For body \(i\) with position \(\vec{x}_i\) and mass \(m_i\):

\[\frac{d^2 \vec{x}_i}{dt^2} = \sum_{j \neq i} G m_j \frac{\vec{x}_j - \vec{x}_i}{\|\vec{x}_j - \vec{x}_i\|^3}, \quad i = 1,2,3\]

where \(G\) is the gravitational constant.

  • Explain why we can no longer assume the Sun is stationary. What changes in the modeling approach?
  • Rewrite this system as a set of first-order ODEs by introducing velocity vectors \(\vec{v}_i = \frac{d\vec{x}_i}{dt}\) for each body.
  • How many scalar equations do you end up with? (Each vector equation in 3D gives three scalar equations.)
  • What initial information is required to fully determine the motion of all three bodies?
  • Bonus. Research and briefly describe why the three-body problem is famous for being “chaotic” and what that means for long-term predictions.2

  1. The derivative of displacement is velocity, and the derivative of velocity is acceleration. ↩

  2. Unlike the two-body problem, the three-body problem has no general closed-form solution and exhibits sensitive dependence on initial conditions—a hallmark of chaos theory, first studied by Henri Poincaré. ↩