Binet Equation and Orbits

Binet Equation and Orbits in advanced mechanics. Theory, formulas, worked examples and applications.

Conceptual context

Directly solving a planet's equation of motion around the Sun, as a function of time, is surprisingly awkward: the radial distance $r$ and the angle $\theta$ change simultaneously in a coupled way, and the resulting differential equation is hard to handle. Jacques Philippe Marie Binet discovered a change of variable that radically simplifies the problem: instead of asking "where is the particle at instant t?", ask "at what radial distance is the particle when it has turned through an angle theta?"

This leaf develops the Binet equation, which expresses the inverse radius $u$=1/$r$ as a function of the angle $\theta$, turning the complete dynamical problem into a purely geometric problem about the orbit's shape.

🟢 Essential level

The essential idea is to change the question: instead of following the motion in time, the trajectory's shape is followed as the angle advances. This shift in perspective is possible because, under any central force, the angular momentum $L$ is exactly conserved, which allows rewriting the time derivative as an angular derivative, completely eliminating time from the problem.

The result is a differential equation in $u$(theta) depending only on the force law F(r) and the angular momentum $L$, with no explicit reference to time whatsoever. For the particular case of a central force proportional to 1/r² (gravitational or Coulombian), this equation becomes linear, and its general solution is exactly the conic sections: ellipses, parabolas, and hyperbolas, the shape of planetary orbits empirically discovered by Kepler two centuries before Newton and Binet provided its complete mathematical explanation.

An example illustrating the method's power: asking "what force law would produce a logarithmic spiral orbit?" is a well-posed geometric problem for the Binet equation (the inverse problem), while solving the same question directly with Newton's equations as a function of time would be much more laborious.

🔵 Formal level

Basic formulas

This first formal layer presents the basic relation that should be recognized, interpreted, and used before attempting longer derivations. The goal is to identify the quantities, their units, and their physical dependence, anticipate how the result changes when each datum varies, and always check that the answer is physically consistent.

The formal level rests on four relations:

Inverse radius

Real student questions

"Why use theta as the variable instead of time, if in the end we want to know where the planet is at each instant?" Because the orbit's geometric shape (an ellipse, for example) is often the most relevant information and is obtained much more easily with the Binet equation; once the shape is known, the temporal position is recovered afterward through angular momentum's conservation.

"Do all central forces produce closed orbits?" No; Bertrand's theorem shows that only forces proportional to 1/r² or to r produce exactly closed bound orbits for any initial condition; any other force law generally produces orbits that precess without ever exactly closing.

Final synthesis

By the end of this leaf, the student should be able to set up the Binet equation for a given central force law, recognize the special 1/r² case producing conic sections, and apply the conserved angular momentum to relate orbital geometry with the underlying temporal dynamics.

The relevant final check is one of consistency: verifying that, substituting the obtained solution u(theta) back into the Binet equation together with the original force law, the equation is exactly satisfied for every value of theta.