The differential equation of the orbit

1. Finding the path without first finding the time

The radial energy equation gives $r(t)$ implicitly, while angular momentum gives $\dot\theta(t)$. Often the desired answer is the path $r(\theta)$ itself. Eliminating time is more direct, particularly when the force is inverse-square.

We assume a time-independent central force, $\ell\ne0$, and work away from the force center. Choose angular orientation so that $\ell=\mu r^2\dot\theta>0$. Since $\dot\theta$ then never vanishes at finite nonzero $r$, angle is a useful parameter along the orbit.

2. The reciprocal-radius substitution

Define $u(\theta)=1/r$. A prime in this chapter means differentiation with respect to $\theta$, whereas a dot still means differentiation with respect to time. Angular momentum gives

Differentiate $r=1/u$ with respect to angle: $dr/d\theta=-u'/u^2$. The chain rule then gives

Differentiate once more, remembering that $u'$ changes through $\theta(t)$:

The angular acceleration term in the radial equation is

Substitute both expressions in $\mu(\ddot r-r\dot\theta^2)=F(r)$:

Thus the differential equation of the orbit, or Binet equation, is

The minus sign is fixed by the signed radial-force convention. An attractive inverse-square force has $F(1/u)=-ku^2$, making the right-hand side positive. Both sides have dimensions of inverse length; $\theta$ is dimensionless.

In potential notation, $dV(1/u)/du=V'(r)(-1/u^2)=F(r)/u^2$. Hence the equivalent form is $u''+u=-(\mu/\ell^2)dV(1/u)/du$.

3. Energy in orbit variables

The radial speed is $-\ell u'/\mu$ and tangential speed is $r\dot\theta=\ell u/\mu$. Their squares give

This first integral relates the constants in a solution of Binet's equation to the mechanical energy. Differentiating it with respect to $\theta$ gives

The bracket vanishes by Binet's equation. This is a consistency check, not a substitute for determining the constants from initial conditions.

Time can be recovered after the orbit is known:

A geometric orbit therefore does not imply constant speed along it.

4. Applications

Example 1 · Straight-line motion with zero force

For $F=0$, Binet's equation is $u''+u=0$. Choose the angular origin so that $u=C\cos\theta$, with $C>0$. Then $r\cos\theta=1/C$. Since $x=r\cos\theta$, this is the straight line $x=b$, where $b=1/C$.

The formula is used only on the angular interval where $r>0$, namely $-\pi/2<\theta<\pi/2$. At the ends the particle is infinitely far away. Its angular momentum about the origin can be nonzero even though no force acts: a line missing the origin has $\ell=\mu bv$.

Example 2 · Recovering the force from an orbit

Suppose $u=C(1+e\cos\theta)$ is known, with $C>0$. Differentiate twice: $u''=-Ce\cos\theta$, so $u''+u=C$. Rearranging Binet's equation yields

The orbit is generated by inverse-square attraction with $k=\ell^2C/\mu$. The orbit's shape alone fixes the radial dependence along the sampled radii; the force scale also needs the mass and angular momentum or time information.

Example 3 · Circular motion from Binet's equation

For a circle of radius $r_c$, $u=u_c=1/r_c$ is constant and $u''=0$. Binet gives

Using $\ell=\mu r_c^2\Omega$ gives $F(r_c)=-\mu r_c\Omega^2$, the centripetal-force condition. A constant reciprocal radius is a special solution, but its existence does not establish stability; that requires perturbing it or examining $V''_{\rm eff}$.

Example 4 · Precession caused by an additional inverse-cube force

Let $F(r)=-k/r^2-\beta/r^3$, with $k>0$ and $\beta$ constant. Then

Suppose $\nu^2=1-\mu\beta/\ell^2>0$. A solution is

Check by differentiating twice: the cosine term cancels between $u''$ and $\nu^2u$, leaving the required constant. For $0<e<1$, radius oscillates between finite limits. Successive periapses occur when $\nu\theta$ increases by $2\pi$, so the angle between them is $2\pi/\nu$.

For small positive $\mu\beta/\ell^2$, expand $(1-z)^{-1/2}\simeq1+z/2$ to obtain a periapsis advance $2\pi(1/\nu-1)\simeq\pi\mu\beta/\ell^2$. This is a Newtonian model of precession, not a derivation of the relativistic planetary correction. The orbit can be bounded without closing after one revolution.

5. The role of initial conditions

At a specified angle $\theta_0$, $u_0=1/r_0$ and $u'_0=-\mu\dot r_0/\ell$. These two values determine the constants of the second-order orbit equation. Angular momentum supplies the remaining time scale, while an arbitrary angular origin sets the orientation of the same geometric path.

Do not use the orbit equation for $\ell=0$, confuse $u'$ with $\dot u$, or keep a mathematically continued branch where the polar radius is negative. Physical radii and accessible angles must be checked after solving the differential equation.

6. A compact derivation record

For a written derivation, the essential chain is $u=1/r$, $\dot\theta=\ell u^2/\mu$, $\dot r=-\ell u'/\mu$, and $\ddot r=-\ell^2u^2u''/\mu^2$. Substitute into the radial Newton equation and divide by $-\ell^2u^2/\mu$. State $\ell\ne0$ and the signed-force convention. Every sign in Binet's equation follows from these four relations.

Exercises

Hints and solutions are collected separately.

1. 18.1 Why is the Binet substitution unavailable for purely radial motion?

2. 18.2 Derive $\dot r=-\ell u^{\prime}/\mu$ for $u=1/r$.

3. 18.3 Derive Binet’s equation with the signed force convention.

4. 18.4 For $F=-k/r^2$, what is the right-hand side of Binet’s equation?

5. 18.5 Show that $u=C\cos\theta$ is a force-free straight line.

6. 18.6 Find the force producing $u=C(1+e\cos\theta)$ at given $\ell$.

7. 18.7 Derive the energy expression in terms of $u$ and $u^{\prime}$.

8. 18.8 Given $r_0=2$, $\dot r_0=-3$, $\mu=1$, $\ell=2$ in consistent units, find initial $u,u^{\prime}$.

9. 18.9 For $u=u_0[1+e\cos(\nu\theta)]$, what angle separates successive periapses?

10. 18.10 For the inverse-square plus inverse-cube force with $\mu\beta/\ell^2=0.04$, estimate the periapsis advance per radial cycle.

11. 18.11 How is time recovered after solving $r(\theta)$?

12. 18.12 Why can a formal solution of Binet’s equation include unphysical intervals?