Analytical mechanics

Motion in a central field, the Kepler problem

The Kepler problem describes motion under an inverse-square gravitational potential. Symmetry reduces it to radial dynamics and classifies trajectories as ellipses, parabolas, or hyperbolas.

In the central potential V(r)=−κ/r, angular momentum is conserved. The orbit lies in a plane and is determined by energy E and angular momentum L.

E=\frac{p_r^2}{2\mu}+\frac{L^2}{2\mu r^2}-\frac{\kappa}{r}, \qquad r(\phi)=\frac{p}{1+e\cos\phi}

Definition: Effective potential and orbit

The reduced mass μ moves with angular momentum L. The effective potential $V_{eff}=L^2/(2μr^2)-κ/r$ combines the centrifugal barrier and attraction. For $E<0$ the Kepler orbit is a bound ellipse; $E=0$ is the escape parabola; $E>0$ gives a hyperbola. These results assume the Newtonian two-body problem without significant perturbations.

Vary eccentricity and true anomaly to compare elliptical, parabolic, and hyperbolic trajectories in the orbital plane. The surface below illustrates the Hamilton–Jacobi principal function S(q,t) for a simplified Kepler example.

From Hamilton–Jacobi to orbits

The Hamilton–Jacobi equation $∂S/∂t+H(q,∂S/∂q,t)=0$ determines the principal action. In a static Kepler field, separation in polar coordinates uses energy and angular momentum as constants; derivatives of S with respect to these constants yield the motion. The displayed S surface is only an illustration of a simplified solution, not a universal surface shared by all Kepler orbits.

Example: Example

A bound Kepler orbit is an ellipse with eccentricity e=0.6 and semimajor axis a=10⁷ m, with the focus at the origin. Find the periapsis and apoapsis distances.

Solution

For a Kepler ellipse, r_min=a(1−e) and r_max=a(1+e). Thus r_min=4×10⁶ m and r_max=1.6×10⁷ m. Their sum is 2a, as expected geometrically.

Quick check

In polar coordinates the effective potential is $V_{\rm eff}(r)=L^2/(2\mu r^2)-\kappa/r$ . The centrifugal term creates a barrier as $r\to0$ , while gravity attracts the particle; the minimum gives a circular orbit of radius $r_0=L^2/(\mu\kappa)$ . For bound energy $E<0$ , the orbit is an ellipse with semimajor axis $a=-\kappa/(2E)$ . At $E=0$ it is parabolic, and for $E>0$ it is hyperbolic. These formulas use the reduced mass $\mu$ for the two-body problem; a fixed central mass is the limiting approximation.

For V(r)=−κ/r, what type of orbit corresponds to total energy E<0?

For an autonomous system, which quantity is conserved along Hamiltonian motion?

References

Herbert Goldstein, Charles Poole, John Safko (2002). Classical Mechanics
L. D. Landau, E. M. Lifshitz (1976). Mechanics