Central Force Fundamentals

Every orbit determined in 2026 runs on this article's machinery. ESA's Gaia mission, whose fourth data release is now mapping the astrometric wobbles of nearly two billion stars, fits binary-star and exoplanet orbits by exactly the two-conserved-scalars logic of Worked Example 2 — measure enough of the orbit to pin $E$ and $L$ (in practice, the orbital elements), and the entire conic follows. Mission designers steering Europa Clipper and JUICE across the outer solar system chain together central-force arcs — each leg a two-body problem about the Sun or a planet — into the patched-conic trajectories that make gravity assists work. The quieter empire is microscopic. Molecular-dynamics codes that simulate proteins, electrolytes, and materials — running on the same GPUs that train neural networks — spend most of their arithmetic evaluating central pair potentials: Lennard-Jones, Coulomb, Yukawa. Each pair interaction is an $f(r)\hat{\mathbf{r}}$, each conserves the pair's angular momentum during a close encounter, and the scattering formulas descend directly from the orbit quadrature derived above. When a 2026-era drug-discovery pipeline scores a candidate molecule's binding, it is summing a billion tiny central forces per timestep — Newton's Proposition 1, vectorized.

Picture two bodies tugging on each other along an invisible string that joins them. Because the pull always runs exactly along the string, neither body can apply a sideways twist about the center — there is no lever arm. A twist is a torque, and torque is what changes angular momentum; with no twist available, the rotational state of the pair is frozen at whatever it was at the start. That frozen vector $\mathbf{L}$ acts like a compass needle fixed in space, and since the position vector must stay perpendicular to it at every instant, the whole dance is locked into the single flat plane the needle defines. A genuinely three-dimensional wander is simply forbidden.

Once the motion is planar and energy is conserved, the angular coordinate $\theta$ stops being a dynamical complication: angular momentum conservation hands you $\dot\theta$ in terms of $r$ for free. All the hard physics migrates into the radial coordinate, which is why a one-dimensional energy graph — the effective potential you will meet properly two articles from now — can capture the entire character of an orbit. Notice what is not assumed. The radial profile $f(r)$ is left completely general at this stage: inverse-square gravity, the linear restoring force $f = -kr$ of an isotropic oscillator, and any power law in between all share the planar, energy-conserving, reducible structure. Only the shape of the resulting orbit distinguishes them. That is the deal this article offers: prove three structural theorems once, and they hold for every central force you will ever meet — in celestial mechanics, in Rutherford's scattering experiment, in the pair potentials of molecular dynamics.

Two conditions must hold simultaneously, and students lose points by checking only one. The force must point along the line joining the bodies, and its magnitude must depend on nothing but the separation. In symbols, with $\mathbf{r}$ the relative coordinate, $r = |\mathbf{r}|$ and $\hat{\mathbf{r}} = \mathbf{r}/r$:

where: $f(r)$ — a scalar function of separation alone, with $f < 0$ attractive (the force points back along $-\hat{\mathbf{r}}$, toward the center) and $f > 0$ repulsive. Gravity has $f(r) = -Gm_1m_2/r^2$; the Coulomb force between like charges has $f(r) = +q_1q_2/4\pi\varepsilon_0 r^2$; an isotropic spring has $f(r) = -kr$. A force like $-(\alpha/r^2)\,\hat{\boldsymbol\theta}$ — magnitude depending on $r$, but pointing tangentially — flunks the direction test: it is not central, it exerts torque, and none of this article's machinery applies to it. The first structural theorem is that condition (1) buys you a potential energy for free. You may remember that a force is conservative when its curl vanishes, or equivalently when it can be written as the gradient of a scalar. For a central force no curl computation is ever needed — the result drops out of one geometric identity:

where in equation (2): $V(r)$ — the potential energy, a function of separation only. Conservativity means the total mechanical energy is a constant of the motion, our first conserved scalar — and it is a structural gift: no property of $f$ beyond centrality was used. Before conservation laws, one honest disclaimer about the word "two-body." Throughout this article the moving object is the fictitious reduced particle: as shown in the two-body reduction, two real bodies interacting through $V(|\mathbf{r}_1 - \mathbf{r}_2|)$ are exactly equivalent to a single particle of reduced mass $\mu = m_1m_2/(m_1+m_2)$ moving in the potential $V(r)$ about a fixed center. Every $\mu$ below is that reduction at work; set $\mu \to m$ whenever the force center is genuinely immovable.

Now the second theorem, the one that makes orbits flat. Torque about the force center is $\boldsymbol\tau = \mathbf{r}\times\mathbf{F}$, and for a central force $\mathbf{F}$ is parallel to $\mathbf{r}$ — a cross product of parallel vectors:

where: the vanishing is exact and instantaneous, holding at every point of every orbit, whatever $f(r)$ is. No lever arm, no twist. And since torque is the rate of change of angular momentum, the angular momentum vector

where: $\mathbf{p} = \mu\dot{\mathbf{r}}$ — the momentum of the reduced particle — is frozen in magnitude and direction. This is a stronger statement than the scalar conservation laws you may be used to: a full vector, three numbers, locked for all time. The direction of $\mathbf{L}$ is what pins the orbital plane; its magnitude is what will feed the radial equation. The formal proof that motion stays planar is short enough to enjoy in full:

With the plane established, set up polar coordinates $(r, \theta)$ in it. The velocity decomposes as $\dot{\mathbf{r}} = \dot r\,\hat{\mathbf{r}} + r\dot\theta\,\hat{\boldsymbol\theta}$ — a radial stretch plus a tangential sweep — and inserting this into (4) collapses the vector conservation law into one scalar workhorse. The $\hat{\mathbf{r}}\times\hat{\mathbf{r}}$ term dies, $\hat{\mathbf{r}}\times\hat{\boldsymbol\theta} = \hat{\mathbf{z}}$ survives:

where: $L = |\mathbf{L}|$ and $\dot\theta$ — the angular velocity in the orbit plane. Read it as a trade: when $r$ shrinks, $\dot\theta$ must grow as $1/r^2$ — the figure-skater pulling in her arms, the comet whipping around perihelion. Equation (5) is the single most-used line in orbital mechanics, because it converts the unknown function $\dot\theta(t)$ into the known expression $L/\mu r^2$, evicting $\theta$ from the dynamics entirely. Equation (5) has a beautiful geometric alias. In time $\mathrm{d}t$ the radius vector sweeps a thin triangle of area $\mathrm{d}A = \tfrac12 r\cdot r\,\mathrm{d}\theta$, so the areal velocity is

where: $\mathrm{d}A/\mathrm{d}t$ — the rate of sweeping out area. Equal areas in equal times: this is Kepler's second law, and notice what the derivation did not use — any property of gravity. Constant areal velocity is the signature of every central force, from Coulomb repulsion to a spring. Kepler read it off Tycho Brahe's Mars data in 1609; Newton recognized it as the fingerprint of a center-seeking force and made it the opening theorem of the Principia. The full story of that law lives in the next article; the Kepler-specific consequences live in the Kepler problem.

Energy is the second conserved scalar, and in the orbit plane it reads: kinetic energy of the radial stretch, plus kinetic energy of the angular sweep, plus potential. There is a concrete number attached to each term for, say, a comet — and the whole art of the subject is what happens when (5) is allowed to rewrite the middle one:

where: the three terms are radial kinetic, tangential kinetic, and potential energy. Now substitute $\dot\theta = L/\mu r^2$ from (5) into the middle term: $\tfrac12\mu r^2\dot\theta^2 = L^2/2\mu r^2$, a function of $r$ alone. The angular kinetic energy has been converted into what is effectively a potential energy:

where: $V_{\mathrm{eff}}(r)$ — the effective potential, and $L^2/2\mu r^2$ — the centrifugal term, the energy cost of conserving angular momentum at small radius. Equation (8) says the radial coordinate moves exactly like a one-dimensional particle in the potential $V_{\mathrm{eff}}$. Turning points sit where $E = V_{\mathrm{eff}}$, circular orbits at its minima, bound and unbound motion on either side of $V_{\mathrm{eff}}(\infty)$ — a full qualitative theory of orbits from one graph, developed properly in the effective potential and the orbit classification. The same substitution in Newton's law gives the radial equation of motion, $\mu\ddot r = -\mathrm{d}V_{\mathrm{eff}}/\mathrm{d}r = -V'(r) + L^2/\mu r^3$, where the positive $L^2/\mu r^3$ term is the outward centrifugal push that keeps any orbit with $L \neq 0$ from ever reaching $r = 0$. The final theorem of this article is that the reduction actually finishes the problem — the whole orbit collapses to two integrals you could hand to a computer, or, for the right $f(r)$, to a table of standard forms:

Which equation answers which question

1 · A satellite's angular momentum and sweep rate (confidence builder)

An $m = 800\ \mathrm{kg}$ satellite is in a circular orbit of radius $r = 7.0\times10^6\ \mathrm{m}$ (about 630 km altitude), where its orbital speed is $v = 7.55\ \mathrm{km/s}$. Take $GM_\oplus = 3.986\times10^{14}\ \mathrm{m^3/s^2}$. Find the torque on it about Earth's center, its angular momentum, its areal velocity, and use the areal velocity to predict the orbital period.

Step 1. Torque: gravity on the satellite points along $-\hat{\mathbf{r}}$, dead at Earth's center, so $\boldsymbol\tau = \mathbf{r}\times\mathbf{F} = \mathbf{0}$ — exactly, at every instant. (Earth is so much heavier that $\mu = m$ to five decimal places; the reduced-particle language collapses to the satellite itself.)

Step 2. Angular momentum: on a circular orbit $\mathbf{v} \perp \mathbf{r}$, so $L = mvr = 800 \times 7.55\times10^3 \times 7.0\times10^6 = \boxed{4.23\times10^{13}\ \mathrm{kg\,m^2/s}}$ — constant for the life of the orbit.

Step 3. Areal velocity from (6): $\mathrm{d}A/\mathrm{d}t = L/2m = 4.23\times10^{13}/1600 = \boxed{2.64\times10^{10}\ \mathrm{m^2/s}}$.

Step 4. Period: one revolution sweeps the full disk area $\pi r^2 = \pi(7.0\times10^6)^2 = 1.54\times10^{14}\ \mathrm{m^2}$, at the constant rate from Step 3: $T = 1.54\times10^{14}/2.64\times10^{10} = \boxed{5.8\times10^3\ \mathrm{s} \approx 97\ \mathrm{min}}$. Cross-check with the standard circular-orbit formula: $T = 2\pi\sqrt{r^3/GM_\oplus} = 2\pi\sqrt{(3.43\times10^{20})/(3.986\times10^{14})} = 5.83\times10^3\ \mathrm{s}$. The areal-velocity route and the dynamics route agree — as they must, since (6) is a theorem, not an approximation.

2 · A comet's aphelion from two conservation laws (exam level)

A comet passes perihelion at $r_p = 1.0\times10^{11}\ \mathrm{m}$ with speed $v_p = 45\ \mathrm{km/s}$, moving perpendicular to the radius vector. Take $GM_\odot = 1.327\times10^{20}\ \mathrm{m^3/s^2}$. Is the orbit bound? Find the aphelion distance and the speed there — without solving any differential equation.

Step 1. Bound or not: the escape speed at $r_p$ is $v_{\mathrm{esc}} = \sqrt{2GM_\odot/r_p} = \sqrt{2 \times 1.327\times10^{20}/10^{11}} = 51.5\ \mathrm{km/s}$. Since $45 < 51.5$, the orbit is bound. Per unit mass, $E/m = \tfrac12 v_p^2 - GM_\odot/r_p = \tfrac12(4.5\times10^4)^2 - 1.327\times10^{20}/10^{11} = 1.013\times10^9 - 1.327\times10^9 = -3.15\times10^8\ \mathrm{J/kg}$ — negative, confirming a bound orbit.

Step 2. Angular momentum per unit mass: at perihelion $\mathbf{v}\perp\mathbf{r}$, so $\ell = L/m = v_p r_p = 4.5\times10^4 \times 1.0\times10^{11} = 4.5\times10^{15}\ \mathrm{m^2/s}$.

Step 3. At any apsis (perihelion or aphelion) the radial velocity vanishes, so the energy equation (8) with $\dot r = 0$ becomes an ordinary quadratic in $r$: $E/m = \ell^2/2r^2 - GM_\odot/r$, i.e. $(E/m)\,r^2 + GM_\odot\,r - \ell^2/2 = 0$. Dividing by $E/m = -3.15\times10^8$:

Step 4. The two roots of this quadratic are precisely the two turning points, and their sum is $4.22\times10^{11}\ \mathrm{m}$. One root is the known perihelion $r_p = 1.0\times10^{11}$, so $r_a = 4.22\times10^{11} - 1.0\times10^{11} = \boxed{3.2\times10^{11}\ \mathrm{m} \approx 2.15\ \mathrm{AU}}$. (Sanity check via the product of roots: $r_p r_a = 3.22\times10^{22}$ ✓.)

Step 5. Speed at aphelion from angular momentum conservation — at an apsis the velocity is again perpendicular to the radius, so $v_a = \ell/r_a = 4.5\times10^{15}/3.22\times10^{11} = \boxed{14.0\ \mathrm{km/s}}$. Energy cross-check: $\tfrac12 v_a^2 - GM_\odot/r_a = 9.77\times10^7 - 4.12\times10^8 = -3.15\times10^8\ \mathrm{J/kg}$ — identical to Step 1. Two conserved scalars pinned the whole geometry, exactly as the reduction theorem promised. (Bonus: the sum of the roots is $GM_\odot/|E/m| = 2a$, giving semi-major axis $a = 2.11\times10^{11}\ \mathrm{m}$ directly.)

Texts. Morin, Introduction to Classical Mechanics, Ch. 7 (Central Forces) · Goldstein, Poole & Safko, Classical Mechanics (3rd ed.), Ch. 3 · Landau & Lifshitz, Mechanics (Vol. 1), §§13–14 · Taylor, Classical Mechanics, Ch. 8.

Provenance. Definitions and reductions follow Morin Ch. 7, Goldstein Ch. 3, Landau–Lifshitz §§13–14, and Taylor Ch. 8. A verifier should confirm: (i) $\nabla r=\hat{\mathbf{r}}$ and hence $f(r)\hat{\mathbf{r}}=-\nabla V$; (ii) the torque vanishes for $\mathbf{F}\parallel\hat{\mathbf{r}}$, giving $\dot{\mathbf{L}}=0$; (iii) $L=\mu r^2\dot\theta$ from the plane-polar velocity decomposition; (iv) the radial energy equation and its quadratures match Landau §14; (v) areal velocity $L/2\mu$; (vi) reduced mass $\mu=m_1m_2/(m_1+m_2)$ and oscillator period $T=2\pi\sqrt{\mu/k}$ are standard textbook results. Worked-example numbers on this page ($L = 4.23\times10^{13}\ \mathrm{kg\,m^2/s}$ and $T \approx 97$ min for the satellite; $r_a = 3.2\times10^{11}$ m and $v_a = 14.0$ km/s for the comet) are illustrative arithmetic recomputed from the boxed formulas.

Builds on: The two-body problem and reduced mass · Angular momentum · Conservative forces and potential energy. Leads to: Angular momentum conservation in central fields · Reduction to the radial equation · The effective potential · Classification of orbits by energy · The Kepler problem.