Lagrange Coefficients (f and g Functions)

Author: Tianjiang Shuo
Website: https://cislunarspace.cn

Definition

In the two-body problem, the position and velocity at any time $t$ can be written as a linear combination of the position $\vec r_0$ and velocity $\vec v_0$ at the initial epoch $t_0$ (Vallado 2022, §2.5; Battin 1999):

\vec r(t) = f\,\vec r_0 + g\,\vec v_0,\qquad \vec v(t) = \dot f\,\vec r_0 + \dot g\,\vec v_0.

The four coefficients $f, g, \dot f, \dot g$ are the Lagrange coefficients (or f and g functions). Lagrange introduced this form for lunar-motion studies in the 18th century; it gives a linear statement of "how to compute the state at one epoch from the state at another in a central gravity field" — Kepler's problem — and underpins two-body propagation, initial orbit determination, and the Lambert problem.

Fundamental Identity

Because the angular momentum $\vec h=\vec r\times\vec v$ is conserved, substituting $\vec r=f\vec r_0+g\vec v_0$ and $\vec v=\dot f\vec r_0+\dot g\vec v_0$ into $\vec r\times\vec v=\vec r_0\times\vec v_0$ and expanding gives

f\dot g - \dot f\,g \equiv 1.\quad(\text{Vallado 2022, Eq. 2-63})

This identity is a free accuracy check in numerical implementations: any closed-form or approximate expression for $f, g, \dot f, \dot g$ must satisfy it, on pain of violating angular-momentum conservation.

Closed Forms by Independent Variable

The explicit form of the Lagrange coefficients depends on "what is known" — typically some orbital-anomaly difference (Vallado 2022, §2.5).

By true-anomaly difference $\Delta\nu$

f = 1-\frac{r}{p}(1-\cos\Delta\nu),\quad g = \frac{r r_0}{\sqrt{\mu p}}\sin\Delta\nu,

\dot f = \sqrt{\frac{\mu}{p}}\tan\!\frac{\Delta\nu}{2}\!\left(\frac{1-\cos\Delta\nu}{p}-\frac{1}{r}-\frac{1}{r_0}\right),\quad \dot g = 1-\frac{r_0}{p}(1-\cos\Delta\nu),

with $p=h^2/\mu=a(1-e^2)$ the semi-latus rectum.

By eccentric-anomaly difference $\Delta E$ (ellipse)

f = 1-\frac{a}{r_0}(1-\cos\Delta E),\quad g = (t-t_0)-\sqrt{\frac{a^3}{\mu}}(\Delta E-\sin\Delta E),

\dot f = -\frac{\sqrt{\mu a}}{r_0 r}\sin\Delta E,\quad \dot g = 1-\frac{a}{r}(1-\cos\Delta E).

$\Delta E$ is fixed implicitly by Kepler's equation $M-M_0=\Delta E-e(\sin E-\sin E_0)$ . This is the most widely used form for elliptic-orbit numerical propagation.

By parabolic anomaly $B$ (parabola)

f = \frac{1-B^2+2BB_0}{1+B_0^2},\quad g = \frac{p^2\Delta B(1+BB_0)}{2h},\quad \dot f = \frac{4h\,\Delta B}{p^2(1+B^2)(1+B_0^2)},\quad \dot g = \frac{1-B_0^2+2BB_0}{1+B^2}.

By hyperbolic-anomaly difference $\Delta H$ (hyperbola)

f = 1-\frac{a}{r_0}(1-\cosh\Delta H),\quad g = (t-t_0)-\sqrt{\frac{(-a)^3}{\mu}}(\sinh\Delta H-\Delta H),

\dot f = -\frac{\sqrt{-\mu a}}{r_0 r}\sinh\Delta H,\quad \dot g = 1-\frac{a}{r}(1-\cosh\Delta H).

By universal variable $\chi$ (unified form; Battin 1987; Vallado 2022 Eq. 2-68)

The universal-variable formulation unifies ellipse, parabola, and hyperbola, avoiding orbit-type branching:

f = 1-\frac{\chi^2}{r_0}c_2,\quad g = (t-t_0)-\frac{\chi^3}{\sqrt{\mu}}c_3,

\dot f = \frac{\sqrt{\mu}}{r_0 r}\chi(\psi c_3-1),\quad \dot g = 1-\frac{\chi^2}{r}c_2,

where $\psi=\chi^2/a$ (signed) and $c_2, c_3$ are Stumpff functions. This form is the core of the universal-variable method.

Series Form (for Initial Orbit Determination)

When only the position magnitude $r_0$ at $t_0$ is known and orbital elements cannot be obtained directly (typical of Gauss's method in initial orbit determination), set $u\triangleq\mu/r_0^3$ and Taylor-expand $\vec r(t)$ about $t_0$ . The resulting series forms for $f$ and $g$ are (Escobal 1965; Vallado 2022, Eq. 2-69):

f = 1-\frac{u}{2}\tau^2-\frac{\dot u}{6}\tau^3-\frac{\ddot u-u^2}{24}\tau^4-\cdots,

g = \tau-\frac{u}{6}\tau^3-\frac{\dot u}{12}\tau^4-\frac{3\ddot u-u^2}{120}\tau^5-\cdots,

with $\tau=t-t_0$ and the derivatives of $u$ evaluated at $t_0$ . The series converges quickly for short $\tau$ (typically a small fraction of the observation arc) and provides a propagation tool using only position information.

Application Notes

Two-body propagation: given $\vec r_0, \vec v_0$ and a flight time, solve Kepler's equation for $\Delta E$ (or $\Delta H$ ) and substitute into the corresponding closed form to obtain $\vec r(t),\vec v(t)$ .
Lambert's problem: Lambert's problem uses $f$ to turn the position relation $\vec r_0\to\vec r$ into a single transcendental equation in the semi-major axis $a$ ; multi-revolution solutions and universal-variable algorithms (Battin-Vaughan, Gooding) are all built on the $f, g$ expressions.
Initial orbit determination: Gauss's method uses the series form of $f, g$ to couple position-velocity at three observation epochs into an iteration over $\vec r_2$ — the classical method for determining an orbit from angles-only (or angles-plus-some-ranges) observations.
Accuracy check: after each propagation step, the residual of $f\dot g-\dot f g-1$ provides a free estimate of integration error.

Common Confusions

Lagrange coefficients vs. Lagrange's planetary equations: the former is the linear transfer between two-body states; the latter are variational equations for the orbital elements under perturbing accelerations (resolved along R/S/W). Same name "Lagrange", different objects.
f, g vs. universal variable: $\chi$ is the independent variable; $f$ and $g$ are coefficients expressed in terms of it. Saying "propagate with the universal-variable method" really means "propagate with the $f, g$ functions written in $\chi$ ".
f, g in Lambert's problem: there, $\vec r_0, \vec r$ are known and $\vec v_0, \vec v$ are sought; from $\vec r=f\vec r_0+g\vec v_0$ solve $\vec v_0=(\vec r-f\vec r_0)/g$ and then $\vec v=\dot f\vec r_0+\dot g\vec v_0$ . This is the role of $f, g$ in boundary-value problems.

References

Vallado, D. A. (2022). Fundamentals of Astrodynamics and Applications, 5th ed., §2.5 (Classical Formulas Using f and g Functions) and §2.6 (Series Forms of f and g). Microcosm Press.
Battin, R. H. (1999). An Introduction to the Mathematics and Methods of Astrodynamics, Revised Edition. AIAA.
Bate, R. R., Mueller, D. D., & White, J. E. (1971). Fundamentals of Astrodynamics. Dover.
Escobal, P. R. (1965, reprinted 1985). Methods of Orbit Determination, 2nd ed. Krieger.
Herrick, S. (1971). Astrodynamics, Vol. 1. Van Nostrand Reinhold.