Weber Two-Body Dynamics via Hypergeometric Series

A reformulation of the Clemente & Assis (1991) analytical solution for Weber electrodynamics in terms of Gauss and Appell hypergeometric functions.

1. Physical Setup

Weber Force Law

The force between two charged particles under Weber's electrodynamics:

$$\mathbf{F}_{1,2} = -U_0 \frac{\hat{r}}{r^2} \left(1 + \frac{r\ddot{r}}{c^2} - \frac{\dot{r}^2}{2c^2}\right)$$

where $U_0 = q_1 q_2$ for electromagnetic interactions, $r$ is the separation, and $c$ is the speed of light.

Conserved Quantities

In the center-of-mass frame with reduced mass $\mu = m_1 m_2/(m_1 + m_2)$:

Angular momentum:

$$L = \mu r^2 \dot{\theta}$$

Weber energy:

$$W = \frac{\mu}{2}\left(\dot{r}^2 + r^2\dot{\theta}^2\right) + \frac{U_0}{r}\left(1 - \frac{\dot{r}^2}{2c^2}\right)$$

2. Core Variables

Characteristic Length Scale

$$K = \frac{U_0}{\mu c^2}$$

This is the Weber correction length—the scale at which velocity-dependent effects become significant.

Dimensionless Radial Coordinate

$$x^2 = 1 - \frac{K}{r}$$

The physical radius is recovered via $r = K/(1 - x^2)$.

Turning Points

The squared turning point coordinates are:

$$x_{1,2}^2 = 1 + \frac{\mu K U_0}{L^2}\left[1 \pm \sqrt{1 + \frac{2WL^2}{\mu U_0^2}}\right]$$

Introducing the dimensionless parameters:

$$\alpha \equiv \frac{\mu K U_0}{L^2} = \frac{U_0^2}{\mu c^2 L^2}, \qquad \beta \equiv 1 + \frac{2WL^2}{\mu U_0^2}$$

the turning points simplify to:

$$x_1^2 = 1 + \alpha(1 + \sqrt{\beta}), \qquad x_2^2 = 1 + \alpha(1 - \sqrt{\beta})$$

Elliptic Modulus

$$k^2 = \frac{x_1^2 - x_2^2}{x_1^2} = \frac{2\alpha\sqrt{\beta}}{1 + \alpha(1 + \sqrt{\beta})}$$

3. Hypergeometric Preliminaries

Gauss Hypergeometric Function

$${}_2F_1(a, b; c; z) = \sum_{n=0}^{\infty} \frac{(a)_n (b)_n}{(c)_n} \frac{z^n}{n!}$$

where $(a)_n = a(a+1)\cdots(a+n-1)$ is the Pochhammer symbol.

Appell $F_1$ Function

For incomplete elliptic integrals, we require the two-variable Appell function:

$$F_1(\alpha; \beta_1, \beta_2; \gamma; x, y) = \sum_{m,n=0}^{\infty} \frac{(\alpha)_{m+n}(\beta_1)_m (\beta_2)_n}{(\gamma)_{m+n}} \frac{x^m y^n}{m!\, n!}$$

4. Complete Elliptic Integrals as Hypergeometric Functions

First Kind

$$K(k) = \frac{\pi}{2} \,{}_2F_1\!\left(\frac{1}{2}, \frac{1}{2}; 1; k^2\right)$$

Power series expansion:

$$K(k) = \frac{\pi}{2} \sum_{n=0}^{\infty} \left[\frac{(2n-1)!!}{(2n)!!}\right]^2 k^{2n}$$

Second Kind

$$E(k) = \frac{\pi}{2} \,{}_2F_1\!\left(-\frac{1}{2}, \frac{1}{2}; 1; k^2\right)$$

Power series expansion:

$$E(k) = \frac{\pi}{2} \left[1 - \sum_{n=1}^{\infty} \left(\frac{(2n-1)!!}{(2n)!!}\right)^2 \frac{k^{2n}}{2n-1}\right]$$

Explicit first terms:

$$E(k) = \frac{\pi}{2}\left(1 - \frac{1}{4}k^2 - \frac{3}{64}k^4 - \frac{5}{256}k^6 - \frac{175}{16384}k^8 - \cdots\right)$$

5. Incomplete Elliptic Integrals

Second Kind via Appell $F_1$

The incomplete elliptic integral of the second kind admits the hypergeometric representation:

$$E(\phi, k) = \sin\phi \cdot F_1\!\left(\frac{1}{2}; -\frac{1}{2}, \frac{1}{2}; \frac{3}{2}; \sin^2\phi,\, k^2\sin^2\phi\right)$$

Alternative Series Representation

Setting $s = \sin\phi$ and $\Delta = \sqrt{1 - k^2 s^2}$:

$$E(\phi, k) = \int_0^\phi \sqrt{1 - k^2\sin^2\theta}\,d\theta = s \cdot \sum_{n=0}^{\infty} \frac{(-1/2)_n}{n!} \cdot \frac{k^{2n}}{2n+1} \cdot {}_2F_1\!\left(-n, \frac{1}{2}; \frac{3}{2}; s^2\right)$$

Practical Expansion for Small $k^2$

$$E(\phi, k) = \phi - \frac{k^2}{4}\left(2\phi - \sin 2\phi\right) - \frac{k^4}{64}\left(6\phi - 4\sin 2\phi + \frac{1}{2}\sin 4\phi\right) + O(k^6)$$

6. Trajectory Solutions in Hypergeometric Form

Attractive Case ($U_0 < 0$)

The angular coordinate as a function of the radial variable:

$$\theta^A(x) = \pm 2|x_1| E(\phi, k)$$

where $\phi = \arcsin\sqrt{(x_1^2 - x^2)/(x_1^2 - x_2^2)}$.

Hypergeometric form:

$$\boxed{\theta^A(x) = \pm 2|x_1| \sin\phi \cdot F_1\!\left(\frac{1}{2}; -\frac{1}{2}, \frac{1}{2}; \frac{3}{2}; \sin^2\phi,\, k^2\sin^2\phi\right)}$$

Repulsive Case ($U_0 > 0$)

$$\theta^R(x) = \pm 2|x_1|\bigl[E(k) - E(\phi, k)\bigr]$$

Hypergeometric form:

$$\boxed{\theta^R(x) = \pm |x_1| \left[\pi\,{}_2F_1\!\left(-\frac{1}{2}, \frac{1}{2}; 1; k^2\right) - 2\sin\phi \cdot F_1\!\left(\frac{1}{2}; -\frac{1}{2}, \frac{1}{2}; \frac{3}{2}; \sin^2\phi,\, k^2\sin^2\phi\right)\right]}$$

7. Perihelion Precession

Full Orbital Period in Angle

For bound orbits (attractive, $W < 0$), one complete radial oscillation spans:

$$\Delta\theta = 4|x_1| E(k)$$

Hypergeometric form:

$$\boxed{\Delta\theta = 2\pi |x_1| \,{}_2F_1\!\left(-\frac{1}{2}, \frac{1}{2}; 1; k^2\right)}$$

Precession Per Orbit

The perihelion advance beyond $2\pi$:

$$\delta\theta = \Delta\theta - 2\pi = 2\pi\left[|x_1|\,{}_2F_1\!\left(-\frac{1}{2}, \frac{1}{2}; 1; k^2\right) - 1\right]$$

Small-$|K|$ Approximation

In the limit of weak Weber corrections ($|K| \ll a$, where $a$ is the semi-major axis):

$$\delta\theta \approx \frac{\pi |K|}{a(1-e^2)}$$

where $e$ is the orbital eccentricity. The full hypergeometric expression captures all orders in $|K|$.

8. Scattering Deflection Angles

For open trajectories ($W \geq 0$), define $\phi^* = \arcsin\sqrt{(x_1^2 - 1)/(x_1^2 - x_2^2)}$.

Attractive Scattering

$$\alpha^A = \frac{4E(\phi^*, k)}{(1 - k^2\sin^2\phi^*)^{1/2}} - \pi$$

Hypergeometric form:

$$\boxed{\alpha^A = \frac{4\sin\phi^*}{(1 - k^2\sin^2\phi^*)^{1/2}} F_1\!\left(\frac{1}{2}; -\frac{1}{2}, \frac{1}{2}; \frac{3}{2}; \sin^2\phi^*,\, k^2\sin^2\phi^*\right) - \pi}$$

Repulsive Scattering

$$\alpha^R = \pi - 4\,\frac{E(k) - E(\phi^*, k)}{(1 - k^2\sin^2\phi^*)^{1/2}}$$

Hypergeometric form:

$$\boxed{\alpha^R = \pi - \frac{2}{(1 - k^2\sin^2\phi^*)^{1/2}}\left[\pi\,{}_2F_1\!\left(-\frac{1}{2}, \frac{1}{2}; 1; k^2\right) - 2\sin\phi^* F_1\!\left(\frac{1}{2}; -\frac{1}{2}, \frac{1}{2}; \frac{3}{2}; \sin^2\phi^*,\, k^2\sin^2\phi^*\right)\right]}$$

9. Recovery of Coulomb Limit

In the limit $c \to \infty$ (equivalently $K \to 0$), we have:

• $|x_1| \to 1$
• $k \to 0$
• ${}_2F_1(-1/2, 1/2; 1; 0) = 1$
• $F_1(1/2; -1/2, 1/2; 3/2; s^2, 0) = {}_2F_1(1/2, -1/2; 3/2; s^2)$

The trajectory equations reduce to the standard Kepler/Rutherford conic sections:

$$r = \frac{L^2/\mu|U_0|}{1 + e\cos\theta}$$

where $e = \sqrt{1 + 2WL^2/\mu U_0^2}$ is the eccentricity.

10. Computational Implementation

Direct Hypergeometric Evaluation

For numerical work, use standard library implementations of ${}_2F_1$ (available in SciPy, GSL, mpmath, etc.).

Complete elliptic integral:

E(k) = (π/2) * hypergeom2F1(-0.5, 0.5, 1.0, k²)

Appell $F_1$ via Numerical Integration

When library support for $F_1$ is unavailable, use the integral representation:

$$F_1(\alpha; \beta_1, \beta_2; \gamma; x, y) = \frac{\Gamma(\gamma)}{\Gamma(\alpha)\Gamma(\gamma-\alpha)} \int_0^1 t^{\alpha-1}(1-t)^{\gamma-\alpha-1}(1-tx)^{-\beta_1}(1-ty)^{-\beta_2}\,dt$$

Series Truncation for Moderate $k^2$

For $k^2 < 0.5$, truncating the ${}_2F_1$ series at $n = 10$ typically gives 10+ digits of accuracy. For $k^2 \to 1$ (near-parabolic orbits), use Landen's transformation or arithmetic-geometric mean methods.

11. Connection to Symplectic Integration

The hypergeometric formulation provides:

1. Exact benchmarks for validating numerical integrators
2. Asymptotic expansions for initializing perturbation methods
3. Analytic derivatives for sensitivity analysis via hypergeometric differentiation formulas

For structure-preserving numerical schemes, the energy $W$ and angular momentum $L$ should be monitored against these exact relations.

References

• Clemente, R. A. & Assis, A. K. T. (1991). Two-Body Problem for Weber-Like Interactions. Int. J. Theor. Phys. 30(4), 537–545.
• Abramowitz, M. & Stegun, I. A. (1972). Handbook of Mathematical Functions, Ch. 15 (Hypergeometric), Ch. 17 (Elliptic).
• NIST Digital Library of Mathematical Functions, https://dlmf.nist.gov/