Closed-Form Solutions in Weber Electrodynamics

Introduction

Weber electrodynamics features a velocity-dependent potential that couples position and velocity, making most solutions involve elliptic integrals. However, several special cases admit elementary closed-form solutions. This document catalogs these cases systematically.

We work in absolute (Gaussian) units unless otherwise specified.

Preliminaries

Reduced Mass Formulation

For a two-body system, we work in the center-of-mass frame with reduced mass:

$\mu = \frac{m_1 m_2}{m_1 + m_2}$

Weber Potential Energy

For attractive charges ($q_1 q_2 = -\kappa$ where $\kappa > 0$):

$U = -\frac{\kappa}{r}\left(1 - \frac{\dot{r}^2}{2c^2}\right)$

For repulsive charges ($q_1 q_2 = +k$ where $k > 0$):

$U = +\frac{k}{r}\left(1 - \frac{\dot{r}^2}{2c^2}\right)$

Key Parameter

Throughout, we define the Weber length scale:

$a = \frac{|q_1 q_2|}{c^2}$

This has dimensions of (mass × length) and characterizes the relativistic correction strength.

Solution 1: Zero-Energy Radial Infall (Attractive)

Setup

• Attractive charges: $q_1 q_2 = -\kappa$
• Zero angular momentum: $L = 0$
• Zero total energy: $E = 0$
• Initial conditions: $r(0) = r_0$, $\dot{r}(0) < 0$ (infalling)

Energy Conservation

$E = \frac{1}{2}\mu\dot{r}^2 - \frac{\kappa}{r} + \frac{\kappa\dot{r}^2}{2c^2 r} = 0$

Solving for $\dot{r}^2$:

$\dot{r}^2 = \frac{2\kappa}{\mu r + a}$

where $a = \kappa/c^2$.

Closed-Form Solution

$\boxed{r(t) = \frac{1}{\mu}\left[\left((\mu r_0 + a)^{3/2} - \frac{3\mu\sqrt{2\kappa}}{2}\,t\right)^{2/3} - a\right]}$

Or equivalently:

$r(t) = \frac{1}{\mu}\left[(A - Bt)^{2/3} - a\right]$

where:

• $A = (\mu r_0 + a)^{3/2}$
• $B = \frac{3\mu\sqrt{2\kappa}}{2}$

Velocity

$\dot{r}(t) = -\sqrt{\frac{2\kappa}{\mu r(t) + a}}$

Collision Time

The particles collide ($r = 0$) at:

$t_{\text{coll}} = \frac{(\mu r_0 + a)^{3/2} - a^{3/2}}{3\mu\sqrt{\kappa/2}}$

Coulomb Limit

As $c \to \infty$ (so $a \to 0$):

$r(t) \to \left(r_0^{3/2} - \frac{3}{2}\sqrt{\frac{2\kappa}{\mu}}\,t\right)^{2/3}$

This is the standard Coulomb $E = 0$ radial solution.

Solution 2: Radial Infall from Rest (Attractive)

Setup

• Attractive charges: $q_1 q_2 = -\kappa$
• Zero angular momentum: $L = 0$
• Particles released from rest at separation $r_0$
• Initial conditions: $r(0) = r_0$, $\dot{r}(0) = 0$

Energy

$E = -\frac{\kappa}{r_0}$

Velocity Equation

$\dot{r}^2 = \frac{2\kappa(1 - r/r_0)}{\mu r + a}$

Time as Function of Position

Define:

• $v = 1 - r/r_0$ (dimensionless progress variable, $v \in [0, 1]$)
• $\beta = \frac{\mu r_0}{\mu r_0 + a}$ (dimensionless parameter, $\beta \in (0, 1)$)

Then the time to fall from $r_0$ to $r$ is:

$\boxed{t(r) = \frac{\mu r_0 + a}{\sqrt{2\kappa\mu r_0}}\left[\arcsin(\sqrt{\beta v}) + \sqrt{\beta v(1 - \beta v)}\right]}$

where $v = 1 - r/r_0$.

Collision Time

At $r = 0$ (so $v = 1$):

$t_{\text{coll}} = \frac{\mu r_0 + a}{\sqrt{2\kappa\mu r_0}}\left[\arcsin(\sqrt{\beta}) + \sqrt{\beta(1 - \beta)}\right]$

Coulomb Limit

As $c \to \infty$, $a \to 0$, $\beta \to 1$:

$t_{\text{coll}} \to r_0\sqrt{\frac{\mu r_0}{2\kappa}} \cdot \frac{\pi}{2} = \frac{\pi}{2}\sqrt{\frac{\mu r_0^3}{2\kappa}}$

This matches the classical Coulomb result (quarter period of a degenerate ellipse).

Solution 3: Circular Orbits

Key Insight

For circular motion, $r = \text{const}$, so $\dot{r} = 0$ and $\ddot{r} = 0$. The Weber force reduces exactly to the Coulomb force!

Force Balance

For attractive charges in circular orbit:

$\mu r\omega^2 = \frac{\kappa}{r^2}$

Angular Velocity (Kepler's Third Law)

$\boxed{\omega^2 = \frac{\kappa}{\mu r^3}}$

This is identical to the Coulomb/Kepler result!

Orbital Period

$T = 2\pi\sqrt{\frac{\mu r^3}{\kappa}}$

Physical Interpretation

Since the radial velocity vanishes for circular orbits, the velocity-dependent Weber correction disappears entirely. Circular orbits in Weber electrodynamics are exact copies of Keplerian circular orbits.

Solution 4: Repulsive Scattering from Rest

Setup

• Repulsive charges: $q_1 q_2 = +k$
• Zero angular momentum: $L = 0$
• Particles released from rest at separation $r_0$
• Initial conditions: $r(0) = r_0$, $\dot{r}(0) = 0$

Energy

$E = +\frac{k}{r_0}$

Velocity Equation

$\dot{r}^2 = \frac{2k(r/r_0 - 1)}{\mu r - a}$

where $a = k/c^2$.

Note: This requires $r > a/\mu$ for physical solutions. Typically $a \ll \mu r_0$, so this is satisfied.

Asymptotic Velocity

As $r \to \infty$:

$\dot{r}_\infty = \sqrt{\frac{2k}{\mu r_0}}$

This is the same as the Coulomb result—the Weber correction affects the trajectory but not the asymptotic speed (conservation of energy).

Time as Function of Position

Define $u = r/r_0$ (so $u \geq 1$), and $\gamma = \frac{a}{\mu r_0}$ (typically $\gamma \ll 1$).

$t(r) = r_0\sqrt{\frac{\mu r_0}{2k}} \int_1^{r/r_0} \sqrt{\frac{u - \gamma}{u - 1}}\, du$

For $\gamma = 0$ (Coulomb limit):

$t = r_0\sqrt{\frac{\mu r_0}{2k}}\left[\sqrt{u(u-1)} + \text{arcosh}(\sqrt{u})\right]_1^{r/r_0}$

For small $\gamma$, the Weber correction adds a perturbative term.

Solution 5: Equilibrium and Critical Points

Radial Equilibrium

For the effective radial potential with angular momentum $L$:

$U_{\text{eff}}(r, \dot{r}) = \frac{L^2}{2\mu r^2} - \frac{\kappa}{r}\left(1 - \frac{\dot{r}^2}{2c^2}\right)$

At circular orbit equilibrium ($\dot{r} = 0$):

$\frac{dU_{\text{eff}}}{dr}\bigg|_{\dot{r}=0} = -\frac{L^2}{\mu r^3} + \frac{\kappa}{r^2} = 0$

$\boxed{r_{\text{circ}} = \frac{L^2}{\mu\kappa}}$

This is identical to the Coulomb result.

Orbital Energy at Circular Orbit

$E_{\text{circ}} = \frac{L^2}{2\mu r_{\text{circ}}^2} - \frac{\kappa}{r_{\text{circ}}} = -\frac{\mu\kappa^2}{2L^2}$

Again, identical to Coulomb.

Solution 6: Small Oscillations About Circular Orbit

Effective Spring Constant

For small radial perturbations $\xi = r - r_{\text{circ}}$ about a circular orbit:

$\ddot{\xi} + \omega_r^2 \xi = 0$

The radial oscillation frequency is:

$\omega_r = \omega_\phi = \sqrt{\frac{\kappa}{\mu r_{\text{circ}}^3}}$

Since $\omega_r = \omega_\phi$, orbits close after one revolution—the ellipse does not precess (to first order).

Precession in Weber (Higher Order)

For general (non-circular) orbits, Weber electrodynamics does predict orbital precession due to the velocity-dependent terms. However, this requires solving the full equations numerically or via perturbation theory.

Solution 7: Head-On Collision Velocity

Maximum Approach Velocity (Attractive)

For particles falling from infinity with $E = 0$:

$\dot{r}^2 = \frac{2\kappa}{\mu r + a}$

As $r \to 0$:

$\dot{r}_{\max} = \sqrt{\frac{2\kappa}{a}} = \sqrt{\frac{2\kappa c^2}{\kappa}} = c\sqrt{2}$

This exceeds the speed of light! This indicates that Weber electrodynamics, as a non-relativistic theory, breaks down at small separations where velocities become relativistic.

Critical Separation

The velocity reaches $c$ at:

$r_c = \frac{\kappa}{\mu c^2} = \frac{a}{\mu}$

For $r < r_c$, the theory predicts superluminal velocities, signaling its breakdown.

Solution 8: Turning Points (Repulsive Scattering with Angular Momentum)

Setup

• Repulsive charges: $q_1 q_2 = +k$
• Angular momentum: $L \neq 0$
• Energy: $E > 0$

Effective Potential

$U_{\text{eff}} = \frac{L^2}{2\mu r^2} + \frac{k}{r}\left(1 - \frac{\dot{r}^2}{2c^2}\right)$

Turning Point Condition

At the distance of closest approach $r_{\min}$, $\dot{r} = 0$:

$E = \frac{L^2}{2\mu r_{\min}^2} + \frac{k}{r_{\min}}$

This is identical to the Coulomb turning point equation! Solving:

$\boxed{r_{\min} = \frac{k + \sqrt{k^2 + 2EL^2/\mu}}{2E}}$

The turning point in Weber electrodynamics (at $\dot{r} = 0$) coincides exactly with the Coulomb prediction.

Summary Table

| Case | Conditions | Solution Type | |------|------------|---------------| | Zero-energy radial infall | $E=0$, $L=0$, attractive | $r(t) = \frac{1}{\mu}[(A-Bt)^{2/3} - a]$ | | Infall from rest | $\dot{r}_0=0$, $L=0$, attractive | $t(r)$ via arcsin (elementary) | | Circular orbits | $\dot{r}=0$, attractive | Same as Kepler | | Repulsive scattering | $\dot{r}_0=0$, $L=0$, repulsive | $\dot{r}_\infty = \sqrt{2k/(\mu r_0)}$ | | Turning points | $\dot{r}=0$, repulsive | Same as Coulomb |

Dimensionless Formulation

Characteristic Scales

For a system with $\kappa = |q_1 q_2|$, we can define:

• Length: $r_0$ (initial separation)
• Time: $\tau_0 = \sqrt{\mu r_0^3 / \kappa}$
• Velocity: $v_0 = \sqrt{\kappa / (\mu r_0)}$

Dimensionless Variables

$\rho = r/r_0, \quad \tau = t/\tau_0, \quad \epsilon = v_0/c = \sqrt{\frac{\kappa}{\mu r_0 c^2}}$

Dimensionless Energy Equation (Radial)

$\tilde{E} = \frac{1}{2}\dot{\rho}^2 - \frac{1}{\rho} + \frac{\epsilon^2 \dot{\rho}^2}{2\rho}$

where $\tilde{E} = E\mu r_0 / \kappa$ and $\dot{\rho} = d\rho/d\tau$.

The parameter $\epsilon$ controls the strength of Weber corrections. For $\epsilon \ll 1$, Coulomb behavior dominates.

Physical Validity Regime

Weber electrodynamics is valid when:

1. $v \ll c$ (non-relativistic velocities)
2. $r \gg a/\mu = |q_1 q_2|/(\mu c^2)$ (above the critical separation)
3. Radiation effects are negligible

For electrons: $a/\mu \approx 2.8 \times 10^{-15}$ m (classical electron radius).

For macroscopic charged objects, Weber corrections are typically negligible.

Appendix: Derivation of Arcsin Solution

Starting from:

$\dot{r}^2 = \frac{2\kappa(1 - r/r_0)}{\mu r + a}$

For infall ($\dot{r} < 0$):

$dt = -\sqrt{\frac{\mu r + a}{2\kappa(1 - r/r_0)}}\,dr$

Substituting $v = 1 - r/r_0$, $r = r_0(1-v)$, $dr = -r_0\,dv$:

$dt = r_0\sqrt{\frac{\mu r_0(1-v) + a}{2\kappa v}}\,dv = r_0\sqrt{\frac{\mu r_0 + a}{2\kappa}}\sqrt{\frac{1 - \beta v}{v}}\,dv$

where $\beta = \mu r_0/(\mu r_0 + a)$.

Using the integral:

$\int_0^v \sqrt{\frac{1 - \beta u}{u}}\,du = \frac{1}{\sqrt{\beta}}\left[\arcsin(\sqrt{\beta v}) + \sqrt{\beta v(1-\beta v)}\right]$

We obtain the closed-form result.

References

1. Weber, W. (1846). Elektrodynamische Maassbestimmungen
2. Assis, A.K.T. (1994). Weber's Electrodynamics
3. Wesley, J.P. (1990). "Weber electrodynamics"