KS Phase-Space Regularization for n-Body Weber Electrodynamics

Abstract

We present a phase-space regularization scheme for the n-body Weber electrodynamics problem in three dimensions. The velocity-dependent Weber potential introduces singularities that are not removed by standard Kustaanheimo-Stiefel (KS) regularization. We show that a quadratic time regularization $dt = r_{ij}^2 \, d\tau$ combined with KS coordinate lifting removes all collision singularities, yielding polynomial equations of motion in the regularized variables.

1. The n-Body Weber System

1.1 Configuration

Consider $n$ charged particles in $\mathbb{R}^3$ with:

• Masses: $m_1, m_2, \ldots, m_n$
• Charges: $q_1, q_2, \ldots, q_n$
• Positions: $\vec{r}_1, \vec{r}_2, \ldots, \vec{r}_n \in \mathbb{R}^3$
• Velocities: $\dot{\vec{r}}_1, \dot{\vec{r}}_2, \ldots, \dot{\vec{r}}_n \in \mathbb{R}^3$

1.2 Pairwise Quantities

For each pair $(i,j)$ with $i < j$:

$\vec{r}_{ij} = \vec{r}_i - \vec{r}_j, \quad r_{ij} = |\vec{r}_{ij}|$

$\dot{\vec{r}}_{ij} = \dot{\vec{r}}_i - \dot{\vec{r}}_j, \quad \dot{r}_{ij} = \frac{\vec{r}_{ij} \cdot \dot{\vec{r}}_{ij}}{r_{ij}}$

1.3 Weber Potential Energy

The total Weber potential energy is:

$U = \sum_{i<j} U_{ij}, \quad U_{ij} = \frac{k_{ij}}{r_{ij}}\left(1 - \frac{\dot{r}_{ij}^2}{2c^2}\right)$

where $k_{ij} = q_i q_j$ (in Gaussian units) or $k_{ij} = \frac{q_i q_j}{4\pi\epsilon_0}$ (in SI units).

1.4 Lagrangian

$L = T - S$

where: $T = \sum_{i=1}^{n} \frac{1}{2}m_i |\dot{\vec{r}}_i|^2$

$S = \sum_{i<j} \frac{k_{ij}}{r_{ij}}\left(1 + \frac{\dot{r}_{ij}^2}{2c^2}\right)$

1.5 Total Energy (Hamiltonian)

$H = T + U = \sum_{i=1}^{n} \frac{1}{2}m_i |\dot{\vec{r}}_i|^2 + \sum_{i<j} \frac{k_{ij}}{r_{ij}}\left(1 - \frac{\dot{r}_{ij}^2}{2c^2}\right)$

2. Why Standard KS Regularization Fails

2.1 The Coulomb Case (Review)

For pure Coulomb interaction $U_{ij}^{\text{Coul}} = k_{ij}/r_{ij}$, the KS transformation:

$\vec{r}_{ij} = A(u_{ij})\bar{u}_{ij}, \quad r_{ij} = |u_{ij}|^2$

with time transformation $dt = r_{ij}\,d\tau$ yields:

$r_{ij} \cdot H = \text{polynomial in } (u, u')$

removing the $1/r_{ij}$ singularity.

2.2 The Weber Obstruction

For Weber, under KS coordinates:

$\dot{r}_{ij} = \frac{2(u_{ij} \cdot \dot{u}_{ij})}{|u_{ij}|^2} = \frac{2(u_{ij} \cdot u'_{ij})}{|u_{ij}|^4}$

where $u' = du/d\tau$. Therefore:

$\frac{\dot{r}_{ij}^2}{r_{ij}} = \frac{4(u_{ij} \cdot u'_{ij})^2}{|u_{ij}|^4 \cdot |u_{ij}|^2} = \frac{4(u_{ij} \cdot u'_{ij})^2}{|u_{ij}|^6}$

Multiplying by $r_{ij} = |u_{ij}|^2$ leaves a residual singularity $\sim 1/|u_{ij}|^4$.

3. Quadratic Time Regularization

3.1 Main Construction

Definition (Pairwise Regularization). For each pair $(i,j)$, introduce:

1. KS coordinates: $u_{ij} \in \mathbb{R}^4$ such that $\vec{r}_{ij} = A(u_{ij})\bar{u}_{ij}$

2. Quadratic fictitious time: a parameter $\tau_{ij}$ satisfying $\frac{dt}{d\tau_{ij}} = r_{ij}^2 = |u_{ij}|^4$

3.2 The KS Matrix

u_1 & -u_2 & -u_3 & u_4 \\ 
u_2 & u_1 & -u_4 & -u_3 \\ 
u_3 & u_4 & u_1 & u_2 
\end{pmatrix}$$

**Properties:**
- $A(u)\bar{u} \in \mathbb{R}^3$ where $\bar{u} = (u_1, u_2, u_3, u_4)^T$
- $|A(u)\bar{u}|^2 = |u|^4$, hence $r_{ij} = |u_{ij}|^2$
- $A(u)^T A(u) = |u|^2 I_4$

---

## 4. Main Result: Regularized n-Body Weber Hamiltonian

### 4.1 Pairwise Regularization Theorem

**Theorem.** *For the two-body Weber problem in relative coordinates, the quadratically regularized Hamiltonian:*

$$\mathcal{K}_{ij} = r_{ij}^2 \cdot H_{ij} = r_{ij}^2 \left(\frac{\mu_{ij}}{2}|\dot{\vec{r}}_{ij}|^2 + \frac{k_{ij}}{r_{ij}} - \frac{k_{ij}\dot{r}_{ij}^2}{2c^2 r_{ij}}\right)$$

*transforms under KS coordinates to:*

$$\boxed{\mathcal{K}_{ij} = 2\mu_{ij}|u_{ij}|^2|u'_{ij}|^2 + k_{ij}|u_{ij}|^2 - \frac{2k_{ij}}{c^2}(u_{ij} \cdot u'_{ij})^2}$$

*which is polynomial in $(u_{ij}, u'_{ij})$ and regular at $u_{ij} = 0$.*

Here $\mu_{ij} = \frac{m_i m_j}{m_i + m_j}$ is the reduced mass and $u'_{ij} = du_{ij}/d\tau_{ij}$.

### 4.2 Energy Constraint

On physical trajectories:

$$\mathcal{K}_{ij} = r_{ij}^2 \cdot E_{ij}$$

where $E_{ij}$ is the (conserved) energy of the pair. At collision ($r_{ij} = 0$):

$$\mathcal{K}_{ij}\big|_{u_{ij}=0} = 0$$

This is automatically satisfied, confirming regularity.

---

## 5. Global n-Body Formulation

### 5.1 Challenge: Multiple Collision Times

Each pair $(i,j)$ has its own natural regularizing time $\tau_{ij}$. For a global formulation, we need a single evolution parameter.

### 5.2 Sundman-Type Global Time Transformation

Introduce a global fictitious time $s$ via:

$$\frac{dt}{ds} = \prod_{i<j} r_{ij}^2 = \prod_{i<j} |u_{ij}|^4$$

or more practically, a **selective regularization**:

$$\frac{dt}{ds} = \left(\min_{i<j} r_{ij}\right)^2$$

### 5.3 Extended Phase Space Formulation

**State space:** For $n$ particles in $\mathbb{R}^3$, introduce:

- Original positions: $\vec{r}_i \in \mathbb{R}^3$ for $i = 1, \ldots, n$
- KS pairwise coordinates: $u_{ij} \in \mathbb{R}^4$ for each pair $i < j$
- Time as coordinate: $t$
- Conjugate momenta: $\vec{p}_i$, $\pi_{ij}$, $p_t = -E$

**Dimension count:**
- Physical: $3n$ positions + $3n$ momenta = $6n$
- KS extended: $3n + 4\binom{n}{2}$ positions + same momenta + constraints

### 5.4 Constraint Structure

**KS bilinear constraints** (gauge freedom): For each pair,
$$\Phi_{ij} = u_{ij,1}\pi_{ij,2} - u_{ij,2}\pi_{ij,1} + u_{ij,3}\pi_{ij,4} - u_{ij,4}\pi_{ij,3} = 0$$

**Consistency constraints**: The KS coordinates must satisfy
$$|u_{ij}|^2 = |\vec{r}_i - \vec{r}_j|$$

**Energy constraint**:
$$\mathcal{H} = H(\vec{r}, \vec{p}) + p_t = 0$$

---

## 6. Regularized Equations of Motion

### 6.1 Single Pair (Relative Coordinates)

Using $\mathcal{K}_{ij}$ as the Hamiltonian with parameter $\tau_{ij}$:

$$\frac{du_{ij}}{d\tau_{ij}} = \frac{\partial \mathcal{K}_{ij}}{\partial \pi_{ij}}$$

$$\frac{d\pi_{ij}}{d\tau_{ij}} = -\frac{\partial \mathcal{K}_{ij}}{\partial u_{ij}}$$

**Explicit form:**

$$\frac{du_{ij,\alpha}}{d\tau_{ij}} = 4\mu_{ij}|u_{ij}|^2 \pi_{ij,\alpha} - \frac{4k_{ij}}{c^2}(u_{ij} \cdot \pi_{ij})u_{ij,\alpha}$$

$$\frac{d\pi_{ij,\alpha}}{d\tau_{ij}} = -4\mu_{ij}|\pi_{ij}|^2 u_{ij,\alpha} - 2k_{ij}u_{ij,\alpha} + \frac{4k_{ij}}{c^2}(u_{ij} \cdot \pi_{ij})\pi_{ij,\alpha}$$

### 6.2 Canonical Momenta Relation

The KS momenta $\pi_{ij}$ relate to physical momenta via:

$$\vec{p}_{ij} = \frac{2A(u_{ij})^T \pi_{ij}}{|u_{ij}|^2}$$

with the inverse (on constraint surface):

$$\pi_{ij} = \frac{1}{2}A(u_{ij})^T \vec{p}_{ij}$$

---

## 7. n-Body Regularized Hamiltonian

### 7.1 Center of Mass Separation

Total momentum: $\vec{P} = \sum_i m_i \dot{\vec{r}}_i$ (conserved)

Center of mass: $\vec{R} = \frac{1}{M}\sum_i m_i \vec{r}_i$, where $M = \sum_i m_i$

The CM motion separates: $H = \frac{|\vec{P}|^2}{2M} + H_{\text{rel}}$

### 7.2 Full Regularized Hamiltonian

**Definition.** The globally regularized n-body Weber Hamiltonian is:

$$\boxed{\mathcal{K}_n = \left(\prod_{i<j} r_{ij}^2\right) \cdot H}$$

Under KS transformation for all pairs:

$$\mathcal{K}_n = \left(\prod_{i<j} |u_{ij}|^4\right) \left[\sum_{i=1}^n \frac{|\vec{p}_i|^2}{2m_i} + \sum_{i<j}\frac{k_{ij}}{|u_{ij}|^2}\left(1 - \frac{2(u_{ij} \cdot u'_{ij})^2}{c^2|u_{ij}|^4}\right)\right]$$

**Expanding and simplifying:**

$$\mathcal{K}_n = \left(\prod_{i<j}|u_{ij}|^4\right)\sum_i \frac{|\vec{p}_i|^2}{2m_i} + \sum_{i<j}\left(\prod_{k<l, (k,l)\neq(i,j)}|u_{kl}|^4\right)\left[k_{ij}|u_{ij}|^2 - \frac{2k_{ij}(u_{ij}\cdot u'_{ij})^2}{c^2}\right]$$

This is **polynomial** in all $u_{ij}$ and $u'_{ij}$.

---

## 8. Practical Implementation

### 8.1 Collision Detection and Chart Switching

For numerical implementation, it is often impractical to maintain $\binom{n}{2}$ sets of KS coordinates globally. Instead:

1. **Monitor pairwise distances** $r_{ij}$ during integration
2. **Switch to KS chart** for pair $(i,j)$ when $r_{ij} < r_{\text{threshold}}$
3. **Regularize locally** using $\tau_{ij}$ time for that pair
4. **Switch back** to physical coordinates after closest approach

### 8.2 Symplectic Integration Considerations

The regularized system has Hamiltonian structure but:
- **Non-separable**: The $(u \cdot \pi)^2$ terms couple positions and momenta
- **Constrained**: KS gauge and energy constraints must be preserved

**Recommended methods:**
- Implicit symplectic methods (e.g., implicit midpoint)
- RATTLE/SHAKE for constraint handling
- Splitting methods adapted for non-separable systems (Jayawardana-Ohsawa)

### 8.3 Recovering Physical Time

Physical time is recovered by integrating:

$$t = t_0 + \int_0^{\tau} \left(\prod_{i<j}|u_{ij}(s)|^4\right) ds$$

or for single-pair regularization:

$$t = t_0 + \int_0^{\tau_{ij}} |u_{ij}(s)|^4 \, ds$$

---

## 9. Conservation Laws

### 9.1 Preserved Quantities

Under the regularized flow, the following are preserved:

1. **Total energy** $E$ (via the constraint $\mathcal{K}_n = \prod r_{ij}^2 \cdot E$)
2. **Total momentum** $\vec{P} = \sum_i \vec{p}_i$
3. **Total angular momentum** $\vec{L} = \sum_i \vec{r}_i \times \vec{p}_i$
4. **KS bilinear constraints** $\Phi_{ij} = 0$

### 9.2 Scaling Property

The Weber Hamiltonian has a modified scaling symmetry compared to pure Coulomb, due to the velocity-dependent terms introducing the constant $c$.

---

## 10. Special Case: Two-Body Problem

For $n = 2$, in center-of-mass frame:

$$\mathcal{K} = 2\mu|u|^2|u'|^2 + k|u|^2 - \frac{2k}{c^2}(u \cdot u')^2 - |u|^4 E$$

**Equations of motion** (with $\pi = \mu u'/|u|^2$ rescaling):

$$u'' = -\frac{k}{2\mu}u + \frac{E}{2\mu}|u|^2 u + \frac{k}{c^2\mu|u|^2}\left[(u' \cdot u')u + 2(u \cdot u')u' - \frac{4(u \cdot u')^2}{|u|^2}u\right]$$

This extends smoothly through $u = 0$.

---

## 11. Limitations and Open Questions

### 11.1 Non-Separability

Unlike Coulomb-KS, the regularized Weber system remains **non-separable**. This means:
- No obvious action-angle variables
- Standard symplectic splitting doesn't apply directly
- May require implicit integrators for structure preservation

### 11.2 Multiple Simultaneous Collisions

The global regularization handles **binary collisions** well. For triple or higher-order simultaneous collisions (measure zero but possible), additional blow-up techniques may be required (cf. McGehee coordinates).

### 11.3 Physical Interpretation of the Gauge Freedom

The KS gauge constraint $\Phi_{ij} = 0$ corresponds to a phase freedom in the spinor representation. For Weber electrodynamics, this may have physical meaning related to the electromagnetic gauge—an unexplored connection.

### 11.4 Relativistic Limit

As $\dot{r}/c \to 0$, the regularized Weber system should reduce to regularized Coulomb. The structure of the $(u \cdot u')^2$ terms ensures this limit is smooth.

---

## 12. Summary

| Aspect | Coulomb | Weber |
|--------|---------|-------|
| Singularity type | $1/r$ | $1/r$ and $\dot{r}^2/r$ |
| KS time transform | $dt = r\,d\tau$ | $dt = r^2\,d\tau$ |
| Regularized form | Polynomial in $(u, u')$ | Polynomial in $(u, u')$ |
| Separability | Separable | **Non-separable** |
| Collision behavior | Elastic bounce | Regularized, smooth extension |

**Main Result:** The n-body Weber system admits a complete regularization via KS coordinates with quadratic time transformation, yielding polynomial equations of motion that extend through binary collisions.

---

## References

1. Kustaanheimo, P., & Stiefel, E. (1965). Perturbation theory of Kepler motion based on spinor regularization. *J. Reine Angew. Math.*, 218, 204-219.

2. Sundman, K. F. (1913). Mémoire sur le problème des trois corps. *Acta Mathematica*, 36, 105-179.

3. Weber, W. (1846). Elektrodynamische Maassbestimmungen. *Abhandlungen der Königl. Sächs. Gesellschaft der Wissenschaften*.

4. Ascher, U. M., & Reich, S. (1999). The midpoint scheme and variants for Hamiltonian systems: advantages and pitfalls. *SIAM J. Sci. Comput.*, 21(3), 1045-1065.

5. Jayawardana, B., & Ohsawa, T. (2023). Symplectic integration of non-separable Hamiltonians. *J. Comput. Phys.*.

---

## Appendix A: KS Transformation Details

### A.1 Quaternion Representation

The KS transformation can be elegantly expressed using quaternions. If $q = u_1 + u_2 i + u_3 j + u_4 k$, then:

$$\vec{r} = q \, \mathbf{k} \, \bar{q}$$

maps $\mathbb{H} \to \mathbb{R}^3$ (imaginary quaternions), with $|\vec{r}| = |q|^2$.

### A.2 Velocity Transformation

$$\dot{\vec{r}} = 2 \, \text{Im}(\dot{q} \, \mathbf{k} \, \bar{q})$$

$$|\dot{\vec{r}}|^2 = 4|q|^2|\dot{q}|^2 - 4(\text{Re}(q\bar{\dot{q}}))^2 = 4|q|^2|\dot{q}|^2 - 4(q \cdot \dot{q})^2$$

This last identity is crucial for the Weber velocity terms.

---

## Appendix B: Constraint Preservation

The Poisson bracket structure ensures that if $\Phi_{ij} = 0$ initially, it remains zero under the Hamiltonian flow. Explicitly:

$$\{Φ_{ij}, \mathcal{K}\} = 0$$

when evaluated on the constraint surface, provided the Hamiltonian respects the SO(2) gauge symmetry of the KS representation.

For the Weber terms, verification requires showing:

$$\{u_1\pi_2 - u_2\pi_1 + u_3\pi_4 - u_4\pi_3, (u \cdot \pi)^2\} = 0$$

which follows from the rotational invariance of $(u \cdot \pi)$.