Hamiltonian Formulation via the PMP
Recall from Part 1 that we want to minimise arc length among horizontal curves in $\mathrm{SE}(2)$:
\[\min \int_0^T \sqrt{u_1^2 + u_2^2}\,dt, \qquad \dot g = u_1 X_1(g) + u_2 X_2(g).\]We use the standard normalisation $u_1^2 + u_2^2 = 1$ (unit-speed parametrisation), which reduces the problem to minimising $T$ — the total arc length.
The Pontryagin Maximum Principle (PMP) introduces a covector $\lambda$ in the cotangent bundle, evolving alongside the state:
\[\lambda \in T^{*}_{g}\,\mathrm{SE}(2).\]In the left-trivialisation provided by the Lie algebra,
\[T^{*}\mathrm{SE}(2) \;\cong\; \mathrm{SE}(2) \times \mathfrak{se}(2)^{*},\]the costate becomes a triple of components $(h_1, h_2, h_3)$ defined by
\[h_i = \langle \lambda,\, X_i(g)\rangle, \qquad i = 1, 2, 3.\]The Hamiltonian for the PMP maximisation condition is
\[H = h_1 u_1 + h_2 u_2 - \nu \sqrt{u_1^2 + u_2^2},\]where $\nu \in {0, \tfrac{1}{2}}$ is the abnormality constant. For normal extremals ($\nu = \tfrac{1}{2}$), maximising over $u_1, u_2$ gives the normal Hamiltonian
\[\mathcal{H}_n = \tfrac{1}{2}\bigl(h_1^2 + h_2^2\bigr).\]The Hamiltonian equations on $\mathfrak{se}(2)^{*}$, via the Lie–Poisson bracket, read
\[\dot h_1 = \{h_1, \mathcal{H}_n\} = h_2 h_3, \quad \dot h_2 = \{h_2, \mathcal{H}_n\} = -h_1 h_3, \quad \dot h_3 = \{h_3, \mathcal{H}_n\} = -h_1 h_2.\]Two integrals reduce this 3D system to a 1D motion. The Hamiltonian itself, $\mathcal{H}_n = \tfrac{1}{2}(h_1^{2} + h_2^{2})$, is conserved because the flow is Hamiltonian. The second is the Casimir of $\mathfrak{se}(2)^{*}$,
\[C \;=\; h_1^{2} + h_3^{2},\]which Poisson-commutes with every smooth function and is therefore automatically preserved by any Hamiltonian flow on the dual algebra.
The Casimir labels the symplectic leaves of the Lie–Poisson bracket — the coadjoint orbits. For $\mathfrak{se}(2)^{*}$ the orbits are the cylinders ${h_1^{2} + h_3^{2} = C}$, on which the Hamiltonian dynamics is genuinely symplectic.
Fix the unit-speed normalisation $\mathcal{H}_n = 1/2$, so $h_1^{2} + h_2^{2} = 1$. The angle $\alpha$ defined by
\[h_1 = \sin\alpha, \qquad h_2 = \cos\alpha\]then evolves as $\dot\alpha = h_3$ (from $\dot h_1 = h_2 h_3$). Using $h_3^{2} = C - h_1^{2} = C - \sin^{2}\alpha$, set $\varphi = 2\alpha$:
\[\dot\varphi^{2} \;=\; 4 h_3^{2} \;=\; (4C - 2) \,+\, 2\cos\varphi.\]Differentiating once more in time gives the pendulum equation.
The Pendulum Equation
The three dynamically distinct regimes of the pendulum correspond directly to the three families of SE(2) geodesics:
| Energy $E$ | Pendulum motion | Elastica family |
|---|---|---|
| $-1 < E < 1$ | oscillation (libration) | inflectional |
| $E = 1$ | separatrix (infinite period) | Euler spiral |
| $E > 1$ | rotation | non-inflectional |
From the pendulum to the curve
The pendulum equation lives on the costate, but the figures below plot the curvature $\kappa(s)$ of the projected plane curve in its own Euclidean arc length $s$. These two are tied together by a quick computation.
In unit-speed parametrisation $h_1^{2}+h_2^{2}=1$, the projected speed is $|h_1| = |\sin(\varphi/2)|$ and the heading derivative is $\dot\theta = h_2 = \cos(\varphi/2)$. Reparametrising by Euclidean arc length $s$ with $ds = |h_1|\,dt$, and using $\dot\varphi = \pm 2 h_3$, a short calculation gives an explicit closed-form $\kappa(s)$ in each regime — the three Jacobi-elliptic curvature profiles below. They satisfy the elastica curvature ODE (the Duffing form equivalent to the pendulum)
\[\kappa''(s) + \tfrac{1}{2}\kappa(s)^{3} - \mu\,\kappa(s) \;=\; 0,\]with the integration constant $\mu$ fixed by the Casimir $C$ — one ODE, three qualitatively different solution branches.
Three Families via Jacobi Elliptic Functions
Inflectional Elastica ($-1 < E < 1$)
The pendulum oscillates; $\varphi(s)$ changes sign. The curvature of the projected plane curve therefore changes sign: these are curves with inflection points.
Setting the energy $E = 2k^{2} - 1$ with $k \in (0, 1)$, the curvature is
\[\boxed{\;\kappa(s) = 2k\,\mathrm{cn}(s \mid k^{2})\;}\]The Jacobi function $\mathrm{cn}(s \mid k^{2})$ has period $4K(k^{2})$ in $s$, so the curvature — and hence the curve shape — repeats with spatial period
\[T_{\kappa} = 4K(k^{2}).\]As $k \to 0$: $\mathrm{cn}(s\mid 0) = \cos s$ and the elastica approximates a cosine-curvature curve (nearly straight). As $k \to 1$: $K(1) = \infty$ and the period diverges — the curve spirals inward without repeating (the Euler spiral limit).
Euler Spiral / Cornu Spiral ($E = 1$, separatrix)
At the separatrix the curvature is
\[\kappa(s) = \frac{2}{\cosh s}.\]This is the Euler–Cornu spiral (also called the clothoid). Its curvature is a smooth bump decaying to zero at both ends; the total turning is $\Delta\theta = 2\pi$. It is the limiting case between oscillating and rotating pendulum, and has infinite period $T_\kappa = \infty$.
The Euler spiral is famous in civil engineering (transition curves for railways) and optics (Fresnel integrals), but here it arises as the unique separatrix geodesic in SE(2).
Non-Inflectional Elastica ($E > 1$)
The pendulum rotates without stopping; $\varphi(s)$ is monotone. Setting $m = 2/(E + 1) \in (0, 1)$ (here $m$ parametrises this family — at $E=1$ the separatrix gives $m=1$, and as $E\to\infty$ we have $m\to 0$):
\[\kappa(s) = 2\,\mathrm{dn}(s \mid m).\]These curves have no inflection points — the curvature never changes sign. The Jacobi function $\mathrm{dn}(s\mid m)$ has period $2K(m)$ in $s$, so the spatial period is
\[T_{\kappa} = 2K(m).\]As $m \to 1$: $\mathrm{dn}(s\mid 1) = \mathrm{sech}(s)$ — the non-inflectional family approaches the Euler spiral from the other side. As $m \to 0$: $\mathrm{dn}(s\mid 0) = 1$ and $\kappa \to 2$ — the curves degenerate into circles of radius $1/2$ (high-energy uniform rotation).
The Complete Elliptic Integral K(k²)
The period formula $T_\kappa = 4K(k^{2})$ is not incidental — $K(k^{2})$ is the exact half-period of $\mathrm{sn}(s\mid k^{2})$ by definition. This makes the elastica period computable to arbitrary precision via the arithmetic–geometric mean (AGM):
\[K(m) = \frac{\pi}{2\,\mathrm{AGM}\!\bigl(1,\;\sqrt{1 - m}\bigr)}, \qquad m = k^{2}.\]The AGM converges quadratically: about 16 significant digits in 6 iterations.
This is precisely what the elliptic package uses:
from elliptic import ellipticK
import numpy as np
k = np.linspace(0, 0.999, 500)
Tk = 4 * ellipticK(k**2) # spatial period of curvature oscillation
For the Jacobi functions themselves:
from elliptic import ellipj
s = np.linspace(-2*K, 2*K, 800)
sn, cn, dn = ellipj(s, k**2)
kappa = 2 * k * cn # curvature of inflectional elastica
The three functions satisfy the differential equations
\[\frac{d}{ds}\mathrm{sn} = \mathrm{cn}\,\mathrm{dn}, \qquad \frac{d}{ds}\mathrm{cn} = -\mathrm{sn}\,\mathrm{dn}, \qquad \frac{d}{ds}\mathrm{dn} = -m\,\mathrm{sn}\,\mathrm{cn},\]and the Pythagorean identities
\[\mathrm{sn}^{2} + \mathrm{cn}^{2} = 1, \qquad \mathrm{dn}^{2} + m\,\mathrm{sn}^{2} = 1.\]These let us differentiate $\kappa(s)$ analytically:
\[\kappa'(s) = -2k\,\mathrm{sn}(s)\,\mathrm{dn}(s),\]which will be essential in Part 3 for locating Maxwell strata.
Integrating the Elastica
Given $\kappa(s)$, the plane curve is recovered by Frenet–Serret integration:
\[\frac{d}{ds}\begin{pmatrix}x\\ y\\ \theta\end{pmatrix} = \begin{pmatrix}\cos\theta\\ \sin\theta\\ \kappa(s)\end{pmatrix}.\]The $\theta$-equation integrates explicitly for the inflectional family:
\[\theta(s) = \theta_0 + 2\arcsin\!\bigl(k\,\mathrm{sn}(s\mid k^{2})\bigr),\]then $x, y$ require a further integration involving $\mathrm{sn}$ and $\mathrm{cn}$ that can be expressed via the incomplete elliptic integral of the second kind $E(s\mid k^{2})$.
In the elliptic package:
from elliptic import elliptic12
phi = np.arcsin(k * sn)
E_vals, F_vals = elliptic12(phi, k**2) # E(φ|k²) and F(φ|k²)
x = 2 * (E_vals - F_vals / 2) # exact formula from Sachkov (2011)
The full closed-form expressions — due to Sachkov (2011) — express every $x(s)$, $y(s)$ of an inflectional elastica as a rational combination of $\mathrm{sn}$, $\mathrm{cn}$, $\mathrm{dn}$, and $E(\cdot\mid k^2)$. No numerical ODE integration is needed.
What the Three Families Look Like
The interactive figure above lets you explore all three families. A few landmarks worth noting:
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$k = 0.1$ (inflectional): nearly straight, very gentle curvature oscillation. The curve barely bends before straightening again.
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$k \approx 0.71$ (inflectional): the “figure-eight” lemniscate — the curve crosses itself once per period and the endpoints of one period coincide. This is the Maxwell stratum for the symmetric geodesics (Part 3).
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$k \to 1^-$ (inflectional → Euler spiral): the period $4K(k^2)$ diverges and the curve spirals inward, winding around two limiting points.
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Euler spiral ($k = 1$): curvature $2/\cosh(s)$, total turning $2\pi$. The two asymptotic directions are parallel but offset — the curve never closes.
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Non-inflectional, $m = 0.3$: like a wavy circle — curvature oscillates but never changes sign; the curve closes after a finite arc length.
Summary and Preview of Part 3
We have shown that every normal SR geodesic on SE(2) has curvature belonging to one of three Jacobi-elliptic families, with the inflectional case $\kappa(s) = 2k\,\mathrm{cn}(s\mid k^2)$ being the generic one. The complete elliptic integral $K(k^2)$ controls the spatial period of the curvature, and the closed-form $x(s), y(s)$ involve elliptic integrals of the second kind.
The next natural question is: which geodesics are globally optimal? A geodesic may be locally length-minimising (no shorter path in a thin tube) while a completely different geodesic of the same length exists. The locus where this happens — where two distinct geodesics of equal length meet — is the Maxwell stratum.
In Part 3 we will show that the Maxwell stratum is governed by the discrete symmetry group of $\mathrm{SE}(2)$, and that the first Maxwell time for an inflectional geodesic with modulus $k$ is exactly
\[t_{\mathrm{MAX}}^{(1)} = \frac{4K(k^{2})}{\omega_0},\]the same period $4K(k^{2})$ that controls the curvature oscillation — but this time as a time, not a length. This identity is the heart of the Maxwell strata proof.
References
- I. Moiseev & Yu. L. Sachkov (2010). "Maxwell strata in sub-Riemannian problem on the group of motions of a plane." ESAIM: COCV 16(2): 380–399. arXiv:0807.4731
- Yu. L. Sachkov (2011). "Cut locus and optimal synthesis in the sub-Riemannian problem on the group of motions of a plane." ESAIM: COCV 17(4): 293–321. arXiv:0903.0727
- Yu. L. Sachkov (2004). "Exponential mapping in generalized Dido's problem." Mat. Sbornik 194(9): 63–90.
- L. Euler (1744). Methodus inveniendi lineas curvas maximi minimive proprietate gaudentes. Lausanne: Marcum-Michaelem Bousquet. Additamentum I (De curvis elasticis).
- M. Abramowitz & I. A. Stegun (1964). Handbook of Mathematical Functions. National Bureau of Standards. §16 (Jacobian Elliptic Functions), §17 (Elliptic Integrals).
- J. Petitot (2003). "The neurogeometry of pinwheels as a sub-Riemannian contact structure." Journal of Physiology–Paris 97(2–3): 265–309.
- L. S. Pontryagin et al. (1962). The Mathematical Theory of Optimal Processes. Wiley–Interscience.