---
title: Quasi-Minimal Connecting Orbits
url: https://www.emergentmind.com/topics/quasi-minimal-connecting-orbits
type: topic
---

# Quasi-Minimal Connecting Orbits

Searching arXiv for the cited papers and closely related work on quasi-minimal/connecting orbits.
“Quasi-minimal connecting orbits” denotes a family of constructions in which an orbit, path, or one-parameter flow connects prescribed geometric or dynamical objects while satisfying a minimality condition that depends on the ambient theory. In quasiaffine spherical geometry, the term is tied to moving from a $G$-orbit in the regular locus to a minimal covering orbit by means of $B$-normalized additive one-parameter subgroups [2512.09906]. In celestial mechanics, it refers to an action minimizer connecting prescribed boundary configurations in the planar three-body problem [1607.00580]. In contact Hamiltonian dynamics, the relevant objects are weak KAM calibrated curves connecting Legendrian graphs [2208.13199]. In high-dimensional computation of heteroclinic connections, “quasi-minimal” denotes a global minimizer of a residual cost functional up to numerical tolerance [2301.11704]. The expression therefore identifies a recurring connection principle rather than a single invariant definition.

## 1. Terminological scope and structural motifs

Across the literature, the connected objects vary substantially. In the algebraic-geometric setting, an additive one-parameter subgroup is the image of a nontrivial homomorphism
\[
\varphi\colon\mathbb G_a\longrightarrow\Aut(X),
\]
and a $B$-root subgroup is such a subgroup normalized by $B$ [2512.09906]. In contact dynamics, a semi-infinite connecting orbit is a curve $\sigma:[0,+\infty)\to\Sigma$ with $\sigma(0)\in\Lambda_{u_0}$ and $\lim_{t\to+\infty}\dist(\sigma(t),\Lambda_{u_-})=0$ [2208.13199]. In the Jacobian-free variational method, one works with a connecting curve $\gamma(s)$ and residual
\[
r(\gamma)\coloneqq f(\gamma(s))-\partial_s\gamma(s),
\]
with cost
\[
J^2[\gamma]=\int_{-\infty}^{+\infty}\|r(\gamma(s))\|^2\,ds
\]
[2301.11704]. In prescribed-energy quasilinear systems, the admissible class imposes $V(q(t))\ge c$ together with asymptotic approach to two disjoint components $\mathcal V_c^-$ and $\mathcal V_c^+$ of a sublevel set [2606.22031].

| Setting | Connected objects | Minimality notion |
|---|---|---|
| Quasiaffine spherical varieties | A $G$-orbit and a minimal covering $G$-orbit | Orbit adjacency via a $B$-root subgroup |
| Planar three-body problem | Collinear/binary-collision boundary and isosceles boundary | Action minimizer |
| Contact Hamiltonian systems | Legendrian graph $\Lambda_{u_0}$ and equilibrium graph $\Lambda_{u_-}$ | Calibrated quasi-minimizer |
| Jacobian-free computation | Two equilibria $u_-,u_+$ | Global minimum of residual cost |
| Quasilinear conservative systems | $\mathcal V_c^-$ and $\mathcal V_c^+$ | Minimal $c$-connection |
| Annular wells and maxima of $V$ | Boundary components or a maximum and an energy level | Minimal multiplicity two |

This suggests that the term is not standardized across subfields. A common misconception is to read “quasi-minimal” uniformly as approximate numerical minimization. That interpretation is accurate for finite-resolution residual descent [2301.11704], but not for exact action minimizers in celestial mechanics [1607.00580], calibrated curves in contact weak KAM theory [2208.13199], or orbit-connecting root-subgroup actions in spherical geometry [2512.09906].

## 2. Quasiaffine spherical varieties and $B$-root connectivity

Let $G$ be a connected reductive algebraic group over an algebraically closed field of characteristic zero, with fixed Borel subgroup $B\subset G$, unipotent radical $U\subset B$, and maximal torus $T\subset B$. An irreducible variety $X$ is a quasiaffine spherical $G$-variety if $X$ carries a regular action of $G$, is quasiaffine, and $B$ has a dense open orbit in $X$. Its regular locus is
\[
X_{\reg}=\{x\in X\mid X\text{ is smooth at }x\}.
\]
A $G$-orbit $Y$ is minimal among those containing a given $Y_0$ in its closure when there is no strictly smaller $G$-orbit strictly between $Y_0$ and $Y$ in the closure ordering [2512.09906].

The main theorem states that if $X$ is a quasiaffine spherical $G$-variety and $Y\subset X_{\reg}$ is any $G$-orbit, then for any minimal $G$-orbit $Y_{\min}$ with
\[
Y\subset\overline{Y_{\min}},
\]
there exists a $B$-root subgroup
\[
\varphi\colon\mathbb G_a\longrightarrow\Aut(X)
\]
whose action carries at least one point of $Y$ into the orbit $Y_{\min}$. Consequently, if $\widetilde G\subset\Aut(X)$ is generated by $G$ and all $B$-root subgroups, then
\[
X_{\reg}=\widetilde G\cdot x\qquad\text{for any }x\in X_{\reg},
\]
so the regular locus is a single $\widetilde G$-orbit [2512.09906].

The proof follows the outline of Shafarevich’s work on affine spherical varieties together with refinements from Knop’s local structure theorem. One embeds $X$ as an open subvariety of the affine $G$-variety $\operatorname{Spec}k[X]$, chooses a $B$-semiinvariant function $f\in k[X]$, and studies the principal open set $X_f=X\setminus\{f=0\}$. Knop’s theorem yields an $L$-stable closed subvariety $Z\subset X_f$ such that
\[
X_f\simeq P\underset{L}{\times}Z\cong P_u\times Z,
\]
where $P\supset B$ is the stabilizer of the line $k\cdot f$, $L\subset P$ is a Levi subgroup, and $P_u$ is its unipotent radical. Under mild hypotheses, including $Y\subset X_{\reg}$, one obtains
\[
Y\cap X_f\simeq P_u\times A,\qquad Z\cap Y\simeq A,
\]
and near $Z\cap Y$ the variety becomes a homogeneous vector bundle
\[
\mathcal N=P\underset{L_0}{\times}N_z\longrightarrow P/L_0\simeq P_u\times A.
\]
Because $P/L_0$ is affine, the space $H^0(P/L_0,\mathcal N)$ contains a nonzero $B$-semiinvariant section $s$, and fiberwise translation
\[
(t;v_y)\longmapsto v_y+t\,s(y)
\]
defines the required regular $\mathbb G_a$-action. The action is then extended from $X_f$ to all of $X$ by multiplying the corresponding locally nilpotent derivation by a sufficiently high power of $f$ [2512.09906].

In the affine toric case $G=B=T$ and
\[
X\simeq\operatorname{Spec}\Bbbk[\sigma^\vee\cap M].
\]
The $T$-root subgroups are in bijection with the lattice edges of $\sigma$, and their action on characters is
\[
\varphi_\alpha(t)\colon \chi^m\longmapsto \chi^m(1+t\,\chi^\alpha)^{\langle m,\alpha^\vee\rangle}.
\]
These connect every torus orbit to the next smaller orbit in the closure ordering. Avdeev–Zhgoon’s theorem generalizes that picture from toric varieties to spherical $G$-varieties by replacing weight cones with colored fans and $T$-roots with $B$-root subgroups [2512.09906].

## 3. Action-minimizing connections in celestial mechanics

In the planar three-body problem with equal masses, the configuration space is
\[
\chi=\Bigl\{q=(q_1,\dots,q_N)\in(\mathbb R^2)^N:\sum_{i=1}^N m_iq_i=0\Bigr\},
\]
with $N=3$ and $m_i=1$. The Lagrangian action on $[0,1]$ is
\[
A[q]=\int_0^1L(q,\dot q)\,dt,\qquad L=K+U,
\]
where
\[
K(\dot q)=\frac12\sum_{i=1}^Nm_i|\dot q_i|^2,\qquad
U(q)=\sum_{1\le i<j\le N}\frac{m_im_j}{|q_i-q_j|}.
\]
The boundary constraints impose a two-point free-boundary problem
\[
q(0)=Q_s(a_1,a_2),\qquad q(1)=Q_e(b_1,b_2),
\]
where $q(0)$ is a collinear configuration on the $x$-axis and $q(1)$ is an isosceles triangle symmetric about the $y$-axis. By Marchal–Chenciner, interior collisions are ruled out; only boundary collisions remain [1607.00580].

The resulting action minimizer
\[
P=\arg\min_{(a,b)\in S}\min_{q(0)=Q_s,\ q(1)=Q_e}A[q]
\]
is called a “quasi-minimal connecting orbit” between the collinear/binary-collision type boundary $Q_s$ and the isosceles boundary $Q_e$. The main theorem identifies $P$ as either a segment of the Schubart orbit, with a $1$–$2$ binary collision at $t=0$ and Euler collinear at $t=1$, or a segment of the Broucke–Hénon orbit, which is collision-free [1607.00580].

The identification uses a layered exclusion argument. Total collisions are ruled out by combining Gordon’s lower bound, which gives action at least $6.69$ for paths with total collision, with a piecewise-defined test path $\hat q$ satisfying $A[\hat q]<3.54$. A one-end free Kepler argument and blow-up analysis exclude binary collision at $t=1$. At $t=0$, only a $1$–$2$ collision can occur. Introducing Jacobi coordinates
\[
Z_1=q_1-q_2,\qquad Z_2=q_3-\frac{q_1+q_2}{2},
\]
the reflection argument compares $(Z_1,Z_2)$ with
\[
\widetilde Z_1=(|Z_{1x}|,|Z_{1y}|),\qquad \widetilde Z_2=(|Z_{2x}|,-|Z_{2y}|),
\]
and uses the fact that the potential decreases strictly in the acute angle between $Z_1$ and $Z_2$ for fixed $|Z_1|,|Z_2|$. If a $1$–$2$ collision occurs at $t=0$ and the endpoint at $t=1$ is Euler collinear, the motion is forced onto the $x$-axis, yielding the $1/4$-Schubart segment. If no collision occurs at $t=0$, first-variation boundary conditions permit a $C^1$ reflection-and-permutation extension to a period-$4$ orbit with $D_2$-symmetry, namely the Broucke–Hénon orbit. Numerical action values
\[
A_{\mathrm{Schubart}}\approx 3.43,\qquad A_{\mathrm{H\acute enon}}\approx 3.46
\]
indicate that the global minimizer is the Schubart orbit [1607.00580].

A related variational program in the planar equal-mass four-body problem studies action minimizers under order constraints connecting a “double isosceles” initial configuration to one of two “isosceles trapezoid” terminal configurations [1710.09959]. The action is
\[
\mathcal A[q]=\int_0^1\Bigl[\frac12\sum|\dot q_i|^2+\sum_{i<j}\frac1{|q_i-q_j|}\Bigr]dt.
\]
Using the level estimate method, the minimizers are shown to be collision-free for
\[
\theta\in(0,0.0539\pi]\quad\text{and}\quad \theta\in(0,0.0664\pi]
\]
in the two families, respectively, and can be extended to periodic or quasi-periodic solutions. The extension uses first-variation identities and reflection-rotation formulas; if $\theta/\pi$ is rational the orbit is periodic, and if irrational it is quasi-periodic [1710.09959]. This provides a closely related class of minimal connecting trajectories governed by free-boundary variational principles.

## 4. Weak KAM and contact Hamiltonian connecting orbits

For contact Hamiltonian systems, the phase space is $\Sigma=J^1M\simeq T^*M\times\mathbb R$ with coordinates $(q,p,u)$ and standard contact form
\[
\alpha=du-p\cdot dq.
\]
Given $H\in C^\infty(\Sigma)$, the contact Hamiltonian vector field $X_H$ is characterized by the coordinate equations
\[
\dot q=\frac{\partial H}{\partial p},\qquad
\dot p=-\frac{\partial H}{\partial q}-p\cdot\frac{\partial H}{\partial u},\qquad
\dot u=p\cdot\frac{\partial H}{\partial p}-H.
\]
A Legendrian graph is the $1$-jet of a $C^1$ function $u:M\to\mathbb R$,
\[
\Lambda_u=\{(q,d_qu(q),u(q)):q\in M\},
\]
and if $H|_{\Lambda_{u_-}}\equiv 0$, then $\Lambda_{u_-}$ is an equilibrium graph [2208.13199].

The variational framework is built from the Legendre transform
\[
L(q,\dot q,u)=\sup_p\{p\cdot\dot q-H(q,p,u)\},
\]
under convexity and superlinearity in $p$. The implicit backward and forward action functions are
\[
h_{q_0,u_0}(q,t)=\inf_{\gamma(0)=q_0,\ \gamma(t)=q}\Bigl\{u_0+\int_0^tL(\gamma(s),\dot\gamma(s),h_{q_0,u_0}(\gamma(s),s))\,ds\Bigr\},
\]
\[
h^{q_0,u_0}(q,t)=\sup_{\gamma(0)=q,\ \gamma(t)=q_0}\Bigl\{u_0-\int_0^tL(\gamma(s),\dot\gamma(s),h^{q_0,u_0}(\gamma(s),t-s))\,ds\Bigr\}.
\]
These induce the semigroups
\[
T_t^-v(q)=\inf_{q_0}h_{q_0,v(q_0)}(q,t),\qquad
T_t^+v(q)=\sup_{q_0}h^{q_0,v(q_0)}(q,t).
\]
Any fixed point $u_-$ of $T_t^-$ is a viscosity solution of
\[
H(q,d_qu_-,u_-)=0
\]
[2208.13199].

If
\[
U(q,t)=T_t^-v(q)
\]
solves
\[
\partial_tU+H(q,\partial_qU,U)=0,\qquad U(\cdot,0)=v,
\]
then any minimizer $\gamma$ for $h_{q_0,v(q_0)}(q,t)$ lifts to a characteristic satisfying
\[
u(s)=U(q(s),s),\qquad p(s)=\partial_{\dot\gamma}L(q(s),\dot\gamma(s),u(s)),
\]
and $\gamma$ is called $U$-calibrated. These calibrated curves are described as the “quasi-minimizers” sought by Aubry–Mather theory [2208.13199].

Two prototype existence results organize the theory. Under uniform convergence
\[
\lim_{t\to\infty}T_t^-u_0=u_-
\]
on $M$, every point of $\Lambda_{u_-}$ is $C^0$-approximated by some finite-time $\phi_H^t$-image of $\Lambda_{u_0}$, and there exists $\sigma_0\in\Lambda_{u_0}$ whose $\omega$-limit lies in $\Lambda_{u_-}$. Under the homotopy criterion, one assumes a $C^1$ homotopy $V(q,s)$ from a strict super- or subsolution to $u_-$; this again forces the same semigroup limit and yields a semi-infinite connecting orbit [2208.13199].

The resulting connecting orbit is $C^1$ and real-analytic whenever $H$ is so. On every finite subinterval it minimizes the contact action among curves with the same endpoints, since it is calibrated by the limiting weak KAM solution. Uniqueness generally fails; one often obtains a family of calibrated rays filling the projected Aubry set. Stability under small $C^2$ perturbations of the initial graph persists when the homotopy conditions are preserved [2208.13199].

## 5. Jacobian-free variational computation of heteroclinic connections

For an autonomous ODE or PDE
\[
\partial_tu=f(u),\qquad f:M\to\mathbb R^n,
\]
with equilibria $u_-$ and $u_+$ satisfying $f(u_\pm)=0$, a true heteroclinic orbit obeys
\[
\lim_{t\to-\infty}u(t)=u_-,\qquad \lim_{t\to+\infty}u(t)=u_+.
\]
Ashtari and Schneider replace shooting by a search over smooth connecting curves $\gamma(s)$, where $s$ is unrelated to the physical time $t$, with
\[
\lim_{s\to\pm\infty}\gamma(s)=u_\pm.
\]
A connecting orbit is one for which
\[
\partial_s\gamma=f(\gamma).
\]
A “quasi-minimal” connecting orbit is any $\gamma_\varepsilon$ for which the cost functional $J[\gamma]$ has been driven to its global minimum, zero in theory, up to numerical tolerance [2301.11704].

The pointwise residual is
\[
r(\gamma)\coloneqq f(\gamma(s))-\partial_s\gamma(s),
\]
and the cost is the $L^2$ norm
\[
J^2[\gamma]=\int_{-\infty}^{+\infty}\|r(\gamma(s))\|^2\,ds.
\]
For spatio-temporal fields,
\[
J^2[u]=\int_{-\infty}^{+\infty}\int_\Omega [f(u)-\partial_su]\cdot[f(u)-\partial_su]\,dx\,ds.
\]
A true connecting orbit has $r\equiv0$ and hence $J=0$ [2301.11704].

Stationarity of $J^2$ leads to adjoint Euler–Lagrange equations. Writing
\[
\delta J^2=2\langle L(u;\delta u),r\rangle=2\langle \delta u,L^\dagger(u;r)\rangle,
\]
the condition is
\[
L^\dagger(u;r)=0
\]
with asymptotic boundary conditions
\[
u(s\to-\infty)=u_-,\qquad u(s\to+\infty)=u_+.
\]
For the Kuramoto–Sivashinsky equation, the residual
\[
r=-\partial_su-u\,\partial_xu-\partial_x^2u-\partial_x^4u
\]
has adjoint operator
\[
L^\dagger(u;r)=\partial_sr+u\,\partial_xr-\partial_x^2r-\partial_x^4r.
\]
Thus the stationary equation becomes
\[
\partial_s[f(u)-\partial_su]+u\,\partial_x[f(u)-\partial_su]-\partial_x^2[f(u)-\partial_su]-\partial_x^4[f(u)-\partial_su]=0
\]
[2301.11704].

The numerical method uses rational Chebyshev collocation in the unbounded $s$-direction,
\[
s_j=s_0+S\cot\!\Bigl(\frac{j\pi}{M+1}\Bigr),\qquad j=1,\dots,M,
\]
with implicit endpoints at $j=0$ and $j=M+1$, and Fourier discretization in space. At iteration $u^k$, one computes
\[
r^k=f(u^k)-D_su^k,\qquad G^k=-L^\dagger(u^k;r^k),
\]
and advances in fictitious time by
\[
u^{k+1}=u^k+\Delta\tau\,G^k.
\]
Convergence is monitored with the arc-length weighted cost
\[
J_{\mathrm{arc}}=\frac{\int |r|\,|\partial_su|\,ds}{\int |\partial_su|\,ds},
\]
and the computation is terminated once
\[
J_{\mathrm{arc}}<10^{-12}.
\]
If $J_{\mathrm{arc}}$ stalls above tolerance as $M,N$ are increased, the descent has settled in a local non-connecting minimum rather than the global one [2301.11704].

The method is Jacobian-free because it avoids assembling a Jacobian matrix of size $O((MN)^2)$; only $O(MN)$ memory is required for $u$, $r$, and $G$. Its robustness stems from deforming an entire curve rather than time-marching a chaotic trajectory, so there is no restriction on the dimension of the unstable manifold at the origin equilibrium and no exponential error amplification associated with direct shooting [2301.11704].

For the Kuramoto–Sivashinsky equation at $L=22$, the equilibria are $E_0=0$ and three nontrivial fixed points $E_1,E_2,E_3$, with unstable manifolds of dimensions $6$, $4$, $2$, and $2$. The reported discretization uses $N=64$ Fourier modes, $M$ between $80$ and $600$ rational-Chebyshev points, $S\approx35$–$60$, and $\Delta\tau=0.01$. The method recovers at least one
\[
E_0\to E_1,\ E_0\to E_2,\ E_0\to E_3,\ E_1\to E_2,\ E_1\to E_3,\ E_2\to E_3,\ E_2\to \gamma(1/4)E_2
\]
orbit, together with two distinct $E_3\to E_2$ orbits, all to machine precision with
\[
J_{\mathrm{arc}}<10^{-12}
\]
[2301.11704].

## 6. Prescribed-energy, homoclinic, and brake-type minimal connections

A broad variational framework for connecting orbits in quasilinear conservative systems studies
\[
(\phi(|\dot q|)\dot q)'=\nabla V(q),\qquad q(t)\in\mathbb R^N,
\]
with kinetic $N$-function
\[
\Phi(s)=\int_0^{|s|}\phi(r)\,r\,dr
\]
and energy
\[
E_q(t)=\phi(|\dot q|)|\dot q|^2-\Phi(|\dot q|)-V(q)=G(|\dot q|)-V(q).
\]
For fixed $c\in\mathbb R$, one assumes that
\[
\mathcal V_c=\{x\in\mathbb R^N:V(x)\le c\}
\]
splits into two nonempty, disjoint, closed, well-separated components
\[
\mathcal V_c^- ,\,\mathcal V_c^+,\qquad \dist(\mathcal V_c^-,\mathcal V_c^+)>0.
\]
The action functional is
\[
J_c(q)=\int_{-\infty}^{+\infty}\bigl[\Phi(|\dot q(t)|)+V(q(t))-c\bigr]dt,
\]
minimized over
\[
\Gamma_c=\Bigl\{q\in W^{1,\Phi}_{\loc}(\mathbb R;\mathbb R^N)\ \Big|\ V(q(t))\ge c\ \forall t,\ \liminf_{t\to-\infty}\dist(q(t),\mathcal V_c^-)=\liminf_{t\to+\infty}\dist(q(t),\mathcal V_c^+)=0\Bigr\}.
\]
Under the structural hypotheses on $\phi$, the splitting condition $(\mathcal V_c)$, and the coercivity assumption $(\mathcal J_c)$, the direct method yields a minimizer $q_0\in\Gamma_c$ with $J_c(q_0)=m_c>0$ [2606.22031].

Defining contact times
\[
\alpha=\sup\{t:q_0(t)\in\mathcal V_c^-\},\qquad
\omega=\inf\{t>\alpha:q_0(t)\in\mathcal V_c^+\},
\]
one restricts to $I=(\alpha,\omega)$, where $V(q_0)>c$. On $I$, $q_0$ satisfies the weak Euler–Lagrange equation
\[
\int_I\Bigl[\phi(|\dot q_0|)\dot q_0\cdot\dot\psi+\nabla V(q_0)\cdot\psi\Bigr]dt=0,
\]
and a time-rescaling argument yields
\[
\int_r^h(V(q_0)-c)\,dt=\int_r^hG(|\dot q_0|)\,dt\qquad \text{for all }r<h\in I.
\]
Hence
\[
E_{q_0}(t)=G(|\dot q_0|)-V(q_0)=-c
\]
almost everywhere on $I$. Regularity arguments then show that $q_0\in C^2(I;\mathbb R^N)$ and is a classical minimal $c$-connection [2606.22031].

The global behavior is classified by the finiteness of $\alpha$ and $\omega$. If both are infinite, one obtains an entire heteroclinic orbit. If exactly one is finite, reflection produces a homoclinic orbit. If both are finite, reflection at both endpoints followed by periodic extension yields a brake-type periodic solution of period
\[
T=2(\omega-\alpha).
\]
If either end-time is infinite, then $c$ must be a critical value of $V$; for regular values of $c$, only brake-type orbits arise [2606.22031].

Representative applications include double-well potentials, Duffing-type systems, and multiple pendulum potentials. In the multiple pendulum case, choosing different partitions of $\mathbb Z^N$ into two well-separated subgroups produces infinitely many distinct minimal connections [2606.22031].

A different use of the quasi-minimal label appears in geometric examples of Giambò–Giannoni–Piccione. They construct autonomous Lagrangian systems in $\mathbb R^N$ with exactly two homoclinics from a unique nondegenerate maximum of the potential energy, and Hamiltonian systems in an annular potential region with exactly two brake orbits connecting the two boundary components of the potential well [1406.0642]. The Lagrangian is
\[
L(q,\dot q)=\frac12|\dot q|^2-V_\varepsilon(q),
\]
with equations
\[
\ddot q+\nabla V_\varepsilon(q)=0,\qquad \frac12|\dot q|^2+V_\varepsilon(q)\equiv E.
\]
The construction uses an interior potential, an exterior potential obtained by an inverse Maupertuis procedure, and a $C^2$ matching across the unit sphere. In the annular case, brake orbits are characterized by
\[
\dot q=0\qquad\text{whenever }\widetilde V_\varepsilon(q)=E.
\]
The corresponding Jacobi formulation employs
\[
g_J=(E-V(x))\,g_0/2.
\]
These examples show that the general lower bound of “at least two” connecting orbits is sharp: one cannot force three or more in full generality without extra symmetry or topology [1406.0642].

This suggests a final conceptual distinction. In [2606.22031], minimality is variational and attached to a constrained action at prescribed energy; in [1406.0642], “quasi-minimal” refers to minimal multiplicity of connecting orbits; in [2512.09906], it refers to adjacent-orbit connectivity in a closure poset; and in [2301.11704], it refers to numerical realization of a global minimum of a residual functional. The shared content is the existence of a connecting mechanism constrained by a minimality principle, but the nature of both “connection” and “minimality” is theory-dependent.

Source: https://www.emergentmind.com/topics/quasi-minimal-connecting-orbits