# American Institute of Mathematical Sciences

September 2017, 9(3): 257-290. doi: 10.3934/jgm.2017011

## About simple variational splines from the Hamiltonian viewpoint

 1 Departamento de Matemática Aplicada, Universidade Federal Fluminense, Rua Mário Santos Braga, S/N, Campus do Valonguinho, 24020-140, Niterói, RJ, Brazil 2 Departamento de Física Matemática, Universidade Federal do Rio de Janeiro, Centro de Tecnologia -Bloco A -Cidade Universitária -Ilha do Fundão, 21941-972 Rio de Janeiro -RJ -Brazil 3 Departamento de Matemática Aplicada, Universidade Federal do Rio de Janeiro, Centro de Tecnologia -Bloco C -Cidade Universitária -Ilha do Fundão, 21941-909 Rio de Janeiro -RJ -Brazil 4 Instituto Nacional de Metrologia, Qualidade e Tecnologia, Divisão de Metrologia em Dinâmica de Fluidos, 25250-020, Xerém, Duque de Caxias -RJ -Brazil

Received  January 2015 Revised  September 2016 Published  June 2017

In this paper, we study simple splines on a Riemannian manifold $Q$ from the point of view of the Pontryagin maximum principle (PMP) in optimal control theory. The control problem consists in finding smooth curves matching two given tangent vectors with the control being the curve's acceleration, while minimizing a given cost functional. We focus on cubic splines (quadratic cost function) and on time-minimal splines (constant cost function) under bounded acceleration. We present a general strategy to solve for the optimal hamiltonian within the PMP framework based on splitting the variables by means of a linear connection. We write down the corresponding hamiltonian equations in intrinsic form and study the corresponding hamiltonian dynamics in the case $Q$ is the $2$-sphere. We also elaborate on possible applications, including landmark cometrics in computational anatomy.

Citation: Paula Balseiro, Teresinha J. Stuchi, Alejandro Cabrera, Jair Koiller. About simple variational splines from the Hamiltonian viewpoint. Journal of Geometric Mechanics, 2017, 9 (3) : 257-290. doi: 10.3934/jgm.2017011
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Energy $h = 0.01$. Regular trajectories.
Energy $h = 0.332412099$. There is a large chaotic zone, with escaping trajectories.
Energy $h = 0.806$. Even larger chaotic/escaping zone. The triangular feature is probably related to a 3:1 torus resonance. Note that we zoomed in with respect toFig. 2.
Nearby energies $h = 0.8065, \,0.818, \, 8189$. Only a small $a$ interval was depicted for better visualization. Which bifurcations took place: pitchfork, period doubling, Hamiltonian Hopf?
The periodic trajectory in the energy level $h = 0.808$. Note the central zone shrinking in the associated surface of section
Invariant tori, seen on a Lagrangian projection in the plane $(a,z)$. Energies $h=0.49494873, \,$ and $h = 0.522397316$.
An invariant torus, seen on a Lagrangian projection in the plane $(a,z)$. $h = 0.586204019$, $\beta=1, \mu= r= 2$.
Top: reconstructed trajectory in the physical sphere, that approaches a neighborhood of an equator. Below: the reduced trajectory emanating from the unstable equilibrium, projected in the $(v,a)$ plane. Note that $v$ is growing quadratically with respect to $a$. The reconstructed trajectory is approaching a neighborhood of an equator. It remains to be seen if it stays there or returns to a vicinity of the reduced equilibrium.
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