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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 |
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.
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