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
References:
[1]

L. Abrunheiro, M. Camarinha and J. Clemente-Gallardo, Cubic polynomials on Lie groups: Reduction of the Hamiltonian system, Journal of Physics A: Mathematical and Theoretical, 44 (2011), 355203, 16pp, URL http://stacks.iop.org/1751-8121/44/i=35/a=355203. doi: 10.1088/1751-8113/44/35/355203.  Google Scholar

[2]

L. Abrunheiro, M. Camarinha and J. Clemente-Gallardo, Corrigendum: Cubic polynomials on Lie groups: Reduction of the Hamiltonian system, Journal of Physics A: Mathematical and Theoretical, 46 (2013), 189501, 2pp, URL http://stacks.iop.org/1751-8121/46/i=18/a=189501. doi: 10.1088/1751-8113/46/18/189501.  Google Scholar

[3]

L. Abrunheiro and M. Camarinha, Optimal control of affine connection control systems from the point of view of Lie algebroids, International Journal of Geometric Methods in Modern Physics, 11 (2014), 1450038, 8pp, URL http://www.worldscientific.com/doi/abs/10.1142/S0219887814500388. doi: 10.1142/S0219887814500388.  Google Scholar

[4]

L. Abrunheiro, M. Camarinha and J. Clemente-Gallardo, Geometric Hamiltonian formulation of a variational problem depending on the covariant acceleration, Conference Papers in Mathematics, 2013 (2013), Article ID 243621, 9 pages, URL http://www.hindawi.com/archive/2013/243621/. doi: 10.1155/2013/243621.  Google Scholar

[5]

. Attri, Development of Models for the Equations of Motion in the Solar System: Implementations and Applications, Master's thesis, Universitat Politécnica de Catalunya Master in Aerospace Science and Technology, 2014, URL http://upcommons.upc.edu/handle/2099.1/22415. Google Scholar

[6]

B. B. KhesinJ. LenellsG. Misiolek and S. C. Preston, Curvatures of Sobolev metrics on diffeomorphism groups, Pure and Applied Mathematics Quarterly, 9 (2013), 291-332.  doi: 10.4310/PAMQ.2013.v9.n2.a2.  Google Scholar

[7]

P. Balseiro, A. Cabrera and J. Koiller, Optimal control on affine bundles, in preparation. Google Scholar

[8]

M. Barbero-Liñán and A. D. Lewis, Geometric interpretations of the symmetric product in affine differential geometry and applications, International Journal of Geometric Methods in Modern Physics, 9 (2012), 1250073, 33pp, URl http://www.worldscientific.com/doi/abs/10.1142/S0219887812500739. doi: 10.1142/S0219887812500739.  Google Scholar

[9]

M. Barbero-Liñán, Characterization of accessibility for affine connection control systems at some points with nonzero velocity, in Proceedings of the IEEE Conference on Decision and Control and European Control Conference, (2011), 6528-6533.   Google Scholar

[10]

M. Barbero-Liñán and M. Sigalotti High-order sufficient conditions for configuration tracking of affine connection control systems, Systems & Control Letters, 59 (2010), 491-503, URl http://www.sciencedirect.com/science/article/pii/S0167691110000757 (http://arxiv.org/abs/1501.04026). doi: 10.1016/j.sysconle.2010.06.011.  Google Scholar

[11]

M. BauerM. BruverisP. Harms and J. Moller-Andersen, A numerical framework for Sobolev metrics on the space of curves, SIAM J. Imaging Sci. (arxiv:1603.03480), SIAM J. Imaging Sci., 10 (2017), 47-73.  doi: 10.1137/16M1066282.  Google Scholar

[12]

M. Bauer, M. Bruveris and P. W. Michor, Why use Sobolev metrics on the space of curves, in Riemannian Computing in Computer Vision, Turaga, P. and Srivastava, A., editors, Springer-Verlag, 11 (2016), 223-255, URL https://arxiv.org/abs/1502.03229, URL https://link.springer.com/book/10.1007/978-3-319-22957-7.  Google Scholar

[13]

M. Bauer, M. Bruveris, P. Harms and J. Møller-Andersen, Curve matching with applications in medical imaging, in MFCA 2015: 5th MICCAI Workshop on Mathematical Foundations of Computational Anatomy, (eds. X. Pennec, S. Joshi, M. Nielsen, T. Fletcher, S. Durrleman and S. Sommer), 2015, 83-94. URL https://arxiv.org/abs/1506.08840, URL http://www-sop.inria.fr/asclepios/events/MFCA15/MFCA15_Proceedings.pdf. Google Scholar

[14]

M. Bauer, M. Bruveris and P. Michor, Overview of the geometries of shape spaces and diffeomorphism groups, Journal of Mathematical Imaging and Vision, 50 (2014), 60-97, URL http://dx.doi.org/10.1007/s10851-013-0490-z. doi: 10.1007/s10851-013-0490-z.  Google Scholar

[15]

M. Bauer, M. Eslitzbichler and M. Grasmair, Landmark-guided elastic shape analysis of human character motions, arxiv: 1502. 07666. Google Scholar

[16]

M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on shape space of surfaces, Journal of Geometric Mechanics, 3 (2011), 389-438, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=7061.  Google Scholar

[17]

M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on shape space, Ⅱ: Weighted sobolev metrics and almost local metrics, Journal of Geometric Mechanics, 4 (2012), 365-383, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=8178.  Google Scholar

[18]

C. Bingham, Review: Geoffrey S. Watson, Statistics on spheres Ann. Statist., 13 (1985), 838-844, RUL http://dx.doi.org/10.1214/aos/1176349566. doi: 10.1214/aos/1176349566.  Google Scholar

[19]

A. BlochD. E. ChangN. Leonard and J. Marsden, Controlled Lagrangians and the stabilization of mechanical systems. Ⅱ. potential shaping, Automatic Control, IEEE Transactions on, 46 (2001), 1556-1571.  doi: 10.1109/9.956051.  Google Scholar

[20]

A. BlochN. Leonard and J. Marsden, Controlled Lagrangians and the stabilization of mechanical systems. Ⅰ. the first matching theorem, Automatic Control, IEEE Transactions on, 45 (2000), 2253-2270.  doi: 10.1109/9.895562.  Google Scholar

[21] A. Bloch, Nonholonomic Mechanics and Control, vol. 24 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York, 2003.  doi: 10.1007/b97376.  Google Scholar
[22]

M. Bruveris, Geometry of Diffeomorphism Groups and Shape Matching, PhD thesis, Department of Mathematics, Imperial College, London, 2012, URL https://www.brunel.ac.uk/mastmmb/pdf/00_thesis.pdf.  Google Scholar

[23]

M. Bruveris and D. Holm, Geometry of image registration: the diffeomorphism group and momentum maps, in Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden, ed. by Chang, D. E., Holm, D. D., Patrick, G., Ratiu, T., vol. 73 of Fields Institute Communications Series, Springer-Verlag, 2015, 19-56. doi: 10.1007/978-1-4939-2441-7_2.  Google Scholar

[24]

F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems, Texts in Applied Mathematics 49, Springer-Verlag, New York, 2005, URL http://www.springer.com/br/book/9780387221953. doi: 10.1007/978-1-4899-7276-7.  Google Scholar

[25]

F. Bullo and A. D. Lewis, Supplementary chapters for Geometric Control of Mechanical Systems, URL http://motion.me.ucsb.edu/book-gcms/. Google Scholar

[26]

C. L. BurnettD.D. Holm and D.M. Meier, Inexact trajectory planning and inverse problems in the Hamilton-Pontryagin framework, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 469 (2014), 20130249, 24 pp.  doi: 10.1098/rspa.2013.0249.  Google Scholar

[27]

A. Castro and J. Koiller, On the dynamic Markov-Dubins problem: From path planning in robotics and biolocomotion to computational anatomy, Regular and Chaotic Dynamics, 18 (2013), 1-20, URL http://dx.doi.org/10.1134/S1560354713010012. doi: 10.1134/S1560354713010012.  Google Scholar

[28]

D. E. Chang, A simple proof of the Pontryagin maximum principle on manifolds, Automatica, 47 (2011), 630-633, URL http://www.sciencedirect.com/science/article/pii/S0005109811000525. doi: 10.1016/j.automatica.2011.01.037.  Google Scholar

[29]

A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Annals of Mathematics. Second Series, 152 (2000), 881-901, URL http://eudml.org/doc/121861. doi: 10.2307/2661357.  Google Scholar

[30]

A. ChertockP. D. Toit and J. E. Marsden, Integration of the EPDiff equation by particle methods, ESAIM: Mathematical Modelling and Numerical Analysis, 46 (2012), 515-534.  doi: 10.1051/m2an/2011054.  Google Scholar

[31]

J. Cortés, S. Martínez, J. P. Ostrowski and H. Zhang, Simple mechanical control systems with constraints and symmetry, SIAM Journal on Control and Optimization, 41 (2002), 851-874, URL http://dx.doi.org/10.1137/S0363012900381741. doi: 10.1137/S0363012900381741.  Google Scholar

[32]

P. Crouch, F. S. Leite and M. Camarinha, A second order Riemannian variational problem from a Hamiltonian perspective, preprint, Centro de Matemática da Universidade de Coimbra, http://hdl.handle.net/10316/11230,1998. Google Scholar

[33]

P. Crouch and F. Leite, The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces, Journal of Dynamical and Control Systems, 1 (1995), 177-202, URL http://dx.doi.org/10.1007/BF02254638. doi: 10.1007/BF02254638.  Google Scholar

[34]

P. Crouch and F. S. Leite, Geometry and the dynamic interpolation problem, in Proceedings of the 1991 American Control Conference, 7 (1991), 1131-1136, URL http://ieeexplore.ieee.org/document/4791552/. Google Scholar

[35]

M. de Léon, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, Journal of Physics A: Mathematical and General, 38 (2005), R241-R308, URL http://stacks.iop.org/0305-4470/38/i=24/a=R01. doi: 10.1088/0305-4470/38/24/R01.  Google Scholar

[36]

N. Desai, S. Ploskonka, L. R. Goodman, C. Austin, J. Goldberg and T. Falcone, Analysis of embryo morphokinetics, multinucleation and cleavage anomalies using continuous time-lapse monitoring in blastocyst transfer cycles, Reproductive Biology and Endocrinology : RB & E, 12 (2014), 54-54, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4074839/. doi: 10.1186/1477-7827-12-54.  Google Scholar

[37]

I. L. Dryden, Statistical analysis on high-dimensional spheres and shape spaces, The Annals of Statistics, 33 (2005), 1643-1665, URL http://www.jstor.org/stable/3448620. doi: 10.1214/009053605000000264.  Google Scholar

[38]

S. Durrleman, T. Fletcher, G. Gerig, M. Niethammer and X. Pennec, STIA 2014 -Spatio-temporal Image Analysis for Longitudinal and Time-Series Image Data, vol. 8682 of Lecture notes in computer science, Springer International Publishing, Cambridge, United States, 2015. URL http://www.springer.com/br/book/9783319149042. Google Scholar

[39]

D. Elworthy, Geometric aspects of diffusions on manifolds, in École d'Été de Probabilités de Saint-Flour ⅩⅤ-ⅩⅦ, 1985-87, Lecture Notes in Mathematics (ed. P. -L. Hennequin), vol. 1362, Springer Berlin Heidelberg, 1988,277-425, URl http://dx.doi.org/10.1007/BFb0086183. doi: 10.1007/BFb0086183.  Google Scholar

[40]

E. Fehlberg, Classical Seventh, Sixth, and Fifth-Order Runge Kutta-Nystrom Formula with Stepsize Control for General Second-Order Differential Equations, Technical report, NASA TR R-432, Washington D. C, 1974, URL https://ntrs.nasa.gov/search.jsp?R=19740026877. Google Scholar

[41]

J. -B. Fiot, H. Raguet, L. Risser, L. D. Cohen, J. Fripp and F. -X. Vialard, Longitudinal deformation models, spatial regularizations and learning strategies to quantify Alzheimer's disease progression, NeuroImage: Clinical, 4 (2014), 718-729, URl http://www.sciencedirect.com/science/article/pii/S2213158214000205. doi: 10.1016/j.nicl.2014.02.002.  Google Scholar

[42] N. FisherT. Lewis and B. Embleton, Statistical Analysis of Spherical Data, Cambridge University Press, 1987.  doi: 10.1017/CBO9780511623059.  Google Scholar
[43]

P. T. Fletcher, Statistical Variability in Nonlinear Spaces: Application to Shape Analysis and DT-MRI, PhD thesis, Department of Computer Science, University of North Carolina, 2004. Google Scholar

[44]

P. T. Fletcher, Geodesic regression and the theory of least squares on Riemannian Manifolds, International Journal of Computer Vision, 105 (2013), 171-185, URL http://dx.doi.org/10.1007/s11263-012-0591-y. doi: 10.1007/s11263-012-0591-y.  Google Scholar

[45]

D. Fortuné, J. A. R. Quintero and C. Vallée, Pontryagin calculus in Riemannian geometry, Geometric Science of Information, Lecture Notes in Computer Science (eds. F. Nielsen, F. Barbaresco and F. Dubois), vol. 9389, Springer International Publishing, 2015,541-549, URL http://dx.doi.org/10.1007/978-3-319-25040-3_58. doi: 10.1007/978-3-319-25040-3_58.  Google Scholar

[46]

B. Francis and M. Maggiore, Flocking and Rendezvous in Distributed Robotics, Springer International Publishing, 2016. doi: 10.1007/978-3-319-24729-8.  Google Scholar

[47]

F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F. -X. Vialard, Invariant higher-order variational problems, Communications in Mathematical Physics, 309 (2012), 413-458, URL http://dx.doi.org/10.1007/s00220-011-1313-y. doi: 10.1007/s00220-011-1313-y.  Google Scholar

[48]

F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F. -X. Vialard, Invariant higher-order variational problems Ⅱ, Journal of Nonlinear Science, 22 (2012), 553-597, URL http://dx.doi.org/10.1007/s00332-012-9137-2. doi: 10.1007/s00332-012-9137-2.  Google Scholar

[49]

P. Harms, Sobolev Metrics on Shape Space of Surfaces, PhD thesis, Universität Wien, 2010, URL https://arxiv.org/abs/1211.3515. Google Scholar

[50]

J. Hinkle, P. Fletcher and S. Joshi, Intrinsic polynomials for regression on riemannian manifolds, Journal of Mathematical Imaging and Vision, 50 (2014), 32-52, URL http://dx.doi.org/10.1007/s10851-013-0489-5. doi: 10.1007/s10851-013-0489-5.  Google Scholar

[51]

J. Hinkle, P. Muralidharan, P. Fletcher and S. Joshi, Polynomial regression on Riemannian manifolds, in Computer Vision -ECCV 2012 (eds. A. Fitzgibbon, S. Lazebnik, P. Perona, Y. Sato and C. Schmid), vol. 7574 of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2012, 1-14, URl http://dx.doi.org/10.1007/978-3-642-33712-3_1. doi: 10.1007/978-3-642-33712-3_1.  Google Scholar

[52]

D. D. Holm, T. Schmah and C. Stoica, Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions, Oxford Texts in Applied and Engineering Mathematics, Oxford University Press, 2009, URL https://global.oup.com/academic/product/geometric-mechanics-and-symmetry-9780199212910?cc=br&lang=en&.  Google Scholar

[53]

K. Hüper, Y. Shen and F. Silva Leite, Geometric splines and interpolation on $S^2$: Numerical experiments, in Proceedings of the 45th IEEE Conference on Decision & Control, San Diego, CA, USA, 2006,6403-6407, URL http://ieeexplore.ieee.org/document/4177774/. Google Scholar

[54]

R. V. Iyer, Pontryagin's minimum principle for simple mechanical systems on Riemannian manifolds and Lie groups, 2005, Http://www.math.ttu.edu/rvenkata/Papers/max-principle.pdf. Google Scholar

[55]

R. V. Iyer, R. Holsapple and D. Doman, Optimal control problems on parallelizable Riemannian manifolds: Theory and applications, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 1-11, URL https://www.esaim-cocv.org/articles/cocv/abs/2006/01/cocv0402/cocv0402.html. doi: 10.1051/cocv:2005026.  Google Scholar

[56]

M. Jóźwikowski, Optimal Control Theory on Almost Lie Algebroids, PhD thesis, Institute of Mathematics, Polish Academy of Sciences, 2011, URL https://arxiv.org/pdf/1111.1549.pdf. Google Scholar

[57]

V. Jurdjevic, The Delauney-Dubins problem, in Geometric Control Theory and Sub-Riemannian Geometry (eds. G. Stefani, U. Boscain, J. -P. Gauthier, A. Sarychev and M. Sigalotti), Springer International Publishing, 5 (2014), 219-239, URl http://dx.doi.org/10.1007/978-3-319-02132-4_14. doi: 10.1007/978-3-319-02132-4_14.  Google Scholar

[58] V. Jurdjevic, Optimal Control and Geometry: Integrable Systems, Cambridge University Press, Cambridge Studies in Advanced Mathematics, 2016.  doi: 10.1017/CBO9781316286852.  Google Scholar
[59]

C. Y. Kaya and J. L. Noakes, Finding interpolating curves minimizing $L^\infty$ acceleration in the euclidean space via optimal control theory, SIAM Journal on Control and Optimization, 51 (2013), 442-464, URl http://dx.doi.org/10.1137/12087880X. doi: 10.1137/12087880X.  Google Scholar

[60]

B. Khesin, J. Lenells, G. Misiołek and S. Preston, Geometry of diffeomorphism groups, complete integrability and geometric statistics, Geometric and Functional Analysis, 23 (2013), 334-366, URL http://dx.doi.org/10.1007/s00039-013-0210-2. doi: 10.1007/s00039-013-0210-2.  Google Scholar

[61]

B. Khesin and R. Wendt, The Geometry of Infinite-Dimensional Groups, vol. 51 of Modern Surveys in Mathematics, 1st edition, Springer-Verlag Berlin Heidelberg, 2009.  Google Scholar

[62]

M. Kummer, On the construction of the reduced phase space of a hamiltonian system with symmetry, Indiana Univ. Math. J., 30 (1981), 281-291.  doi: 10.1512/iumj.1981.30.30022.  Google Scholar

[63]

M. Kummer, Realizations of the reduced phase space of a hamiltonian system with symmetry, in Local and Global Methods of Nonlinear Dynamics, Lecture Notes in Physics (eds. A. W. Sáenz, W. W. Zachary and R. Cawley), vol. 252, Springer Berlin Heidelberg, 1986, 32-39, URL http://dx.doi.org/10.1007/BFb0018326. doi: 10.1007/BFb0018326.  Google Scholar

[64]

J. Langer and D. A. Singer, The total squared curvature of closed curves, J. Differential Geom., 20 (1984), 1-22, URl http://projecteuclid.org/euclid.jdg/1214438990. doi: 10.4310/jdg/1214438990.  Google Scholar

[65]

A. D. Lewis, The Affine Structure of Jet Bundles, Technical report, 2005, URL www.mast.queensu.ca/~ andrew/notes/pdf/2005a.pdf. Google Scholar

[66]

A. D. Lewis, Aspects of Geometric Mechanics and Control of Mechanical Systems, PhD thesis, Caltech, 1995, URL http://www.mast.queensu.ca/andrew/papers/pdf/1995f.pdf. Google Scholar

[67]

Configuration controllability of simple mechanical control systems, SIAM Review, 41 (1999), 555-574, URL http://dx.doi.org/10.1137/S0036144599351065. doi: 10.1137/S0036144599351065.  Google Scholar

[68]

M. Lewis, D. Offin, P. -L. Buono and M. Kovacic, Instability of the periodic hip-hop orbit in the $2n$-body problem with equal masses, Discrete and Continuous Dynamical Systems, 33 (2013), 1137-1155, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=7827.  Google Scholar

[69]

P. Libermann, Lie algebroids in mechanics, Archivum Mathematicum, 32 (1966), 147-162.   Google Scholar

[70]

Z. LinB. Francis and M. Maggiore, Getting mobile autonomous robots to rendezvous, n Control of Uncertain Systems: Modelling, Approximation, and Design (eds. B. Francis, M. Smith and J. E. Willems), 329 (2006), 119-137.  doi: 10.1007/11664550_7.  Google Scholar

[71]

M. Micheli, P. W. Michor and D. Mumford, Sectional curvature in terms of the cometric, with applications to the Riemannian manifolds of landmarks, SIAM Journal on Imaging Sciences, 5 (2012), 394-433, URL http://dx.doi.org/10.1137/10081678X. doi: 10.1137/10081678X.  Google Scholar

[72]

M. Micheli, P. W. Michor and D. Mumford, Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds, Izvestiya: Mathematics, 77 (2013), 541-570, URL http://stacks.iop.org/1064-5632/77/i=3/a=541. doi: 10.4213/im7966.  Google Scholar

[73]

P. W. Michor, Manifolds of mappings and shapes (http://arxiv.org/abs/1505.02359), in The legacy of Bernhard Riemann after one hundred and fifty years (eds. L. Ji, F. Oort and S. Yau), vol. 35 of ALM 35, International Press Boston, 2016,459-486.  Google Scholar

[74]

J. Morales Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Modern Birkhäuser Classics, Springer-Verlag, 1999. doi: 10.1007/978-3-0348-8718-2.  Google Scholar

[75]

D. Mumford and P. W. Michor, On Euler's equation and 'EPDiff', Journal of Geometric Mechanics, 5 (2013), 319-344, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=9003. doi: 10.3934/jgm.2013.5.319.  Google Scholar

[76]

P. Muralidharan, J. Fishbaugh, H. J. Johnson, S. Durrleman, J. S. Paulsen, G. Gerig and P. T. Fletcher, Diffeomorphic shape trajectories for improved longitudinal segmentation and statistics, Springer International Publishing, https://www.ncbi.nlm.nih.gov/pubmed/25320781, 2014, 49-56, URL http://dx.doi.org/10.1007/978-3-319-10443-0_7. Google Scholar

[77]

M. Niethammer, Y. Huang and F. -X. Vialard, Geodesic regression for image time-series, in Medical Image Computing and Computer-Assisted Intervention -MICCAI 2011 (eds. G. Fichtinger, A. Martel and T. Peters), vol. 6892 of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2011,655-662, URL http://dx.doi.org/10.1007/978-3-642-23629-7_80. doi: 10.1007/978-3-642-23629-7_80.  Google Scholar

[78]

L. Noakes, Spherical splines, in Geometric Properties for Incomplete Data, ed. by R. Klette, R. Kozera, R., L. Noakes and J. Weickert, Computational Imaging and Vision, Springer-Verlag, 31 (2006), 77-101. URL https://link.springer.com/chapter/10.1007/1-4020-3858-8_5. Google Scholar

[79]

L. Noakes, Approximating near-geodesic natural cubic splines, Communications in Mathematical Sciences, 12 (2014), 1409-1425.  doi: 10.4310/CMS.2014.v12.n8.a2.  Google Scholar

[80]

L. Noakes, Minimum $L^\infty$ accelerations in Riemannian manifolds, Advances in Computational Mathematics, 40 (2014), 839-863, URL http://dx.doi.org/10.1007/s10444-013-9329-9. doi: 10.1007/s10444-013-9329-9.  Google Scholar

[81]

L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces, IMA Journal of Mathematical Control and Information, 6 (1989), 465-473, URL http://imamci.oxfordjournals.org/content/6/4/465.abstract. doi: 10.1093/imamci/6.4.46.  Google Scholar

[82]

M. Pauley and L. Noakes, Cubics and negative curvature, Differ. Geom. Appl., 30 (2012), 694-701.  doi: 10.1016/j.difgeo.2012.09.004.  Google Scholar

[83]

D. Pavlov, Geometric Discretization of the EPDiff Equations, ArXiv e-prints 1503. 03935. Google Scholar

[84]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience, 1962.  Google Scholar

[85]

T. Popiel, Higher order geodesics in Lie groups, Mathematics of Control, Signals, and Systems, 19 (2007), 235-253, URL http://dx.doi.org/10.1007/s00498-007-0012-x. doi: 10.1007/s00498-007-0012-x.  Google Scholar

[86] R. T. Rockafellar, Convex Analysis, Cambridge University Press, Princeton Landmarks in Mathematics and Physics, 1997.   Google Scholar
[87] M. Ross, A Primer on Pontryagin's Principle in Optimal Control, Collegiate Publishers, 2009.   Google Scholar
[88]

C. Samir, P. -A. Absil, A. Srivastava and E. Klassen, A gradient-descent method for curve fitting on Riemannian manifolds, Foundations of Computational Mathematics, 12 (2012), 49-73, URL http://dx.doi.org/10.1007/s10208-011-9091-7. doi: 10.1007/s10208-011-9091-7.  Google Scholar

[89]

N. Singh and M. Niethammer, Splines for diffeomorphic image regression, in Medical Image Computing and Computer-Assisted Intervention -MICCAI 2014 (eds. P. Golland, N. Hata, C. Barillot, J. Hornegger and R. Howe), vol. 8674 of Lecture Notes in Computer Science, Springer International Publishing, 2014,121-129, URL http://dx.doi.org/10.1007/978-3-319-10470-6_16. doi: 10.1007/978-3-319-10470-6_16.  Google Scholar

[90]

N. Singh, F. -X. Vialard and M. Niethammer, Splines for diffeomorphisms, Medical Image Analysis, 25 (2015), 56-71, URL http://dx.doi.org/10.1016/j.media.2015.04.012. doi: 10.1016/j.media.2015.04.012.  Google Scholar

[91]

S. SmithM. Broucke and B. Francis, Curve shortening and the rendezvous problem for mobile autonomous robots, Automatic Control, IEEE Transactions on, 52 (2007), 1154-1159.  doi: 10.1109/TAC.2007.899024.  Google Scholar

[92]

M. H. SohnJ. L. McKay and L. H. Ting, Defining feasible bounds on muscle activation in a redundant biomechanical task: Practical implications of redundancy, J Biomech, 46 (2013), 1363-1368.  doi: 10.1016/j.jbiomech.2013.01.020.  Google Scholar

[93]

S. Sommer, Anisotropic distributions on manifolds: Template estimation and most probable paths, in Information Processing in Medical Imaging, Lecture Notes in Computer Science (eds. S. Ourselin, D. C. Alexander, C. -F. Westin and M. J. Cardoso), vol. 9123, Springer International Publishing, 2015,193-204, URL http://dx.doi.org/10.1007/978-3-319-19992-4_15. doi: 10.1007/978-3-319-19992-4_15.  Google Scholar

[94]

S. Sommer, Evolution equations with anisotropic distributions and diffusion PCA, in Geometric Science of Information (eds. F. Nielsen and F. Barbaresco), vol. 9389, Lecture Notes in Computer Science, Springer International Publishing, 2015, 3-11, URL http://dx.doi.org/10.1007/978-3-319-25040-3_1. doi: 10.1007/978-3-319-25040-3_1.  Google Scholar

[95]

F. Steinke, M. Hein and B. Schölkopf, Nonparametric regression between general Riemannian manifolds, SIAM Journal on Imaging Sciences, 3 (2010), 527-563, URL http://dx.doi.org/10.1137/080744189. doi: 10.1137/080744189.  Google Scholar

[96]

T. Stuchi, T. Balseiro, J. Koiller and A. Cabrera, Minimal time splines on the sphere, in Proceedings of the Sixth IST-IME Meeting(September 5-9 2016, Lisbon), in honor of Waldyr Oliva, to appear in São Paulo J. of Mathematical Sciences. Google Scholar

[97]

L. H. TingS. A. ChvatalS. A. Safavynia and J. L. McKay, Review and perspective: Neuromechanical considerations for predicting muscle activation patterns for movement, Int J Numer Method Biomed Eng, 28 (2012), 1003-1014.  doi: 10.1002/cnm.2485.  Google Scholar

[98]

B. Wang, W. Liu, M. Prastawa, A. Irimia, P. M. Vespa, J. D. van Horn, P. T. Fletcher and G. Gerig, 4d active cut: An interactive tool for pathological anatomy modeling, Proceedings / IEEE International Symposium on Biomedical Imaging: from nano to macro. IEEE International Symposium on Biomedical Imaging, 2014 (2014), 529-532, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4209480/. doi: 10.1109/ISBI.2014.6867925.  Google Scholar

[99]

G. S. Watson, Statistics on Spheres, The University of Arkansas lecture notes in the mathematical sciences (Book 6), Wiley-Interscience, 1983.  Google Scholar

[100]

A. D. Weinstein, The cut locus and conjugate locus of a riemannian manifold, Annals of Mathematics, 87 (1968), 29-41.  doi: 10.2307/1970592.  Google Scholar

[101]

A. D. Weinstein, Lagrangian mechanics and groupoids, in Mechanics Day, Fields Institute Communications Series, American Mathematical Society, 7 (1996), 207-231.   Google Scholar

[102]

L. Younes, Shapes and Diffeomorphisms, vol. 171 of Applied Mathematical Sciences, Springer-Verlag, 2010. doi: 10.1007/978-3-642-12055-8.  Google Scholar

show all references

References:
[1]

L. Abrunheiro, M. Camarinha and J. Clemente-Gallardo, Cubic polynomials on Lie groups: Reduction of the Hamiltonian system, Journal of Physics A: Mathematical and Theoretical, 44 (2011), 355203, 16pp, URL http://stacks.iop.org/1751-8121/44/i=35/a=355203. doi: 10.1088/1751-8113/44/35/355203.  Google Scholar

[2]

L. Abrunheiro, M. Camarinha and J. Clemente-Gallardo, Corrigendum: Cubic polynomials on Lie groups: Reduction of the Hamiltonian system, Journal of Physics A: Mathematical and Theoretical, 46 (2013), 189501, 2pp, URL http://stacks.iop.org/1751-8121/46/i=18/a=189501. doi: 10.1088/1751-8113/46/18/189501.  Google Scholar

[3]

L. Abrunheiro and M. Camarinha, Optimal control of affine connection control systems from the point of view of Lie algebroids, International Journal of Geometric Methods in Modern Physics, 11 (2014), 1450038, 8pp, URL http://www.worldscientific.com/doi/abs/10.1142/S0219887814500388. doi: 10.1142/S0219887814500388.  Google Scholar

[4]

L. Abrunheiro, M. Camarinha and J. Clemente-Gallardo, Geometric Hamiltonian formulation of a variational problem depending on the covariant acceleration, Conference Papers in Mathematics, 2013 (2013), Article ID 243621, 9 pages, URL http://www.hindawi.com/archive/2013/243621/. doi: 10.1155/2013/243621.  Google Scholar

[5]

. Attri, Development of Models for the Equations of Motion in the Solar System: Implementations and Applications, Master's thesis, Universitat Politécnica de Catalunya Master in Aerospace Science and Technology, 2014, URL http://upcommons.upc.edu/handle/2099.1/22415. Google Scholar

[6]

B. B. KhesinJ. LenellsG. Misiolek and S. C. Preston, Curvatures of Sobolev metrics on diffeomorphism groups, Pure and Applied Mathematics Quarterly, 9 (2013), 291-332.  doi: 10.4310/PAMQ.2013.v9.n2.a2.  Google Scholar

[7]

P. Balseiro, A. Cabrera and J. Koiller, Optimal control on affine bundles, in preparation. Google Scholar

[8]

M. Barbero-Liñán and A. D. Lewis, Geometric interpretations of the symmetric product in affine differential geometry and applications, International Journal of Geometric Methods in Modern Physics, 9 (2012), 1250073, 33pp, URl http://www.worldscientific.com/doi/abs/10.1142/S0219887812500739. doi: 10.1142/S0219887812500739.  Google Scholar

[9]

M. Barbero-Liñán, Characterization of accessibility for affine connection control systems at some points with nonzero velocity, in Proceedings of the IEEE Conference on Decision and Control and European Control Conference, (2011), 6528-6533.   Google Scholar

[10]

M. Barbero-Liñán and M. Sigalotti High-order sufficient conditions for configuration tracking of affine connection control systems, Systems & Control Letters, 59 (2010), 491-503, URl http://www.sciencedirect.com/science/article/pii/S0167691110000757 (http://arxiv.org/abs/1501.04026). doi: 10.1016/j.sysconle.2010.06.011.  Google Scholar

[11]

M. BauerM. BruverisP. Harms and J. Moller-Andersen, A numerical framework for Sobolev metrics on the space of curves, SIAM J. Imaging Sci. (arxiv:1603.03480), SIAM J. Imaging Sci., 10 (2017), 47-73.  doi: 10.1137/16M1066282.  Google Scholar

[12]

M. Bauer, M. Bruveris and P. W. Michor, Why use Sobolev metrics on the space of curves, in Riemannian Computing in Computer Vision, Turaga, P. and Srivastava, A., editors, Springer-Verlag, 11 (2016), 223-255, URL https://arxiv.org/abs/1502.03229, URL https://link.springer.com/book/10.1007/978-3-319-22957-7.  Google Scholar

[13]

M. Bauer, M. Bruveris, P. Harms and J. Møller-Andersen, Curve matching with applications in medical imaging, in MFCA 2015: 5th MICCAI Workshop on Mathematical Foundations of Computational Anatomy, (eds. X. Pennec, S. Joshi, M. Nielsen, T. Fletcher, S. Durrleman and S. Sommer), 2015, 83-94. URL https://arxiv.org/abs/1506.08840, URL http://www-sop.inria.fr/asclepios/events/MFCA15/MFCA15_Proceedings.pdf. Google Scholar

[14]

M. Bauer, M. Bruveris and P. Michor, Overview of the geometries of shape spaces and diffeomorphism groups, Journal of Mathematical Imaging and Vision, 50 (2014), 60-97, URL http://dx.doi.org/10.1007/s10851-013-0490-z. doi: 10.1007/s10851-013-0490-z.  Google Scholar

[15]

M. Bauer, M. Eslitzbichler and M. Grasmair, Landmark-guided elastic shape analysis of human character motions, arxiv: 1502. 07666. Google Scholar

[16]

M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on shape space of surfaces, Journal of Geometric Mechanics, 3 (2011), 389-438, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=7061.  Google Scholar

[17]

M. Bauer, P. Harms and P. W. Michor, Sobolev metrics on shape space, Ⅱ: Weighted sobolev metrics and almost local metrics, Journal of Geometric Mechanics, 4 (2012), 365-383, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=8178.  Google Scholar

[18]

C. Bingham, Review: Geoffrey S. Watson, Statistics on spheres Ann. Statist., 13 (1985), 838-844, RUL http://dx.doi.org/10.1214/aos/1176349566. doi: 10.1214/aos/1176349566.  Google Scholar

[19]

A. BlochD. E. ChangN. Leonard and J. Marsden, Controlled Lagrangians and the stabilization of mechanical systems. Ⅱ. potential shaping, Automatic Control, IEEE Transactions on, 46 (2001), 1556-1571.  doi: 10.1109/9.956051.  Google Scholar

[20]

A. BlochN. Leonard and J. Marsden, Controlled Lagrangians and the stabilization of mechanical systems. Ⅰ. the first matching theorem, Automatic Control, IEEE Transactions on, 45 (2000), 2253-2270.  doi: 10.1109/9.895562.  Google Scholar

[21] A. Bloch, Nonholonomic Mechanics and Control, vol. 24 of Interdisciplinary Applied Mathematics, Springer-Verlag, New York, 2003.  doi: 10.1007/b97376.  Google Scholar
[22]

M. Bruveris, Geometry of Diffeomorphism Groups and Shape Matching, PhD thesis, Department of Mathematics, Imperial College, London, 2012, URL https://www.brunel.ac.uk/mastmmb/pdf/00_thesis.pdf.  Google Scholar

[23]

M. Bruveris and D. Holm, Geometry of image registration: the diffeomorphism group and momentum maps, in Geometry, Mechanics, and Dynamics: The Legacy of Jerry Marsden, ed. by Chang, D. E., Holm, D. D., Patrick, G., Ratiu, T., vol. 73 of Fields Institute Communications Series, Springer-Verlag, 2015, 19-56. doi: 10.1007/978-1-4939-2441-7_2.  Google Scholar

[24]

F. Bullo and A. D. Lewis, Geometric Control of Mechanical Systems, Texts in Applied Mathematics 49, Springer-Verlag, New York, 2005, URL http://www.springer.com/br/book/9780387221953. doi: 10.1007/978-1-4899-7276-7.  Google Scholar

[25]

F. Bullo and A. D. Lewis, Supplementary chapters for Geometric Control of Mechanical Systems, URL http://motion.me.ucsb.edu/book-gcms/. Google Scholar

[26]

C. L. BurnettD.D. Holm and D.M. Meier, Inexact trajectory planning and inverse problems in the Hamilton-Pontryagin framework, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 469 (2014), 20130249, 24 pp.  doi: 10.1098/rspa.2013.0249.  Google Scholar

[27]

A. Castro and J. Koiller, On the dynamic Markov-Dubins problem: From path planning in robotics and biolocomotion to computational anatomy, Regular and Chaotic Dynamics, 18 (2013), 1-20, URL http://dx.doi.org/10.1134/S1560354713010012. doi: 10.1134/S1560354713010012.  Google Scholar

[28]

D. E. Chang, A simple proof of the Pontryagin maximum principle on manifolds, Automatica, 47 (2011), 630-633, URL http://www.sciencedirect.com/science/article/pii/S0005109811000525. doi: 10.1016/j.automatica.2011.01.037.  Google Scholar

[29]

A. Chenciner and R. Montgomery, A remarkable periodic solution of the three-body problem in the case of equal masses, Annals of Mathematics. Second Series, 152 (2000), 881-901, URL http://eudml.org/doc/121861. doi: 10.2307/2661357.  Google Scholar

[30]

A. ChertockP. D. Toit and J. E. Marsden, Integration of the EPDiff equation by particle methods, ESAIM: Mathematical Modelling and Numerical Analysis, 46 (2012), 515-534.  doi: 10.1051/m2an/2011054.  Google Scholar

[31]

J. Cortés, S. Martínez, J. P. Ostrowski and H. Zhang, Simple mechanical control systems with constraints and symmetry, SIAM Journal on Control and Optimization, 41 (2002), 851-874, URL http://dx.doi.org/10.1137/S0363012900381741. doi: 10.1137/S0363012900381741.  Google Scholar

[32]

P. Crouch, F. S. Leite and M. Camarinha, A second order Riemannian variational problem from a Hamiltonian perspective, preprint, Centro de Matemática da Universidade de Coimbra, http://hdl.handle.net/10316/11230,1998. Google Scholar

[33]

P. Crouch and F. Leite, The dynamic interpolation problem: On Riemannian manifolds, Lie groups, and symmetric spaces, Journal of Dynamical and Control Systems, 1 (1995), 177-202, URL http://dx.doi.org/10.1007/BF02254638. doi: 10.1007/BF02254638.  Google Scholar

[34]

P. Crouch and F. S. Leite, Geometry and the dynamic interpolation problem, in Proceedings of the 1991 American Control Conference, 7 (1991), 1131-1136, URL http://ieeexplore.ieee.org/document/4791552/. Google Scholar

[35]

M. de Léon, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, Journal of Physics A: Mathematical and General, 38 (2005), R241-R308, URL http://stacks.iop.org/0305-4470/38/i=24/a=R01. doi: 10.1088/0305-4470/38/24/R01.  Google Scholar

[36]

N. Desai, S. Ploskonka, L. R. Goodman, C. Austin, J. Goldberg and T. Falcone, Analysis of embryo morphokinetics, multinucleation and cleavage anomalies using continuous time-lapse monitoring in blastocyst transfer cycles, Reproductive Biology and Endocrinology : RB & E, 12 (2014), 54-54, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4074839/. doi: 10.1186/1477-7827-12-54.  Google Scholar

[37]

I. L. Dryden, Statistical analysis on high-dimensional spheres and shape spaces, The Annals of Statistics, 33 (2005), 1643-1665, URL http://www.jstor.org/stable/3448620. doi: 10.1214/009053605000000264.  Google Scholar

[38]

S. Durrleman, T. Fletcher, G. Gerig, M. Niethammer and X. Pennec, STIA 2014 -Spatio-temporal Image Analysis for Longitudinal and Time-Series Image Data, vol. 8682 of Lecture notes in computer science, Springer International Publishing, Cambridge, United States, 2015. URL http://www.springer.com/br/book/9783319149042. Google Scholar

[39]

D. Elworthy, Geometric aspects of diffusions on manifolds, in École d'Été de Probabilités de Saint-Flour ⅩⅤ-ⅩⅦ, 1985-87, Lecture Notes in Mathematics (ed. P. -L. Hennequin), vol. 1362, Springer Berlin Heidelberg, 1988,277-425, URl http://dx.doi.org/10.1007/BFb0086183. doi: 10.1007/BFb0086183.  Google Scholar

[40]

E. Fehlberg, Classical Seventh, Sixth, and Fifth-Order Runge Kutta-Nystrom Formula with Stepsize Control for General Second-Order Differential Equations, Technical report, NASA TR R-432, Washington D. C, 1974, URL https://ntrs.nasa.gov/search.jsp?R=19740026877. Google Scholar

[41]

J. -B. Fiot, H. Raguet, L. Risser, L. D. Cohen, J. Fripp and F. -X. Vialard, Longitudinal deformation models, spatial regularizations and learning strategies to quantify Alzheimer's disease progression, NeuroImage: Clinical, 4 (2014), 718-729, URl http://www.sciencedirect.com/science/article/pii/S2213158214000205. doi: 10.1016/j.nicl.2014.02.002.  Google Scholar

[42] N. FisherT. Lewis and B. Embleton, Statistical Analysis of Spherical Data, Cambridge University Press, 1987.  doi: 10.1017/CBO9780511623059.  Google Scholar
[43]

P. T. Fletcher, Statistical Variability in Nonlinear Spaces: Application to Shape Analysis and DT-MRI, PhD thesis, Department of Computer Science, University of North Carolina, 2004. Google Scholar

[44]

P. T. Fletcher, Geodesic regression and the theory of least squares on Riemannian Manifolds, International Journal of Computer Vision, 105 (2013), 171-185, URL http://dx.doi.org/10.1007/s11263-012-0591-y. doi: 10.1007/s11263-012-0591-y.  Google Scholar

[45]

D. Fortuné, J. A. R. Quintero and C. Vallée, Pontryagin calculus in Riemannian geometry, Geometric Science of Information, Lecture Notes in Computer Science (eds. F. Nielsen, F. Barbaresco and F. Dubois), vol. 9389, Springer International Publishing, 2015,541-549, URL http://dx.doi.org/10.1007/978-3-319-25040-3_58. doi: 10.1007/978-3-319-25040-3_58.  Google Scholar

[46]

B. Francis and M. Maggiore, Flocking and Rendezvous in Distributed Robotics, Springer International Publishing, 2016. doi: 10.1007/978-3-319-24729-8.  Google Scholar

[47]

F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F. -X. Vialard, Invariant higher-order variational problems, Communications in Mathematical Physics, 309 (2012), 413-458, URL http://dx.doi.org/10.1007/s00220-011-1313-y. doi: 10.1007/s00220-011-1313-y.  Google Scholar

[48]

F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F. -X. Vialard, Invariant higher-order variational problems Ⅱ, Journal of Nonlinear Science, 22 (2012), 553-597, URL http://dx.doi.org/10.1007/s00332-012-9137-2. doi: 10.1007/s00332-012-9137-2.  Google Scholar

[49]

P. Harms, Sobolev Metrics on Shape Space of Surfaces, PhD thesis, Universität Wien, 2010, URL https://arxiv.org/abs/1211.3515. Google Scholar

[50]

J. Hinkle, P. Fletcher and S. Joshi, Intrinsic polynomials for regression on riemannian manifolds, Journal of Mathematical Imaging and Vision, 50 (2014), 32-52, URL http://dx.doi.org/10.1007/s10851-013-0489-5. doi: 10.1007/s10851-013-0489-5.  Google Scholar

[51]

J. Hinkle, P. Muralidharan, P. Fletcher and S. Joshi, Polynomial regression on Riemannian manifolds, in Computer Vision -ECCV 2012 (eds. A. Fitzgibbon, S. Lazebnik, P. Perona, Y. Sato and C. Schmid), vol. 7574 of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2012, 1-14, URl http://dx.doi.org/10.1007/978-3-642-33712-3_1. doi: 10.1007/978-3-642-33712-3_1.  Google Scholar

[52]

D. D. Holm, T. Schmah and C. Stoica, Geometric Mechanics and Symmetry: From Finite to Infinite Dimensions, Oxford Texts in Applied and Engineering Mathematics, Oxford University Press, 2009, URL https://global.oup.com/academic/product/geometric-mechanics-and-symmetry-9780199212910?cc=br&lang=en&.  Google Scholar

[53]

K. Hüper, Y. Shen and F. Silva Leite, Geometric splines and interpolation on $S^2$: Numerical experiments, in Proceedings of the 45th IEEE Conference on Decision & Control, San Diego, CA, USA, 2006,6403-6407, URL http://ieeexplore.ieee.org/document/4177774/. Google Scholar

[54]

R. V. Iyer, Pontryagin's minimum principle for simple mechanical systems on Riemannian manifolds and Lie groups, 2005, Http://www.math.ttu.edu/rvenkata/Papers/max-principle.pdf. Google Scholar

[55]

R. V. Iyer, R. Holsapple and D. Doman, Optimal control problems on parallelizable Riemannian manifolds: Theory and applications, ESAIM: Control, Optimisation and Calculus of Variations, 12 (2006), 1-11, URL https://www.esaim-cocv.org/articles/cocv/abs/2006/01/cocv0402/cocv0402.html. doi: 10.1051/cocv:2005026.  Google Scholar

[56]

M. Jóźwikowski, Optimal Control Theory on Almost Lie Algebroids, PhD thesis, Institute of Mathematics, Polish Academy of Sciences, 2011, URL https://arxiv.org/pdf/1111.1549.pdf. Google Scholar

[57]

V. Jurdjevic, The Delauney-Dubins problem, in Geometric Control Theory and Sub-Riemannian Geometry (eds. G. Stefani, U. Boscain, J. -P. Gauthier, A. Sarychev and M. Sigalotti), Springer International Publishing, 5 (2014), 219-239, URl http://dx.doi.org/10.1007/978-3-319-02132-4_14. doi: 10.1007/978-3-319-02132-4_14.  Google Scholar

[58] V. Jurdjevic, Optimal Control and Geometry: Integrable Systems, Cambridge University Press, Cambridge Studies in Advanced Mathematics, 2016.  doi: 10.1017/CBO9781316286852.  Google Scholar
[59]

C. Y. Kaya and J. L. Noakes, Finding interpolating curves minimizing $L^\infty$ acceleration in the euclidean space via optimal control theory, SIAM Journal on Control and Optimization, 51 (2013), 442-464, URl http://dx.doi.org/10.1137/12087880X. doi: 10.1137/12087880X.  Google Scholar

[60]

B. Khesin, J. Lenells, G. Misiołek and S. Preston, Geometry of diffeomorphism groups, complete integrability and geometric statistics, Geometric and Functional Analysis, 23 (2013), 334-366, URL http://dx.doi.org/10.1007/s00039-013-0210-2. doi: 10.1007/s00039-013-0210-2.  Google Scholar

[61]

B. Khesin and R. Wendt, The Geometry of Infinite-Dimensional Groups, vol. 51 of Modern Surveys in Mathematics, 1st edition, Springer-Verlag Berlin Heidelberg, 2009.  Google Scholar

[62]

M. Kummer, On the construction of the reduced phase space of a hamiltonian system with symmetry, Indiana Univ. Math. J., 30 (1981), 281-291.  doi: 10.1512/iumj.1981.30.30022.  Google Scholar

[63]

M. Kummer, Realizations of the reduced phase space of a hamiltonian system with symmetry, in Local and Global Methods of Nonlinear Dynamics, Lecture Notes in Physics (eds. A. W. Sáenz, W. W. Zachary and R. Cawley), vol. 252, Springer Berlin Heidelberg, 1986, 32-39, URL http://dx.doi.org/10.1007/BFb0018326. doi: 10.1007/BFb0018326.  Google Scholar

[64]

J. Langer and D. A. Singer, The total squared curvature of closed curves, J. Differential Geom., 20 (1984), 1-22, URl http://projecteuclid.org/euclid.jdg/1214438990. doi: 10.4310/jdg/1214438990.  Google Scholar

[65]

A. D. Lewis, The Affine Structure of Jet Bundles, Technical report, 2005, URL www.mast.queensu.ca/~ andrew/notes/pdf/2005a.pdf. Google Scholar

[66]

A. D. Lewis, Aspects of Geometric Mechanics and Control of Mechanical Systems, PhD thesis, Caltech, 1995, URL http://www.mast.queensu.ca/andrew/papers/pdf/1995f.pdf. Google Scholar

[67]

Configuration controllability of simple mechanical control systems, SIAM Review, 41 (1999), 555-574, URL http://dx.doi.org/10.1137/S0036144599351065. doi: 10.1137/S0036144599351065.  Google Scholar

[68]

M. Lewis, D. Offin, P. -L. Buono and M. Kovacic, Instability of the periodic hip-hop orbit in the $2n$-body problem with equal masses, Discrete and Continuous Dynamical Systems, 33 (2013), 1137-1155, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=7827.  Google Scholar

[69]

P. Libermann, Lie algebroids in mechanics, Archivum Mathematicum, 32 (1966), 147-162.   Google Scholar

[70]

Z. LinB. Francis and M. Maggiore, Getting mobile autonomous robots to rendezvous, n Control of Uncertain Systems: Modelling, Approximation, and Design (eds. B. Francis, M. Smith and J. E. Willems), 329 (2006), 119-137.  doi: 10.1007/11664550_7.  Google Scholar

[71]

M. Micheli, P. W. Michor and D. Mumford, Sectional curvature in terms of the cometric, with applications to the Riemannian manifolds of landmarks, SIAM Journal on Imaging Sciences, 5 (2012), 394-433, URL http://dx.doi.org/10.1137/10081678X. doi: 10.1137/10081678X.  Google Scholar

[72]

M. Micheli, P. W. Michor and D. Mumford, Sobolev metrics on diffeomorphism groups and the derived geometry of spaces of submanifolds, Izvestiya: Mathematics, 77 (2013), 541-570, URL http://stacks.iop.org/1064-5632/77/i=3/a=541. doi: 10.4213/im7966.  Google Scholar

[73]

P. W. Michor, Manifolds of mappings and shapes (http://arxiv.org/abs/1505.02359), in The legacy of Bernhard Riemann after one hundred and fifty years (eds. L. Ji, F. Oort and S. Yau), vol. 35 of ALM 35, International Press Boston, 2016,459-486.  Google Scholar

[74]

J. Morales Ruiz, Differential Galois Theory and Non-Integrability of Hamiltonian Systems, Modern Birkhäuser Classics, Springer-Verlag, 1999. doi: 10.1007/978-3-0348-8718-2.  Google Scholar

[75]

D. Mumford and P. W. Michor, On Euler's equation and 'EPDiff', Journal of Geometric Mechanics, 5 (2013), 319-344, URL http://aimsciences.org/journals/displayArticlesnew.jsp?paperID=9003. doi: 10.3934/jgm.2013.5.319.  Google Scholar

[76]

P. Muralidharan, J. Fishbaugh, H. J. Johnson, S. Durrleman, J. S. Paulsen, G. Gerig and P. T. Fletcher, Diffeomorphic shape trajectories for improved longitudinal segmentation and statistics, Springer International Publishing, https://www.ncbi.nlm.nih.gov/pubmed/25320781, 2014, 49-56, URL http://dx.doi.org/10.1007/978-3-319-10443-0_7. Google Scholar

[77]

M. Niethammer, Y. Huang and F. -X. Vialard, Geodesic regression for image time-series, in Medical Image Computing and Computer-Assisted Intervention -MICCAI 2011 (eds. G. Fichtinger, A. Martel and T. Peters), vol. 6892 of Lecture Notes in Computer Science, Springer Berlin Heidelberg, 2011,655-662, URL http://dx.doi.org/10.1007/978-3-642-23629-7_80. doi: 10.1007/978-3-642-23629-7_80.  Google Scholar

[78]

L. Noakes, Spherical splines, in Geometric Properties for Incomplete Data, ed. by R. Klette, R. Kozera, R., L. Noakes and J. Weickert, Computational Imaging and Vision, Springer-Verlag, 31 (2006), 77-101. URL https://link.springer.com/chapter/10.1007/1-4020-3858-8_5. Google Scholar

[79]

L. Noakes, Approximating near-geodesic natural cubic splines, Communications in Mathematical Sciences, 12 (2014), 1409-1425.  doi: 10.4310/CMS.2014.v12.n8.a2.  Google Scholar

[80]

L. Noakes, Minimum $L^\infty$ accelerations in Riemannian manifolds, Advances in Computational Mathematics, 40 (2014), 839-863, URL http://dx.doi.org/10.1007/s10444-013-9329-9. doi: 10.1007/s10444-013-9329-9.  Google Scholar

[81]

L. Noakes, G. Heinzinger and B. Paden, Cubic splines on curved spaces, IMA Journal of Mathematical Control and Information, 6 (1989), 465-473, URL http://imamci.oxfordjournals.org/content/6/4/465.abstract. doi: 10.1093/imamci/6.4.46.  Google Scholar

[82]

M. Pauley and L. Noakes, Cubics and negative curvature, Differ. Geom. Appl., 30 (2012), 694-701.  doi: 10.1016/j.difgeo.2012.09.004.  Google Scholar

[83]

D. Pavlov, Geometric Discretization of the EPDiff Equations, ArXiv e-prints 1503. 03935. Google Scholar

[84]

L. S. Pontryagin, V. G. Boltyanskii, R. V. Gamkrelidze and E. F. Mishchenko, The Mathematical Theory of Optimal Processes, Interscience, 1962.  Google Scholar

[85]

T. Popiel, Higher order geodesics in Lie groups, Mathematics of Control, Signals, and Systems, 19 (2007), 235-253, URL http://dx.doi.org/10.1007/s00498-007-0012-x. doi: 10.1007/s00498-007-0012-x.  Google Scholar

[86] R. T. Rockafellar, Convex Analysis, Cambridge University Press, Princeton Landmarks in Mathematics and Physics, 1997.   Google Scholar
[87] M. Ross, A Primer on Pontryagin's Principle in Optimal Control, Collegiate Publishers, 2009.   Google Scholar
[88]

C. Samir, P. -A. Absil, A. Srivastava and E. Klassen, A gradient-descent method for curve fitting on Riemannian manifolds, Foundations of Computational Mathematics, 12 (2012), 49-73, URL http://dx.doi.org/10.1007/s10208-011-9091-7. doi: 10.1007/s10208-011-9091-7.  Google Scholar

[89]

N. Singh and M. Niethammer, Splines for diffeomorphic image regression, in Medical Image Computing and Computer-Assisted Intervention -MICCAI 2014 (eds. P. Golland, N. Hata, C. Barillot, J. Hornegger and R. Howe), vol. 8674 of Lecture Notes in Computer Science, Springer International Publishing, 2014,121-129, URL http://dx.doi.org/10.1007/978-3-319-10470-6_16. doi: 10.1007/978-3-319-10470-6_16.  Google Scholar

[90]

N. Singh, F. -X. Vialard and M. Niethammer, Splines for diffeomorphisms, Medical Image Analysis, 25 (2015), 56-71, URL http://dx.doi.org/10.1016/j.media.2015.04.012. doi: 10.1016/j.media.2015.04.012.  Google Scholar

[91]

S. SmithM. Broucke and B. Francis, Curve shortening and the rendezvous problem for mobile autonomous robots, Automatic Control, IEEE Transactions on, 52 (2007), 1154-1159.  doi: 10.1109/TAC.2007.899024.  Google Scholar

[92]

M. H. SohnJ. L. McKay and L. H. Ting, Defining feasible bounds on muscle activation in a redundant biomechanical task: Practical implications of redundancy, J Biomech, 46 (2013), 1363-1368.  doi: 10.1016/j.jbiomech.2013.01.020.  Google Scholar

[93]

S. Sommer, Anisotropic distributions on manifolds: Template estimation and most probable paths, in Information Processing in Medical Imaging, Lecture Notes in Computer Science (eds. S. Ourselin, D. C. Alexander, C. -F. Westin and M. J. Cardoso), vol. 9123, Springer International Publishing, 2015,193-204, URL http://dx.doi.org/10.1007/978-3-319-19992-4_15. doi: 10.1007/978-3-319-19992-4_15.  Google Scholar

[94]

S. Sommer, Evolution equations with anisotropic distributions and diffusion PCA, in Geometric Science of Information (eds. F. Nielsen and F. Barbaresco), vol. 9389, Lecture Notes in Computer Science, Springer International Publishing, 2015, 3-11, URL http://dx.doi.org/10.1007/978-3-319-25040-3_1. doi: 10.1007/978-3-319-25040-3_1.  Google Scholar

[95]

F. Steinke, M. Hein and B. Schölkopf, Nonparametric regression between general Riemannian manifolds, SIAM Journal on Imaging Sciences, 3 (2010), 527-563, URL http://dx.doi.org/10.1137/080744189. doi: 10.1137/080744189.  Google Scholar

[96]

T. Stuchi, T. Balseiro, J. Koiller and A. Cabrera, Minimal time splines on the sphere, in Proceedings of the Sixth IST-IME Meeting(September 5-9 2016, Lisbon), in honor of Waldyr Oliva, to appear in São Paulo J. of Mathematical Sciences. Google Scholar

[97]

L. H. TingS. A. ChvatalS. A. Safavynia and J. L. McKay, Review and perspective: Neuromechanical considerations for predicting muscle activation patterns for movement, Int J Numer Method Biomed Eng, 28 (2012), 1003-1014.  doi: 10.1002/cnm.2485.  Google Scholar

[98]

B. Wang, W. Liu, M. Prastawa, A. Irimia, P. M. Vespa, J. D. van Horn, P. T. Fletcher and G. Gerig, 4d active cut: An interactive tool for pathological anatomy modeling, Proceedings / IEEE International Symposium on Biomedical Imaging: from nano to macro. IEEE International Symposium on Biomedical Imaging, 2014 (2014), 529-532, URL http://www.ncbi.nlm.nih.gov/pmc/articles/PMC4209480/. doi: 10.1109/ISBI.2014.6867925.  Google Scholar

[99]

G. S. Watson, Statistics on Spheres, The University of Arkansas lecture notes in the mathematical sciences (Book 6), Wiley-Interscience, 1983.  Google Scholar

[100]

A. D. Weinstein, The cut locus and conjugate locus of a riemannian manifold, Annals of Mathematics, 87 (1968), 29-41.  doi: 10.2307/1970592.  Google Scholar

[101]

A. D. Weinstein, Lagrangian mechanics and groupoids, in Mechanics Day, Fields Institute Communications Series, American Mathematical Society, 7 (1996), 207-231.   Google Scholar

[102]

L. Younes, Shapes and Diffeomorphisms, vol. 171 of Applied Mathematical Sciences, Springer-Verlag, 2010. doi: 10.1007/978-3-642-12055-8.  Google Scholar

Figure 1.  Energy $h = 0.01$. Regular trajectories.
Figure 2.  Energy $h = 0.332412099$. There is a large chaotic zone, with escaping trajectories.
Figure 3.  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.
Figure 4.  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?
Figure 5.  The periodic trajectory in the energy level $h = 0.808$. Note the central zone shrinking in the associated surface of section
Figure 6.  Invariant tori, seen on a Lagrangian projection in the plane $(a,z)$. Energies $ h=0.49494873, \, $ and $h = 0.522397316$.
Figure 7.  An invariant torus, seen on a Lagrangian projection in the plane $(a,z)$. $h = 0.586204019$, $\beta=1, \mu= r= 2$.
Figure 8.  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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