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The projective symplectic geometry of higher order variational problems: Minimality conditions
1. | Departamento de Matemática, UFPR, Setor de Ciências Exatas, Centro Politécnico, Caixa Postal 019081, CEP 81531-990, Curitiba, Brazil, Brazil |
References:
[1] |
A. Agrachev, D. Barilari and L. Rizzi, Curvature: A variational approach,, Memoirs of the AMS, (). Google Scholar |
[2] |
J. C. Álvarez Paiva and C. E. Durán, Geometric invariants of fanning curves,, Adv. in Appl. Math., 42 (2009), 290.
doi: 10.1016/j.aam.2006.07.008. |
[3] |
I. M. Anderson, The Variational Bicomplex,, Cambridge University Press, (). Google Scholar |
[4] |
G. Cimmino, Estensione dell'identita di Picone alia pi generale equazione differenziale lineare ordinaria autoaggiuntar,, R. Accad. Naz. Lincei, 28 (1939), 354. Google Scholar |
[5] |
W. A. Coppel, Disconjugacy,, Springer-Verlag, (1971).
|
[6] |
M. Crampin, W. Sarlet and F. Cantrijn, Higher-order differential equations and higher-order Lagrangian mechanics,, Math. Proc. Cambridge Philos. Soc., 99 (1986), 565.
doi: 10.1017/S0305004100064501. |
[7] |
C. E. Durán and C. Peixoto, Geometry of fanning curves in divisible Grassmannians,, Differ. Geom. Appl., (). Google Scholar |
[8] |
C. E. Durán, J. C. Eidam and D. Otero, The projective symplectic geometry of higher order variational problems: Index theory,, work in progress., (). Google Scholar |
[9] |
M. S. P. Eastham, The Picone identity for self-adjoint differential equations of even order,, Mathematika, 20 (1973), 197.
doi: 10.1112/S0025579300004769. |
[10] |
S. Easwaran, Quadratic functionals of $n$-th order,, Canad. Math. Bull., 19 (1976), 159.
doi: 10.4153/CMB-1976-024-6. |
[11] |
I. Gelfand and S. Fomin, Calculus of Variations,, Dover, (2000). Google Scholar |
[12] |
M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities,, Springer-Verlag, (1973).
|
[13] |
R. Giambò, F. Giannoni and P. Piccione, Optimal control on riemannian manifolds by interpolation,, Math. Control Signals Systems, 16 (2004), 278.
doi: 10.1007/s00498-003-0139-3. |
[14] |
M. A. Javaloyes and H. Vitório, Zermelo navigation in pseudo-Finsler metrics,, preprint, (). Google Scholar |
[15] |
K. Kreith, A picone identity for first order differential systems,, J. Math. Anal. Appl., 31 (1970), 297.
doi: 10.1016/0022-247X(70)90024-7. |
[16] |
A. Kriegl and P. Michor, The Convenient Setting of Global Analysis,, Math. Surveys and Monogr., (1997).
doi: 10.1090/surv/053. |
[17] |
W. Leighton, Quadratic functionals of second order,, Trans. Amer. Math. Soc., 151 (1970), 309.
doi: 10.1090/S0002-9947-1970-0264485-1. |
[18] |
M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory,, Elsevier, (1985). Google Scholar |
[19] |
F. Mercuri, P. Piccione and D. V. Tausk, Stability of the conjugate index, degenerate conjugate points and the Maslov index in semi-Riemannian geometry,, Pacific J. Math., 206 (2002), 375.
doi: 10.2140/pjm.2002.206.375. |
[20] |
G. Paternain, Geodesic Flows,, Birkhauser, (1999).
doi: 10.1007/978-1-4612-1600-1. |
[21] |
R. Palais, Morse theory on hilbert manifolds,, Topology, 2 (1963), 299.
doi: 10.1016/0040-9383(63)90013-2. |
[22] |
R. Palais, The Morse lemma for Banach spaces,, Bull. Amer. Math. Soc., 75 (1969), 968.
doi: 10.1090/S0002-9904-1969-12318-9. |
[23] |
R. Palais, Foundations of Global Non-linear Analysis,, Benjamin and Co., (1968).
|
[24] |
M. Picone, Sui valori eccezionali di un parametro da cui dipende un'equazione differenziale lineare del secondo ordine,, Ann. Scuola Norm. Sup. Pisa, 28 (1910), 1. Google Scholar |
[25] |
P. D. Prieto-Martínez and N. Romón-Roy, Higher-order mechanics: Variational principles and other topics,, J. Geom. Mech., 5 (2013), 493.
doi: 10.3934/jgm.2013.5.493. |
[26] |
R. Ruggiero, Dynamics and Global Geometry of Manifolds without Conjugate Points,, Sociedade Brasileira de Matemática, (2007).
|
[27] |
D. J. Saunders, The Geometry of Jet Bundles,, London Math. Soc. Lecture Note Ser., (1989).
doi: 10.1017/CBO9780511526411. |
[28] |
W. Tulczyjew, Sur la différentiele de Lagrange,, C. R. Math. Acad. Sci. Paris, 280 (1975), 1295.
|
[29] |
E. J. Wilczynski, Projective Differential Geometry of Curves and Ruled Surfaces,, Chelsea Publishing Co., (1962).
|
[30] |
I. Zelenko and C. Li, Differential geometry of curves in Lagrange Grassmannians with given Young diagram,, Differential Geom. Appl., 27 (2009), 723.
doi: 10.1016/j.difgeo.2009.07.002. |
show all references
References:
[1] |
A. Agrachev, D. Barilari and L. Rizzi, Curvature: A variational approach,, Memoirs of the AMS, (). Google Scholar |
[2] |
J. C. Álvarez Paiva and C. E. Durán, Geometric invariants of fanning curves,, Adv. in Appl. Math., 42 (2009), 290.
doi: 10.1016/j.aam.2006.07.008. |
[3] |
I. M. Anderson, The Variational Bicomplex,, Cambridge University Press, (). Google Scholar |
[4] |
G. Cimmino, Estensione dell'identita di Picone alia pi generale equazione differenziale lineare ordinaria autoaggiuntar,, R. Accad. Naz. Lincei, 28 (1939), 354. Google Scholar |
[5] |
W. A. Coppel, Disconjugacy,, Springer-Verlag, (1971).
|
[6] |
M. Crampin, W. Sarlet and F. Cantrijn, Higher-order differential equations and higher-order Lagrangian mechanics,, Math. Proc. Cambridge Philos. Soc., 99 (1986), 565.
doi: 10.1017/S0305004100064501. |
[7] |
C. E. Durán and C. Peixoto, Geometry of fanning curves in divisible Grassmannians,, Differ. Geom. Appl., (). Google Scholar |
[8] |
C. E. Durán, J. C. Eidam and D. Otero, The projective symplectic geometry of higher order variational problems: Index theory,, work in progress., (). Google Scholar |
[9] |
M. S. P. Eastham, The Picone identity for self-adjoint differential equations of even order,, Mathematika, 20 (1973), 197.
doi: 10.1112/S0025579300004769. |
[10] |
S. Easwaran, Quadratic functionals of $n$-th order,, Canad. Math. Bull., 19 (1976), 159.
doi: 10.4153/CMB-1976-024-6. |
[11] |
I. Gelfand and S. Fomin, Calculus of Variations,, Dover, (2000). Google Scholar |
[12] |
M. Golubitsky and V. Guillemin, Stable Mappings and Their Singularities,, Springer-Verlag, (1973).
|
[13] |
R. Giambò, F. Giannoni and P. Piccione, Optimal control on riemannian manifolds by interpolation,, Math. Control Signals Systems, 16 (2004), 278.
doi: 10.1007/s00498-003-0139-3. |
[14] |
M. A. Javaloyes and H. Vitório, Zermelo navigation in pseudo-Finsler metrics,, preprint, (). Google Scholar |
[15] |
K. Kreith, A picone identity for first order differential systems,, J. Math. Anal. Appl., 31 (1970), 297.
doi: 10.1016/0022-247X(70)90024-7. |
[16] |
A. Kriegl and P. Michor, The Convenient Setting of Global Analysis,, Math. Surveys and Monogr., (1997).
doi: 10.1090/surv/053. |
[17] |
W. Leighton, Quadratic functionals of second order,, Trans. Amer. Math. Soc., 151 (1970), 309.
doi: 10.1090/S0002-9947-1970-0264485-1. |
[18] |
M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory,, Elsevier, (1985). Google Scholar |
[19] |
F. Mercuri, P. Piccione and D. V. Tausk, Stability of the conjugate index, degenerate conjugate points and the Maslov index in semi-Riemannian geometry,, Pacific J. Math., 206 (2002), 375.
doi: 10.2140/pjm.2002.206.375. |
[20] |
G. Paternain, Geodesic Flows,, Birkhauser, (1999).
doi: 10.1007/978-1-4612-1600-1. |
[21] |
R. Palais, Morse theory on hilbert manifolds,, Topology, 2 (1963), 299.
doi: 10.1016/0040-9383(63)90013-2. |
[22] |
R. Palais, The Morse lemma for Banach spaces,, Bull. Amer. Math. Soc., 75 (1969), 968.
doi: 10.1090/S0002-9904-1969-12318-9. |
[23] |
R. Palais, Foundations of Global Non-linear Analysis,, Benjamin and Co., (1968).
|
[24] |
M. Picone, Sui valori eccezionali di un parametro da cui dipende un'equazione differenziale lineare del secondo ordine,, Ann. Scuola Norm. Sup. Pisa, 28 (1910), 1. Google Scholar |
[25] |
P. D. Prieto-Martínez and N. Romón-Roy, Higher-order mechanics: Variational principles and other topics,, J. Geom. Mech., 5 (2013), 493.
doi: 10.3934/jgm.2013.5.493. |
[26] |
R. Ruggiero, Dynamics and Global Geometry of Manifolds without Conjugate Points,, Sociedade Brasileira de Matemática, (2007).
|
[27] |
D. J. Saunders, The Geometry of Jet Bundles,, London Math. Soc. Lecture Note Ser., (1989).
doi: 10.1017/CBO9780511526411. |
[28] |
W. Tulczyjew, Sur la différentiele de Lagrange,, C. R. Math. Acad. Sci. Paris, 280 (1975), 1295.
|
[29] |
E. J. Wilczynski, Projective Differential Geometry of Curves and Ruled Surfaces,, Chelsea Publishing Co., (1962).
|
[30] |
I. Zelenko and C. Li, Differential geometry of curves in Lagrange Grassmannians with given Young diagram,, Differential Geom. Appl., 27 (2009), 723.
doi: 10.1016/j.difgeo.2009.07.002. |
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