The projective symplectic geometry of higher order variational problems: Minimality conditions

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September 2016, 8(3): 305-322. doi: 10.3934/jgm.2016009

The projective symplectic geometry of higher order variational problems: Minimality conditions

Carlos Durán ^1, and Diego Otero ^1,

1.	Departamento de Matemática, UFPR, Setor de Ciências Exatas, Centro Politécnico, Caixa Postal 019081, CEP 81531-990, Curitiba, Brazil, Brazil

Received October 2015 Revised April 2016 Published September 2016

Abstract
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We associate curves of isotropic, Lagrangian and coisotropic subspaces to higher order, one parameter variational problems. Minimality and conjugacy properties of extremals are described in terms self-intersections of these curves.

Keywords: isotropic subspaces, Lagrangian Grassmannian., Calculus of variations.

Mathematics Subject Classification: Primary: 49K05; Secondary: 53D1.

Citation: Carlos Durán, Diego Otero. The projective symplectic geometry of higher order variational problems: Minimality conditions. Journal of Geometric Mechanics, 2016, 8 (3) : 305-322. doi: 10.3934/jgm.2016009

References:

[1]	A. Agrachev, D. Barilari and L. Rizzi, Curvature: A variational approach,, Memoirs of the AMS, ().
[2]	J. C. Álvarez Paiva and C. E. Durán, Geometric invariants of fanning curves, Adv. in Appl. Math., 42 (2009), 290-312. doi: 10.1016/j.aam.2006.07.008.
[3]	I. M. Anderson, The Variational Bicomplex,, Cambridge University Press, ().
[4]	G. Cimmino, Estensione dell'identita di Picone alia pi generale equazione differenziale lineare ordinaria autoaggiuntar, R. Accad. Naz. Lincei, 28 (1939), 354-364.
[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), 565-587. doi: 10.1017/S0305004100064501.
[7]	C. E. Durán and C. Peixoto, Geometry of fanning curves in divisible Grassmannians,, Differ. Geom. Appl., ().
[8]	C. E. Durán, J. C. Eidam and D. Otero, The projective symplectic geometry of higher order variational problems: Index theory,, work in progress., ().
[9]	M. S. P. Eastham, The Picone identity for self-adjoint differential equations of even order, Mathematika, 20 (1973), 197-200. doi: 10.1112/S0025579300004769.
[10]	S. Easwaran, Quadratic functionals of $n$-th order, Canad. Math. Bull., 19 (1976), 159-167. doi: 10.4153/CMB-1976-024-6.
[11]	I. Gelfand and S. Fomin, Calculus of Variations, Dover, 2000.
[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), 278-296. doi: 10.1007/s00498-003-0139-3.
[14]	M. A. Javaloyes and H. Vitório, Zermelo navigation in pseudo-Finsler metrics,, preprint, ().
[15]	K. Kreith, A picone identity for first order differential systems, J. Math. Anal. Appl., 31 (1970), 297-308. 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), 309-322. doi: 10.1090/S0002-9947-1970-0264485-1.
[18]	M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory, Elsevier, 1985.
[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), 375-400. 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), 299-340. doi: 10.1016/0040-9383(63)90013-2.
[22]	R. Palais, The Morse lemma for Banach spaces, Bull. Amer. Math. Soc., 75 (1969), 968-971. doi: 10.1090/S0002-9904-1969-12318-9.
[23]	R. Palais, Foundations of Global Non-linear Analysis, Benjamin and Co., New York, 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), 1-141.
[25]	P. D. Prieto-Martínez and N. Romón-Roy, Higher-order mechanics: Variational principles and other topics, J. Geom. Mech., 5 (2013), 493-510. doi: 10.3934/jgm.2013.5.493.
[26]	R. Ruggiero, Dynamics and Global Geometry of Manifolds without Conjugate Points, Sociedade Brasileira de Matemática, Rio de Janeiro, 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), 1295-1298.
[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), 723-742. doi: 10.1016/j.difgeo.2009.07.002.

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References:

[1]	A. Agrachev, D. Barilari and L. Rizzi, Curvature: A variational approach,, Memoirs of the AMS, ().
[2]	J. C. Álvarez Paiva and C. E. Durán, Geometric invariants of fanning curves, Adv. in Appl. Math., 42 (2009), 290-312. doi: 10.1016/j.aam.2006.07.008.
[3]	I. M. Anderson, The Variational Bicomplex,, Cambridge University Press, ().
[4]	G. Cimmino, Estensione dell'identita di Picone alia pi generale equazione differenziale lineare ordinaria autoaggiuntar, R. Accad. Naz. Lincei, 28 (1939), 354-364.
[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), 565-587. doi: 10.1017/S0305004100064501.
[7]	C. E. Durán and C. Peixoto, Geometry of fanning curves in divisible Grassmannians,, Differ. Geom. Appl., ().
[8]	C. E. Durán, J. C. Eidam and D. Otero, The projective symplectic geometry of higher order variational problems: Index theory,, work in progress., ().
[9]	M. S. P. Eastham, The Picone identity for self-adjoint differential equations of even order, Mathematika, 20 (1973), 197-200. doi: 10.1112/S0025579300004769.
[10]	S. Easwaran, Quadratic functionals of $n$-th order, Canad. Math. Bull., 19 (1976), 159-167. doi: 10.4153/CMB-1976-024-6.
[11]	I. Gelfand and S. Fomin, Calculus of Variations, Dover, 2000.
[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), 278-296. doi: 10.1007/s00498-003-0139-3.
[14]	M. A. Javaloyes and H. Vitório, Zermelo navigation in pseudo-Finsler metrics,, preprint, ().
[15]	K. Kreith, A picone identity for first order differential systems, J. Math. Anal. Appl., 31 (1970), 297-308. 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), 309-322. doi: 10.1090/S0002-9947-1970-0264485-1.
[18]	M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory, Elsevier, 1985.
[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), 375-400. 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), 299-340. doi: 10.1016/0040-9383(63)90013-2.
[22]	R. Palais, The Morse lemma for Banach spaces, Bull. Amer. Math. Soc., 75 (1969), 968-971. doi: 10.1090/S0002-9904-1969-12318-9.
[23]	R. Palais, Foundations of Global Non-linear Analysis, Benjamin and Co., New York, 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), 1-141.
[25]	P. D. Prieto-Martínez and N. Romón-Roy, Higher-order mechanics: Variational principles and other topics, J. Geom. Mech., 5 (2013), 493-510. doi: 10.3934/jgm.2013.5.493.
[26]	R. Ruggiero, Dynamics and Global Geometry of Manifolds without Conjugate Points, Sociedade Brasileira de Matemática, Rio de Janeiro, 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), 1295-1298.
[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), 723-742. doi: 10.1016/j.difgeo.2009.07.002.

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