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

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

Received  October 2015 Revised  April 2016 Published  September 2016

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.
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.

show all references

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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