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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

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, ().   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.  Google Scholar

[3]

I. M. Anderson, The Variational Bicomplex,, Cambridge University Press, ().   Google Scholar

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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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. Easwaran, Quadratic functionals of $n$-th order,, Canad. Math. Bull., 19 (1976), 159.  doi: 10.4153/CMB-1976-024-6.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

A. Kriegl and P. Michor, The Convenient Setting of Global Analysis,, Math. Surveys and Monogr., (1997).  doi: 10.1090/surv/053.  Google Scholar

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W. Leighton, Quadratic functionals of second order,, Trans. Amer. Math. Soc., 151 (1970), 309.  doi: 10.1090/S0002-9947-1970-0264485-1.  Google Scholar

[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.  Google Scholar

[20]

G. Paternain, Geodesic Flows,, Birkhauser, (1999).  doi: 10.1007/978-1-4612-1600-1.  Google Scholar

[21]

R. Palais, Morse theory on hilbert manifolds,, Topology, 2 (1963), 299.  doi: 10.1016/0040-9383(63)90013-2.  Google Scholar

[22]

R. Palais, The Morse lemma for Banach spaces,, Bull. Amer. Math. Soc., 75 (1969), 968.  doi: 10.1090/S0002-9904-1969-12318-9.  Google Scholar

[23]

R. Palais, Foundations of Global Non-linear Analysis,, Benjamin and Co., (1968).   Google Scholar

[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.  Google Scholar

[26]

R. Ruggiero, Dynamics and Global Geometry of Manifolds without Conjugate Points,, Sociedade Brasileira de Matemática, (2007).   Google Scholar

[27]

D. J. Saunders, The Geometry of Jet Bundles,, London Math. Soc. Lecture Note Ser., (1989).  doi: 10.1017/CBO9780511526411.  Google Scholar

[28]

W. Tulczyjew, Sur la différentiele de Lagrange,, C. R. Math. Acad. Sci. Paris, 280 (1975), 1295.   Google Scholar

[29]

E. J. Wilczynski, Projective Differential Geometry of Curves and Ruled Surfaces,, Chelsea Publishing Co., (1962).   Google Scholar

[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.  Google Scholar

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.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[10]

S. Easwaran, Quadratic functionals of $n$-th order,, Canad. Math. Bull., 19 (1976), 159.  doi: 10.4153/CMB-1976-024-6.  Google Scholar

[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).   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

A. Kriegl and P. Michor, The Convenient Setting of Global Analysis,, Math. Surveys and Monogr., (1997).  doi: 10.1090/surv/053.  Google Scholar

[17]

W. Leighton, Quadratic functionals of second order,, Trans. Amer. Math. Soc., 151 (1970), 309.  doi: 10.1090/S0002-9947-1970-0264485-1.  Google Scholar

[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.  Google Scholar

[20]

G. Paternain, Geodesic Flows,, Birkhauser, (1999).  doi: 10.1007/978-1-4612-1600-1.  Google Scholar

[21]

R. Palais, Morse theory on hilbert manifolds,, Topology, 2 (1963), 299.  doi: 10.1016/0040-9383(63)90013-2.  Google Scholar

[22]

R. Palais, The Morse lemma for Banach spaces,, Bull. Amer. Math. Soc., 75 (1969), 968.  doi: 10.1090/S0002-9904-1969-12318-9.  Google Scholar

[23]

R. Palais, Foundations of Global Non-linear Analysis,, Benjamin and Co., (1968).   Google Scholar

[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.  Google Scholar

[26]

R. Ruggiero, Dynamics and Global Geometry of Manifolds without Conjugate Points,, Sociedade Brasileira de Matemática, (2007).   Google Scholar

[27]

D. J. Saunders, The Geometry of Jet Bundles,, London Math. Soc. Lecture Note Ser., (1989).  doi: 10.1017/CBO9780511526411.  Google Scholar

[28]

W. Tulczyjew, Sur la différentiele de Lagrange,, C. R. Math. Acad. Sci. Paris, 280 (1975), 1295.   Google Scholar

[29]

E. J. Wilczynski, Projective Differential Geometry of Curves and Ruled Surfaces,, Chelsea Publishing Co., (1962).   Google Scholar

[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.  Google Scholar

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