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. 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.

[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., ().

[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. 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).

[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, ().

[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).

[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.

[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, ().

[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, ().

[4]

G. Cimmino, Estensione dell'identita di Picone alia pi generale equazione differenziale lineare ordinaria autoaggiuntar,, R. Accad. Naz. Lincei, 28 (1939), 354.

[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., ().

[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. 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).

[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, ().

[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).

[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.

[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.

[1]

Daniel Faraco, Jan Kristensen. Compactness versus regularity in the calculus of variations. Discrete & Continuous Dynamical Systems - B, 2012, 17 (2) : 473-485. doi: 10.3934/dcdsb.2012.17.473

[2]

Bernard Dacorogna, Giovanni Pisante, Ana Margarida Ribeiro. On non quasiconvex problems of the calculus of variations. Discrete & Continuous Dynamical Systems - A, 2005, 13 (4) : 961-983. doi: 10.3934/dcds.2005.13.961

[3]

Felix Sadyrbaev. Nonlinear boundary value problems of the calculus of variations. Conference Publications, 2003, 2003 (Special) : 760-770. doi: 10.3934/proc.2003.2003.760

[4]

Delfim F. M. Torres. Proper extensions of Noether's symmetry theorem for nonsmooth extremals of the calculus of variations. Communications on Pure & Applied Analysis, 2004, 3 (3) : 491-500. doi: 10.3934/cpaa.2004.3.491

[5]

Nuno R. O. Bastos, Rui A. C. Ferreira, Delfim F. M. Torres. Necessary optimality conditions for fractional difference problems of the calculus of variations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 417-437. doi: 10.3934/dcds.2011.29.417

[6]

Agnieszka B. Malinowska, Delfim F. M. Torres. Euler-Lagrange equations for composition functionals in calculus of variations on time scales. Discrete & Continuous Dynamical Systems - A, 2011, 29 (2) : 577-593. doi: 10.3934/dcds.2011.29.577

[7]

Gisella Croce, Nikos Katzourakis, Giovanni Pisante. $\mathcal{D}$-solutions to the system of vectorial Calculus of Variations in $L^\infty$ via the singular value problem. Discrete & Continuous Dynamical Systems - A, 2017, 37 (12) : 6165-6181. doi: 10.3934/dcds.2017266

[8]

Ioan Bucataru, Matias F. Dahl. Semi-basic 1-forms and Helmholtz conditions for the inverse problem of the calculus of variations. Journal of Geometric Mechanics, 2009, 1 (2) : 159-180. doi: 10.3934/jgm.2009.1.159

[9]

Hans Josef Pesch. Carathéodory's royal road of the calculus of variations: Missed exits to the maximum principle of optimal control theory. Numerical Algebra, Control & Optimization, 2013, 3 (1) : 161-173. doi: 10.3934/naco.2013.3.161

[10]

Ivar Ekeland. From Frank Ramsey to René Thom: A classical problem in the calculus of variations leading to an implicit differential equation. Discrete & Continuous Dynamical Systems - A, 2010, 28 (3) : 1101-1119. doi: 10.3934/dcds.2010.28.1101

[11]

Nikos Katzourakis. Nonuniqueness in vector-valued calculus of variations in $L^\infty$ and some Linear elliptic systems. Communications on Pure & Applied Analysis, 2015, 14 (1) : 313-327. doi: 10.3934/cpaa.2015.14.313

[12]

Michael Cranston, Benjamin Gess, Michael Scheutzow. Weak synchronization for isotropic flows. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3003-3014. doi: 10.3934/dcdsb.2016084

[13]

Fabrizio Colombo, Graziano Gentili, Irene Sabadini and Daniele C. Struppa. A functional calculus in a noncommutative setting. Electronic Research Announcements, 2007, 14: 60-68. doi: 10.3934/era.2007.14.60

[14]

Frank Sottile. The special Schubert calculus is real. Electronic Research Announcements, 1999, 5: 35-39.

[15]

Guillaume Bal, Ian Langmore, François Monard. Inverse transport with isotropic sources and angularly averaged measurements. Inverse Problems & Imaging, 2008, 2 (1) : 23-42. doi: 10.3934/ipi.2008.2.23

[16]

Nicola Sansonetto, Daniele Sepe. Twisted isotropic realisations of twisted Poisson structures. Journal of Geometric Mechanics, 2013, 5 (2) : 233-256. doi: 10.3934/jgm.2013.5.233

[17]

Géry de Saxcé. Modelling contact with isotropic and anisotropic friction by the bipotential approach. Discrete & Continuous Dynamical Systems - S, 2016, 9 (2) : 409-425. doi: 10.3934/dcdss.2016004

[18]

Mauro Fabrizio, Claudio Giorgi, Angelo Morro. Isotropic-nematic phase transitions in liquid crystals. Discrete & Continuous Dynamical Systems - S, 2011, 4 (3) : 565-579. doi: 10.3934/dcdss.2011.4.565

[19]

Chi-Kwong Fok. Picard group of isotropic realizations of twisted Poisson manifolds. Journal of Geometric Mechanics, 2016, 8 (2) : 179-197. doi: 10.3934/jgm.2016003

[20]

Marc Briane. Isotropic realizability of electric fields around critical points. Discrete & Continuous Dynamical Systems - B, 2014, 19 (2) : 353-372. doi: 10.3934/dcdsb.2014.19.353

2016 Impact Factor: 0.857

Metrics

  • PDF downloads (1)
  • HTML views (0)
  • Cited by (2)

Other articles
by authors

[Back to Top]