March  2018, 10(1): 69-92. doi: 10.3934/jgm.2018003

The projective Cartan-Klein geometry of the Helmholtz conditions

Setor de Ciências Exatas, Centro Politécnico, Caixa Postal 019081, CEP 81531-990, Curitiba, PR, Brazil

* Corresponding author

Received  April 2016 Revised  August 2017 Published  December 2017

We show that the Helmholtz conditions characterizing differential equations arising from variational problems can be expressed in terms of invariants of curves in a suitable Grassmann manifold.

Citation: Carlos Durán, Diego Otero. The projective Cartan-Klein geometry of the Helmholtz conditions. Journal of Geometric Mechanics, 2018, 10 (1) : 69-92. doi: 10.3934/jgm.2018003
References:
[1]

A. Agrachev, D. Barilari and L. Rizzi, Curvature: A variational approach, preprint, arXiv: 1306. 5318v5. Google Scholar

[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.  Google Scholar

[3]

I. Anderson, Introduction to variational bicomplex, Contemp. Math., 132 (1992), 51-73.   Google Scholar

[4]

M. de León and P. R. Rodrigues, The inverse problem of Lagrangian dynamics for higher-order differential equations -a geometrical approach, Inverse Prob., 8 (1992), 525-540.  doi: 10.1088/0266-5611/8/4/006.  Google Scholar

[5]

C. E. Durán and D. Otero, The projective symplectic geometry of higher order variational problems: Minimality conditions, J. Geom. Mech., 8 (2016), 305-322.  doi: 10.3934/jgm.2016009.  Google Scholar

[6]

C. E. Durán and C. Peixoto, Geometry of fanning curves in divisible grassmannians, Diff. Geom. Appl., 49 (2016), 447-472.  doi: 10.1016/j.difgeo.2016.10.001.  Google Scholar

[7]

C. E. Durán and H. Vitório, Moving planes, Jacobi curves and the dynamical approach to Finsler geometry, European Journal of Mathematics, 3 (2017), 1245-1274.   Google Scholar

[8]

H. Flanders, The Schwarzian as a curvature, J. Differential Geometry, 4 (1970), 515-519.  doi: 10.4310/jdg/1214429647.  Google Scholar

[9]

S. R. Garcia, The eigenstructure of complex symmetric operators, Recent Advances in Operator Theory, Operator Theory: Advances and Applications, 179 (2008), 169-183.   Google Scholar

[10]

I. Gelfand and S. Fomin, Calculus of Variations, Silverman Prentice-Hall, Inc., Englewood Cliffs, N. J. 1963.  Google Scholar

[11]

P. Piccione and D. V. Tausk, A students' guide to symplectic spaces, Grassmannians and Maslov index, Publicações Matemáticas do IMPA, Rio de Janeiro, 2008.  Google Scholar

[12]

W. SarletE. Engels and L. Y. Bahar, Time-dependent linear systems derivable from a variational principle, Int. J. Eng. Sci., 20 (1982), 55-66.  doi: 10.1016/0020-7225(82)90072-6.  Google Scholar

[13]

P. Vassiliou and I. Lisle, Geometric Approaches to Differential Equations, Australian Mathematical Society Lecture Series, Cambridge University Press, 2000. Google Scholar

[14]

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

show all references

References:
[1]

A. Agrachev, D. Barilari and L. Rizzi, Curvature: A variational approach, preprint, arXiv: 1306. 5318v5. Google Scholar

[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.  Google Scholar

[3]

I. Anderson, Introduction to variational bicomplex, Contemp. Math., 132 (1992), 51-73.   Google Scholar

[4]

M. de León and P. R. Rodrigues, The inverse problem of Lagrangian dynamics for higher-order differential equations -a geometrical approach, Inverse Prob., 8 (1992), 525-540.  doi: 10.1088/0266-5611/8/4/006.  Google Scholar

[5]

C. E. Durán and D. Otero, The projective symplectic geometry of higher order variational problems: Minimality conditions, J. Geom. Mech., 8 (2016), 305-322.  doi: 10.3934/jgm.2016009.  Google Scholar

[6]

C. E. Durán and C. Peixoto, Geometry of fanning curves in divisible grassmannians, Diff. Geom. Appl., 49 (2016), 447-472.  doi: 10.1016/j.difgeo.2016.10.001.  Google Scholar

[7]

C. E. Durán and H. Vitório, Moving planes, Jacobi curves and the dynamical approach to Finsler geometry, European Journal of Mathematics, 3 (2017), 1245-1274.   Google Scholar

[8]

H. Flanders, The Schwarzian as a curvature, J. Differential Geometry, 4 (1970), 515-519.  doi: 10.4310/jdg/1214429647.  Google Scholar

[9]

S. R. Garcia, The eigenstructure of complex symmetric operators, Recent Advances in Operator Theory, Operator Theory: Advances and Applications, 179 (2008), 169-183.   Google Scholar

[10]

I. Gelfand and S. Fomin, Calculus of Variations, Silverman Prentice-Hall, Inc., Englewood Cliffs, N. J. 1963.  Google Scholar

[11]

P. Piccione and D. V. Tausk, A students' guide to symplectic spaces, Grassmannians and Maslov index, Publicações Matemáticas do IMPA, Rio de Janeiro, 2008.  Google Scholar

[12]

W. SarletE. Engels and L. Y. Bahar, Time-dependent linear systems derivable from a variational principle, Int. J. Eng. Sci., 20 (1982), 55-66.  doi: 10.1016/0020-7225(82)90072-6.  Google Scholar

[13]

P. Vassiliou and I. Lisle, Geometric Approaches to Differential Equations, Australian Mathematical Society Lecture Series, Cambridge University Press, 2000. Google Scholar

[14]

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

[1]

Jaume Llibre, Luci Any Roberto. On the periodic solutions of a class of Duffing differential equations. Discrete & Continuous Dynamical Systems - A, 2013, 33 (1) : 277-282. doi: 10.3934/dcds.2013.33.277

[2]

Nizami A. Gasilov. Solving a system of linear differential equations with interval coefficients. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2739-2747. doi: 10.3934/dcdsb.2020203

[3]

Xianming Liu, Guangyue Han. A Wong-Zakai approximation of stochastic differential equations driven by a general semimartingale. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2499-2508. doi: 10.3934/dcdsb.2020192

[4]

Nhu N. Nguyen, George Yin. Stochastic partial differential equation models for spatially dependent predator-prey equations. Discrete & Continuous Dynamical Systems - B, 2020, 25 (1) : 117-139. doi: 10.3934/dcdsb.2019175

[5]

Bin Pei, Yong Xu, Yuzhen Bai. Convergence of p-th mean in an averaging principle for stochastic partial differential equations driven by fractional Brownian motion. Discrete & Continuous Dynamical Systems - B, 2020, 25 (3) : 1141-1158. doi: 10.3934/dcdsb.2019213

[6]

Xiaoming Wang. Quasi-periodic solutions for a class of second order differential equations with a nonlinear damping term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (3) : 543-556. doi: 10.3934/dcdss.2017027

[7]

Gheorghe Craciun, Abhishek Deshpande, Hyejin Jenny Yeon. Quasi-toric differential inclusions. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2343-2359. doi: 10.3934/dcdsb.2020181

[8]

Madalina Petcu, Roger Temam. The one dimensional shallow water equations with Dirichlet boundary conditions on the velocity. Discrete & Continuous Dynamical Systems - S, 2011, 4 (1) : 209-222. doi: 10.3934/dcdss.2011.4.209

[9]

Wolf-Jüergen Beyn, Janosch Rieger. The implicit Euler scheme for one-sided Lipschitz differential inclusions. Discrete & Continuous Dynamical Systems - B, 2010, 14 (2) : 409-428. doi: 10.3934/dcdsb.2010.14.409

[10]

Abdulrazzaq T. Abed, Azzam S. Y. Aladool. Applying particle swarm optimization based on Padé approximant to solve ordinary differential equation. Numerical Algebra, Control & Optimization, 2021  doi: 10.3934/naco.2021008

[11]

Zengyun Wang, Jinde Cao, Zuowei Cai, Lihong Huang. Finite-time stability of impulsive differential inclusion: Applications to discontinuous impulsive neural networks. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2677-2692. doi: 10.3934/dcdsb.2020200

[12]

Yuri Chekanov, Felix Schlenk. Notes on monotone Lagrangian twist tori. Electronic Research Announcements, 2010, 17: 104-121. doi: 10.3934/era.2010.17.104

[13]

Zhiming Guo, Zhi-Chun Yang, Xingfu Zou. Existence and uniqueness of positive solution to a non-local differential equation with homogeneous Dirichlet boundary condition---A non-monotone case. Communications on Pure & Applied Analysis, 2012, 11 (5) : 1825-1838. doi: 10.3934/cpaa.2012.11.1825

[14]

Alexander A. Davydov, Massimo Giulietti, Stefano Marcugini, Fernanda Pambianco. Linear nonbinary covering codes and saturating sets in projective spaces. Advances in Mathematics of Communications, 2011, 5 (1) : 119-147. doi: 10.3934/amc.2011.5.119

[15]

Samir Adly, Oanh Chau, Mohamed Rochdi. Solvability of a class of thermal dynamical contact problems with subdifferential conditions. Numerical Algebra, Control & Optimization, 2012, 2 (1) : 91-104. doi: 10.3934/naco.2012.2.91

[16]

Elvise Berchio, Filippo Gazzola, Dario Pierotti. Nodal solutions to critical growth elliptic problems under Steklov boundary conditions. Communications on Pure & Applied Analysis, 2009, 8 (2) : 533-557. doi: 10.3934/cpaa.2009.8.533

[17]

Valery Y. Glizer. Novel Conditions of Euclidean space controllability for singularly perturbed systems with input delay. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 307-320. doi: 10.3934/naco.2020027

[18]

Rabiaa Ouahabi, Nasr-Eddine Hamri. Design of new scheme adaptive generalized hybrid projective synchronization for two different chaotic systems with uncertain parameters. Discrete & Continuous Dynamical Systems - B, 2021, 26 (5) : 2361-2370. doi: 10.3934/dcdsb.2020182

[19]

Sergi Simon. Linearised higher variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4827-4854. doi: 10.3934/dcds.2014.34.4827

[20]

María J. Garrido-Atienza, Bohdan Maslowski, Jana  Šnupárková. Semilinear stochastic equations with bilinear fractional noise. Discrete & Continuous Dynamical Systems - B, 2016, 21 (9) : 3075-3094. doi: 10.3934/dcdsb.2016088

2019 Impact Factor: 0.649

Metrics

  • PDF downloads (138)
  • HTML views (427)
  • Cited by (0)

Other articles
by authors

[Back to Top]