American Institute of Mathematical Sciences

Advanced
July  2002, 8(3): 697-724. doi: 10.3934/dcds.2002.8.697

Morse theory for the travel time brachistochrones in stationary spacetimes

Fabio Giannoni 1, , Paolo Piccione 2, and  Daniel V. Tausk 2,

1. 

Dipartimento di Matematica e Fisica, Universitá di Camerino, Italy
2. 

Departamento de Matemática, Universidade de São Paulo, Brazil, Brazil

Received  January 2001 Revised  November 2001 Published  April 2002

The travel time brachistochrone curves in a general relativistic framework are timelike curves, satisfying a suitable conservation law with respect to a an observer field, that are stationary points of the travel time functional. We develop a global variational theory for brachistochrones joining an event p and the worldline of an observer $\gamma$ in a stationary spacetime $\mathcal M$.
Keywords: Morse theory, variational theory..
Mathematics Subject Classification: 58E05, 53C22, 83Cx.
Citation: Fabio Giannoni, Paolo Piccione, Daniel V. Tausk. Morse theory for the travel time brachistochrones in stationary spacetimes. Discrete and Continuous Dynamical Systems, 2002, 8 (3) : 697-724. doi: 10.3934/dcds.2002.8.697
[1]

Philip Schrader. Morse theory for elastica. Journal of Geometric Mechanics, 2016, 8 (2) : 235-256. doi: 10.3934/jgm.2016006
[2]

Jijiang Sun, Shiwang Ma. Nontrivial solutions for Kirchhoff type equations via Morse theory. Communications on Pure and Applied Analysis, 2014, 13 (2) : 483-494. doi: 10.3934/cpaa.2014.13.483
[3]

Takeshi Fukao, Nobuyuki Kenmochi. Abstract theory of variational inequalities and Lagrange multipliers. Conference Publications, 2013, 2013 (special) : 237-246. doi: 10.3934/proc.2013.2013.237
[4]

Silvia Cingolani, Mónica Clapp. Symmetric semiclassical states to a magnetic nonlinear Schrödinger equation via equivariant Morse theory. Communications on Pure and Applied Analysis, 2010, 9 (5) : 1263-1281. doi: 10.3934/cpaa.2010.9.1263
[5]

Qiong Meng, X. H. Tang. Multiple solutions of second-order ordinary differential equation via Morse theory. Communications on Pure and Applied Analysis, 2012, 11 (3) : 945-958. doi: 10.3934/cpaa.2012.11.945
[6]

Massimiliano Ferrara, Giovanni Molica Bisci, Binlin Zhang. Existence of weak solutions for non-local fractional problems via Morse theory. Discrete and Continuous Dynamical Systems - B, 2014, 19 (8) : 2483-2499. doi: 10.3934/dcdsb.2014.19.2483
[7]

Claudianor O. Alves, Rodrigo C. M. Nemer, Sergio H. Monari Soares. The use of the Morse theory to estimate the number of nontrivial solutions of a nonlinear Schrödinger equation with a magnetic field. Communications on Pure and Applied Analysis, 2021, 20 (1) : 449-465. doi: 10.3934/cpaa.2020276
[8]

Carlos F. Daganzo. On the variational theory of traffic flow: well-posedness, duality and applications. Networks and Heterogeneous Media, 2006, 1 (4) : 601-619. doi: 10.3934/nhm.2006.1.601
[9]

Suleyman Tek. Some classes of surfaces in $\mathbb{R}^3$ and $\M_3$ arising from soliton theory and a variational principle. Conference Publications, 2009, 2009 (Special) : 761-770. doi: 10.3934/proc.2009.2009.761
[10]

Robert M. Strain. Coordinates in the relativistic Boltzmann theory. Kinetic and Related Models, 2011, 4 (1) : 345-359. doi: 10.3934/krm.2011.4.345
[11]

Krešimir Burazin, Marko Vrdoljak. Homogenisation theory for Friedrichs systems. Communications on Pure and Applied Analysis, 2014, 13 (3) : 1017-1044. doi: 10.3934/cpaa.2014.13.1017
[12]

Ryszard Rudnicki. An ergodic theory approach to chaos. Discrete and Continuous Dynamical Systems, 2015, 35 (2) : 757-770. doi: 10.3934/dcds.2015.35.757
[13]

Thierry de la Rue. An introduction to joinings in ergodic theory. Discrete and Continuous Dynamical Systems, 2006, 15 (1) : 121-142. doi: 10.3934/dcds.2006.15.121
[14]

Mads R. Bisgaard. Mather theory and symplectic rigidity. Journal of Modern Dynamics, 2019, 15: 165-207. doi: 10.3934/jmd.2019018
[15]

Luis Barreira. Dimension theory of flows: A survey. Discrete and Continuous Dynamical Systems - B, 2015, 20 (10) : 3345-3362. doi: 10.3934/dcdsb.2015.20.3345
[16]

Jianjun Tian, Xiao-Song Lin. Colored coalescent theory. Conference Publications, 2005, 2005 (Special) : 833-845. doi: 10.3934/proc.2005.2005.833
[17]

Augusto Visintin. An extension of the Fitzpatrick theory. Communications on Pure and Applied Analysis, 2014, 13 (5) : 2039-2058. doi: 10.3934/cpaa.2014.13.2039
[18]

Badam Ulemj, Enkhbat Rentsen, Batchimeg Tsendpurev. Application of survival theory in taxation. Journal of Industrial and Management Optimization, 2021, 17 (5) : 2573-2578. doi: 10.3934/jimo.2020083
[19]

Andrew D. Lewis, David R. Tyner. Geometric Jacobian linearization and LQR theory. Journal of Geometric Mechanics, 2010, 2 (4) : 397-440. doi: 10.3934/jgm.2010.2.397
[20]

Jonathan H. Tu, Clarence W. Rowley, Dirk M. Luchtenburg, Steven L. Brunton, J. Nathan Kutz. On dynamic mode decomposition: Theory and applications. Journal of Computational Dynamics, 2014, 1 (2) : 391-421. doi: 10.3934/jcd.2014.1.391

2020 Impact Factor: 1.392

Tools

Metrics

  • PDF downloads (65)
  • HTML views (0)
  • Cited by (1)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy