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Higher-order mechanics: Variational principles and other topics
1. | Departamento de Matemática Aplicada IV, Universitat Politècnica de Catalunya-BarcelonaTech, Campus Norte, Ed. C-3. C/ Jordi Girona 1, E-08034 Barcelona, Spain, Spain |
References:
[1] |
V. Aldaya and J. A. de Azcárraga, Variational principles on $r-th$ order jets of fibre bundles in field theory,, J. Math. Phys., 19 (1978), 1869.
doi: 10.1063/1.523904. |
[2] |
M. Barbero-Liñán, A. Echeverría-Enrí quez, D. Martín de Diego, M. C. Muñ oz-Lecanda and N. Román-Roy, Unified formalism for non-autonomous mechanical systems,, J. Math. Phys., 49 (2008). Google Scholar |
[3] |
M. Barbero-Liñán, A. Echeverría-Enrí quit, D. Martín de Diego, M. C. Muñoz-Lecanda and N. Román-Roy, Skinner-Rusk unified formalism for optimal control systems and applications,, J. Phys. A, 40 (2007), 12071.
doi: 10.1088/1751-8113/40/40/005. |
[4] |
C. M. Campos, M. de León, D. Martín de Diego and J. Vankerschaver, Unambigous formalism for higher-order Lagrangian field theories,, J. Phys. A, 42 (2009).
doi: 10.1088/1751-8113/42/47/475207. |
[5] |
F. Cantrijn, M. Crampin and W. Sarlet, Higher-order differential equations and higher-order Lagrangian mechanics,, Math. Proc. Cambridge Philos. Soc., 99 (1986), 565.
doi: 10.1017/S0305004100064501. |
[6] |
L. Colombo, D. Marín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometric approach,, J. Math. Phys., 51 (2010).
doi: 10.1063/1.3456158. |
[7] |
J. Cortés, S. Martínez and F. Cantrijn, Skinner-Rusk approach to time-dependent mechanics,, Phys. Lett. A, 300 (2002), 250.
doi: 10.1016/S0375-9601(02)00777-6. |
[8] |
M. de León, J. Marín-Solano, J. C. Marrero, M. C. Muñoz-Lecanda and N. Román-Roy, Singular Lagrangian systems on jet bundles,, Fortschr. Phys., 50 (2002), 105.
doi: 10.1002/1521-3978(200203)50:2<105::AID-PROP105>3.0.CO;2-N. |
[9] |
M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Math. Studies, (1985). Google Scholar |
[10] |
M. de León and P. R. Rodrigues, Higher-order almost tangent geometry and non-autonomous Lagrangian dynamics,, in Proc. Winter School on Geometry and Physics (Srní, (1987), 157. Google Scholar |
[11] |
A. Echeverría-Enríquez, M. de León, M. C. Muñoz-Lecanda and N. Román-Roy, Extended Hamiltonian systems in multisymplectic field theories,, J. Math. Phys., 48 (2007).
doi: 10.1063/1.2801875. |
[12] |
A. Echeverría-Enríquez, C. López, J. Marín-Solano, M. C. Muñoz-Lecanda and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for field theory,, J. Math. Phys., 45 (2004), 360.
doi: 10.1063/1.1628384. |
[13] |
P. L. García, The Poincaré-Cartan invariant in the calculus of variations,, in Symposia Mathematica, (1973), 219.
|
[14] |
P. L. García and J. Muñoz, On the geometrical structure of higher order variational calculus,, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 117 (1983), 127.
|
[15] |
H. Goldschmidt and S. Sternberg, The Hamilton-Cartan formalism in the calculus of variations,, Ann. Inst. Fourier (Grenoble), 23 (1973), 203.
doi: 10.5802/aif.451. |
[16] |
M. J. Gotay, J. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints,, J. Math. Phys., 19 (1978), 2388.
doi: 10.1063/1.523597. |
[17] |
X. Gràcia, J. M. Pons and N. Román-Roy, Higher-order Lagrangian systems: Geometric-structures, dynamics and constraints,, J. Math. Phys., 32 (1991), 2744.
doi: 10.1063/1.529066. |
[18] |
X. Gràcia, J. M. Pons and N. Román-Roy, Higher-order conditions for singular Lagrangian systems,, J. Phys. A: Math. Gen., 25 (1992), 1981. Google Scholar |
[19] |
O. Krupková, Higher-order mechanical systems with constraints,, J. Math. Phys., 41 (2000), 5304.
doi: 10.1063/1.533411. |
[20] |
P. D. Prieto-Martínez and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for autonomous higher-order dynamical systems,, J. Phys. A, 44 (2011).
doi: 10.1088/1751-8113/44/38/385203. |
[21] |
P. D. Prieto-Martínez and N. Román-Roy, Unified formalism for higher-order non-autonomous dynamical systems,, J. Math. Phys., 53 (2012).
doi: 10.1063/1.3692326. |
[22] |
D. J. Saunders, An alternative approach to the Cartan form in Lagrangian field theories,, J. Phys. A, 20 (1987), 339.
doi: 10.1088/0305-4470/20/2/019. |
[23] |
D. J. Saunders, The Geometry of Jet Bundles,, London Math. Soc., (1989).
doi: 10.1017/CBO9780511526411. |
[24] |
R. Skinner and R. Rusk, Generalized Hamiltonian dynamics. I. Formulation on $T*Q \oplus TQ$,, J. Math. Phys., 24 (1983), 2589.
doi: 10.1063/1.525654. |
[25] |
L. Vitagliano, The Lagrangian-Hamiltonian formalism for higher-order field theories,, J. Geom. Phys., 60 (2010), 857.
doi: 10.1016/j.geomphys.2010.02.003. |
show all references
References:
[1] |
V. Aldaya and J. A. de Azcárraga, Variational principles on $r-th$ order jets of fibre bundles in field theory,, J. Math. Phys., 19 (1978), 1869.
doi: 10.1063/1.523904. |
[2] |
M. Barbero-Liñán, A. Echeverría-Enrí quez, D. Martín de Diego, M. C. Muñ oz-Lecanda and N. Román-Roy, Unified formalism for non-autonomous mechanical systems,, J. Math. Phys., 49 (2008). Google Scholar |
[3] |
M. Barbero-Liñán, A. Echeverría-Enrí quit, D. Martín de Diego, M. C. Muñoz-Lecanda and N. Román-Roy, Skinner-Rusk unified formalism for optimal control systems and applications,, J. Phys. A, 40 (2007), 12071.
doi: 10.1088/1751-8113/40/40/005. |
[4] |
C. M. Campos, M. de León, D. Martín de Diego and J. Vankerschaver, Unambigous formalism for higher-order Lagrangian field theories,, J. Phys. A, 42 (2009).
doi: 10.1088/1751-8113/42/47/475207. |
[5] |
F. Cantrijn, M. Crampin and W. Sarlet, Higher-order differential equations and higher-order Lagrangian mechanics,, Math. Proc. Cambridge Philos. Soc., 99 (1986), 565.
doi: 10.1017/S0305004100064501. |
[6] |
L. Colombo, D. Marín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometric approach,, J. Math. Phys., 51 (2010).
doi: 10.1063/1.3456158. |
[7] |
J. Cortés, S. Martínez and F. Cantrijn, Skinner-Rusk approach to time-dependent mechanics,, Phys. Lett. A, 300 (2002), 250.
doi: 10.1016/S0375-9601(02)00777-6. |
[8] |
M. de León, J. Marín-Solano, J. C. Marrero, M. C. Muñoz-Lecanda and N. Román-Roy, Singular Lagrangian systems on jet bundles,, Fortschr. Phys., 50 (2002), 105.
doi: 10.1002/1521-3978(200203)50:2<105::AID-PROP105>3.0.CO;2-N. |
[9] |
M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Math. Studies, (1985). Google Scholar |
[10] |
M. de León and P. R. Rodrigues, Higher-order almost tangent geometry and non-autonomous Lagrangian dynamics,, in Proc. Winter School on Geometry and Physics (Srní, (1987), 157. Google Scholar |
[11] |
A. Echeverría-Enríquez, M. de León, M. C. Muñoz-Lecanda and N. Román-Roy, Extended Hamiltonian systems in multisymplectic field theories,, J. Math. Phys., 48 (2007).
doi: 10.1063/1.2801875. |
[12] |
A. Echeverría-Enríquez, C. López, J. Marín-Solano, M. C. Muñoz-Lecanda and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for field theory,, J. Math. Phys., 45 (2004), 360.
doi: 10.1063/1.1628384. |
[13] |
P. L. García, The Poincaré-Cartan invariant in the calculus of variations,, in Symposia Mathematica, (1973), 219.
|
[14] |
P. L. García and J. Muñoz, On the geometrical structure of higher order variational calculus,, Atti Accad. Sci. Torino Cl. Sci. Fis. Mat. Natur., 117 (1983), 127.
|
[15] |
H. Goldschmidt and S. Sternberg, The Hamilton-Cartan formalism in the calculus of variations,, Ann. Inst. Fourier (Grenoble), 23 (1973), 203.
doi: 10.5802/aif.451. |
[16] |
M. J. Gotay, J. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints,, J. Math. Phys., 19 (1978), 2388.
doi: 10.1063/1.523597. |
[17] |
X. Gràcia, J. M. Pons and N. Román-Roy, Higher-order Lagrangian systems: Geometric-structures, dynamics and constraints,, J. Math. Phys., 32 (1991), 2744.
doi: 10.1063/1.529066. |
[18] |
X. Gràcia, J. M. Pons and N. Román-Roy, Higher-order conditions for singular Lagrangian systems,, J. Phys. A: Math. Gen., 25 (1992), 1981. Google Scholar |
[19] |
O. Krupková, Higher-order mechanical systems with constraints,, J. Math. Phys., 41 (2000), 5304.
doi: 10.1063/1.533411. |
[20] |
P. D. Prieto-Martínez and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for autonomous higher-order dynamical systems,, J. Phys. A, 44 (2011).
doi: 10.1088/1751-8113/44/38/385203. |
[21] |
P. D. Prieto-Martínez and N. Román-Roy, Unified formalism for higher-order non-autonomous dynamical systems,, J. Math. Phys., 53 (2012).
doi: 10.1063/1.3692326. |
[22] |
D. J. Saunders, An alternative approach to the Cartan form in Lagrangian field theories,, J. Phys. A, 20 (1987), 339.
doi: 10.1088/0305-4470/20/2/019. |
[23] |
D. J. Saunders, The Geometry of Jet Bundles,, London Math. Soc., (1989).
doi: 10.1017/CBO9780511526411. |
[24] |
R. Skinner and R. Rusk, Generalized Hamiltonian dynamics. I. Formulation on $T*Q \oplus TQ$,, J. Math. Phys., 24 (1983), 2589.
doi: 10.1063/1.525654. |
[25] |
L. Vitagliano, The Lagrangian-Hamiltonian formalism for higher-order field theories,, J. Geom. Phys., 60 (2010), 857.
doi: 10.1016/j.geomphys.2010.02.003. |
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