Higher-order mechanics: Variational principles and other topics

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December 2013, 5(4): 493-510. doi: 10.3934/jgm.2013.5.493

Higher-order mechanics: Variational principles and other topics

Pedro D. Prieto-Martínez ^1, and Narciso Román-Roy ^1,

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

Received March 2013 Revised March 2013 Published December 2013

Abstract
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After reviewing the Lagrangian-Hamiltonian unified formalism (i.e, the Skinner-Rusk formalism) for higher-order (non-autonomous) dynamical systems, we state a unified geometrical version of the Variational Principles which allows us to derive the Lagrangian and Hamiltonian equations for these kinds of systems. Then, the standard Lagrangian and Hamiltonian formulations of these principles and the corresponding dynamical equations are recovered from this unified framework..

Keywords: Skinner-Rusk formulation, Variational Principles, non-autonomous sytems., Lagrangian and Hamiltonian formalisms, Higher-order mechanics.

Mathematics Subject Classification: Primary: 35A15, 37B55; Secondary: 70H5.

Citation: Pedro D. Prieto-Martínez, Narciso Román-Roy. Higher-order mechanics: Variational principles and other topics. Journal of Geometric Mechanics, 2013, 5 (4) : 493-510. doi: 10.3934/jgm.2013.5.493

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).
[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).
[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.
[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.
[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 $TQ \oplus TQ$,, 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).
[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).
[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.
[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.
[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 $TQ \oplus TQ$,, 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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