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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

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), 1869-1875. doi: 10.1063/1.523904.  Google Scholar

[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), 062902. 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), 12071-12093. doi: 10.1088/1751-8113/40/40/005.  Google Scholar

[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), 475207, 24 pp. doi: 10.1088/1751-8113/42/47/475207.  Google Scholar

[5]

F. Cantrijn, M. Crampin and W. Sarlet, Higher-order differential equations and higher-order Lagrangian mechanics, Math. Proc. Cambridge Philos. Soc., 99 (1986), 565-587. doi: 10.1017/S0305004100064501.  Google Scholar

[6]

L. Colombo, D. Marín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometric approach, J. Math. Phys., 51 (2010), 083519, 24 pp. doi: 10.1063/1.3456158.  Google Scholar

[7]

J. Cortés, S. Martínez and F. Cantrijn, Skinner-Rusk approach to time-dependent mechanics, Phys. Lett. A, 300 (2002), 250-258. doi: 10.1016/S0375-9601(02)00777-6.  Google Scholar

[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), 105-169. doi: 10.1002/1521-3978(200203)50:2<105::AID-PROP105>3.0.CO;2-N.  Google Scholar

[9]

M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory, North-Holland Math. Studies, 112, Elsevier Science Publishers B. V., Amsterdam, 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), Rend. Circ. Mat. Palermo, 2, 1987, 157-171. 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), 112901, 30 pp. doi: 10.1063/1.2801875.  Google Scholar

[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), 360-380. doi: 10.1063/1.1628384.  Google Scholar

[13]

P. L. García, The Poincaré-Cartan invariant in the calculus of variations, in Symposia Mathematica, Vol. XIV (Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973), Academic Press, London, 1974, 219-246.  Google Scholar

[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), 127-147.  Google Scholar

[15]

H. Goldschmidt and S. Sternberg, The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier (Grenoble), 23 (1973), 203-267. doi: 10.5802/aif.451.  Google Scholar

[16]

M. J. Gotay, J. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints, J. Math. Phys., 19(11) (1978), 2388-2399. doi: 10.1063/1.523597.  Google Scholar

[17]

X. Gràcia, J. M. Pons and N. Román-Roy, Higher-order Lagrangian systems: Geometric-structures, dynamics and constraints, J. Math. Phys., 32 (1991), 2744-2763. doi: 10.1063/1.529066.  Google Scholar

[18]

X. Gràcia, J. M. Pons and N. Román-Roy, Higher-order conditions for singular Lagrangian systems, J. Phys. A: Math. Gen., 25 (1992), 1981-2004. Google Scholar

[19]

O. Krupková, Higher-order mechanical systems with constraints, J. Math. Phys., 41 (2000), 5304-5324. doi: 10.1063/1.533411.  Google Scholar

[20]

P. D. Prieto-Martínez and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for autonomous higher-order dynamical systems, J. Phys. A, 44 (2011), 385203, 35 pp. doi: 10.1088/1751-8113/44/38/385203.  Google Scholar

[21]

P. D. Prieto-Martínez and N. Román-Roy, Unified formalism for higher-order non-autonomous dynamical systems, J. Math. Phys., 53 (2012), 032901, 38 pp. doi: 10.1063/1.3692326.  Google Scholar

[22]

D. J. Saunders, An alternative approach to the Cartan form in Lagrangian field theories, J. Phys. A, 20 (1987), 339-349. doi: 10.1088/0305-4470/20/2/019.  Google Scholar

[23]

D. J. Saunders, The Geometry of Jet Bundles, London Math. Soc., Lect. Notes Series, 142, Cambridge Univ. Press, Cambridge, 1989. doi: 10.1017/CBO9780511526411.  Google Scholar

[24]

R. Skinner and R. Rusk, Generalized Hamiltonian dynamics. I. Formulation on $T*Q \oplus TQ$, J. Math. Phys., 24 (1983), 2589-2594. doi: 10.1063/1.525654.  Google Scholar

[25]

L. Vitagliano, The Lagrangian-Hamiltonian formalism for higher-order field theories, J. Geom. Phys., 60 (2010), 857-873. doi: 10.1016/j.geomphys.2010.02.003.  Google Scholar

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), 1869-1875. doi: 10.1063/1.523904.  Google Scholar

[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), 062902. 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), 12071-12093. doi: 10.1088/1751-8113/40/40/005.  Google Scholar

[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), 475207, 24 pp. doi: 10.1088/1751-8113/42/47/475207.  Google Scholar

[5]

F. Cantrijn, M. Crampin and W. Sarlet, Higher-order differential equations and higher-order Lagrangian mechanics, Math. Proc. Cambridge Philos. Soc., 99 (1986), 565-587. doi: 10.1017/S0305004100064501.  Google Scholar

[6]

L. Colombo, D. Marín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometric approach, J. Math. Phys., 51 (2010), 083519, 24 pp. doi: 10.1063/1.3456158.  Google Scholar

[7]

J. Cortés, S. Martínez and F. Cantrijn, Skinner-Rusk approach to time-dependent mechanics, Phys. Lett. A, 300 (2002), 250-258. doi: 10.1016/S0375-9601(02)00777-6.  Google Scholar

[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), 105-169. doi: 10.1002/1521-3978(200203)50:2<105::AID-PROP105>3.0.CO;2-N.  Google Scholar

[9]

M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory, North-Holland Math. Studies, 112, Elsevier Science Publishers B. V., Amsterdam, 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), Rend. Circ. Mat. Palermo, 2, 1987, 157-171. 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), 112901, 30 pp. doi: 10.1063/1.2801875.  Google Scholar

[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), 360-380. doi: 10.1063/1.1628384.  Google Scholar

[13]

P. L. García, The Poincaré-Cartan invariant in the calculus of variations, in Symposia Mathematica, Vol. XIV (Convegno di Geometria Simplettica e Fisica Matematica, INDAM, Rome, 1973), Academic Press, London, 1974, 219-246.  Google Scholar

[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), 127-147.  Google Scholar

[15]

H. Goldschmidt and S. Sternberg, The Hamilton-Cartan formalism in the calculus of variations, Ann. Inst. Fourier (Grenoble), 23 (1973), 203-267. doi: 10.5802/aif.451.  Google Scholar

[16]

M. J. Gotay, J. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints, J. Math. Phys., 19(11) (1978), 2388-2399. doi: 10.1063/1.523597.  Google Scholar

[17]

X. Gràcia, J. M. Pons and N. Román-Roy, Higher-order Lagrangian systems: Geometric-structures, dynamics and constraints, J. Math. Phys., 32 (1991), 2744-2763. doi: 10.1063/1.529066.  Google Scholar

[18]

X. Gràcia, J. M. Pons and N. Román-Roy, Higher-order conditions for singular Lagrangian systems, J. Phys. A: Math. Gen., 25 (1992), 1981-2004. Google Scholar

[19]

O. Krupková, Higher-order mechanical systems with constraints, J. Math. Phys., 41 (2000), 5304-5324. doi: 10.1063/1.533411.  Google Scholar

[20]

P. D. Prieto-Martínez and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for autonomous higher-order dynamical systems, J. Phys. A, 44 (2011), 385203, 35 pp. doi: 10.1088/1751-8113/44/38/385203.  Google Scholar

[21]

P. D. Prieto-Martínez and N. Román-Roy, Unified formalism for higher-order non-autonomous dynamical systems, J. Math. Phys., 53 (2012), 032901, 38 pp. doi: 10.1063/1.3692326.  Google Scholar

[22]

D. J. Saunders, An alternative approach to the Cartan form in Lagrangian field theories, J. Phys. A, 20 (1987), 339-349. doi: 10.1088/0305-4470/20/2/019.  Google Scholar

[23]

D. J. Saunders, The Geometry of Jet Bundles, London Math. Soc., Lect. Notes Series, 142, Cambridge Univ. Press, Cambridge, 1989. doi: 10.1017/CBO9780511526411.  Google Scholar

[24]

R. Skinner and R. Rusk, Generalized Hamiltonian dynamics. I. Formulation on $T*Q \oplus TQ$, J. Math. Phys., 24 (1983), 2589-2594. doi: 10.1063/1.525654.  Google Scholar

[25]

L. Vitagliano, The Lagrangian-Hamiltonian formalism for higher-order field theories, J. Geom. Phys., 60 (2010), 857-873. doi: 10.1016/j.geomphys.2010.02.003.  Google Scholar

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