American Institute of Mathematical Sciences

Advanced
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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

O. Krupková, Higher-order mechanical systems with constraints, J. Math. Phys., 41 (2000), 5304-5324. 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), 385203, 35 pp. 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), 032901, 38 pp. 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), 339-349. doi: 10.1088/0305-4470/20/2/019.

[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.

[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.

[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.

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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[19]

O. Krupková, Higher-order mechanical systems with constraints, J. Math. Phys., 41 (2000), 5304-5324. 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), 385203, 35 pp. 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), 032901, 38 pp. 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), 339-349. doi: 10.1088/0305-4470/20/2/019.

[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.

[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.

[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.

[1]

Eduardo Martínez. Higher-order variational calculus on Lie algebroids. Journal of Geometric Mechanics, 2015, 7 (1) : 81-108. doi: 10.3934/jgm.2015.7.81
[2]

Janusz Grabowski, Katarzyna Grabowska, Paweł Urbański. Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings. Journal of Geometric Mechanics, 2014, 6 (4) : 503-526. doi: 10.3934/jgm.2014.6.503
[3]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether's theorem for higher-order variational problems of Herglotz type. Conference Publications, 2015, 2015 (special) : 990-999. doi: 10.3934/proc.2015.990
[4]

Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higher-order Lagrange-Poincaré equations. Discrete and Continuous Dynamical Systems, 2019, 39 (1) : 309-344. doi: 10.3934/dcds.2019013
[5]

Michał Jóźwikowski, Mikołaj Rotkiewicz. Bundle-theoretic methods for higher-order variational calculus. Journal of Geometric Mechanics, 2014, 6 (1) : 99-120. doi: 10.3934/jgm.2014.6.99
[6]

Leonardo Colombo, David Martín de Diego. Higher-order variational problems on lie groups and optimal control applications. Journal of Geometric Mechanics, 2014, 6 (4) : 451-478. doi: 10.3934/jgm.2014.6.451
[7]

Yuri B. Suris. Variational formulation of commuting Hamiltonian flows: Multi-time Lagrangian 1-forms. Journal of Geometric Mechanics, 2013, 5 (3) : 365-379. doi: 10.3934/jgm.2013.5.365
[8]

Kazuyuki Yagasaki. Higher-order Melnikov method and chaos for two-degree-of-freedom Hamiltonian systems with saddle-centers. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 387-402. doi: 10.3934/dcds.2011.29.387
[9]

Simão P. S. Santos, Natália Martins, Delfim F. M. Torres. Noether currents for higher-order variational problems of Herglotz type with time delay. Discrete and Continuous Dynamical Systems - S, 2018, 11 (1) : 91-102. doi: 10.3934/dcdss.2018006
[10]

Ahmed Y. Abdallah, Rania T. Wannan. Second order non-autonomous lattice systems and their uniform attractors. Communications on Pure and Applied Analysis, 2019, 18 (4) : 1827-1846. doi: 10.3934/cpaa.2019085
[11]

Delia Schiera. Existence and non-existence results for variational higher order elliptic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (10) : 5145-5161. doi: 10.3934/dcds.2018227
[12]

Qilin Wang, Liu He, Shengjie Li. Higher-order weak radial epiderivatives and non-convex set-valued optimization problems. Journal of Industrial and Management Optimization, 2019, 15 (2) : 465-480. doi: 10.3934/jimo.2018051
[13]

Kui Li, Zhitao Zhang. Liouville-type theorem for higher-order Hardy-Hénon system. Communications on Pure and Applied Analysis, 2021, 20 (11) : 3851-3869. doi: 10.3934/cpaa.2021134
[14]

Guillaume Duval, Andrzej J. Maciejewski. Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations. Discrete and Continuous Dynamical Systems, 2014, 34 (11) : 4589-4615. doi: 10.3934/dcds.2014.34.4589
[15]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete and Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703
[16]

Michael Khanevsky. Non-autonomous curves on surfaces. Journal of Modern Dynamics, 2021, 17: 305-317. doi: 10.3934/jmd.2021010
[17]

T. Tachim Medjo. A non-autonomous 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external force and its global attractor. Communications on Pure and Applied Analysis, 2011, 10 (2) : 415-433. doi: 10.3934/cpaa.2011.10.415
[18]

Chjan C. Lim, Junping Shi. The role of higher vorticity moments in a variational formulation of Barotropic flows on a rotating sphere. Discrete and Continuous Dynamical Systems - B, 2009, 11 (3) : 717-740. doi: 10.3934/dcdsb.2009.11.717
[19]

Regina Martínez, Carles Simó. Non-integrability of the degenerate cases of the Swinging Atwood's Machine using higher order variational equations. Discrete and Continuous Dynamical Systems, 2011, 29 (1) : 1-24. doi: 10.3934/dcds.2011.29.1
[20]

Weisheng Niu, Yao Xu. Convergence rates in homogenization of higher-order parabolic systems. Discrete and Continuous Dynamical Systems, 2018, 38 (8) : 4203-4229. doi: 10.3934/dcds.2018183

2020 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (64)
  • HTML views (0)
  • Cited by (5)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy