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. 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). 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. 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). 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. 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). 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. 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. 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, (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. 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. doi: 10.1063/1.1628384. Google Scholar

[13]

P. L. García, The Poincaré-Cartan invariant in the calculus of variations,, in Symposia Mathematica, (1973), 219. 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. Google Scholar

[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. 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 (1978), 2388. 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. 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. Google Scholar

[19]

O. Krupková, Higher-order mechanical systems with constraints,, J. Math. Phys., 41 (2000), 5304. 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). 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). 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. doi: 10.1088/0305-4470/20/2/019. Google Scholar

[23]

D. J. Saunders, The Geometry of Jet Bundles,, London Math. Soc., (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. 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. 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. 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). 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. 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). 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. 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). 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. 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. 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, (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. 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. doi: 10.1063/1.1628384. Google Scholar

[13]

P. L. García, The Poincaré-Cartan invariant in the calculus of variations,, in Symposia Mathematica, (1973), 219. 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. Google Scholar

[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. 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 (1978), 2388. 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. 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. Google Scholar

[19]

O. Krupková, Higher-order mechanical systems with constraints,, J. Math. Phys., 41 (2000), 5304. 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). 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). 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. doi: 10.1088/0305-4470/20/2/019. Google Scholar

[23]

D. J. Saunders, The Geometry of Jet Bundles,, London Math. Soc., (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. 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. doi: 10.1016/j.geomphys.2010.02.003. Google Scholar

[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]

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
[5]

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
[6]

Anthony Bloch, Leonardo Colombo, Fernando Jiménez. The variational discretization of the constrained higher-order Lagrange-Poincaré equations. Discrete & Continuous Dynamical Systems - A, 2019, 39 (1) : 309-344. doi: 10.3934/dcds.2019013
[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 & Continuous Dynamical Systems - A, 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 & 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 & 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 & Continuous Dynamical Systems - A, 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 & Management Optimization, 2019, 15 (2) : 465-480. doi: 10.3934/jimo.2018051
[13]

Guillaume Duval, Andrzej J. Maciejewski. Integrability of Hamiltonian systems with homogeneous potentials of degrees $\pm 2$. An application of higher order variational equations. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4589-4615. doi: 10.3934/dcds.2014.34.4589
[14]

Alexandre N. Carvalho, José A. Langa, James C. Robinson. Non-autonomous dynamical systems. Discrete & Continuous Dynamical Systems - B, 2015, 20 (3) : 703-747. doi: 10.3934/dcdsb.2015.20.703
[15]

T. Tachim Medjo. A non-autonomous 3D Lagrangian averaged Navier-Stokes-$\alpha$ model with oscillating external force and its global attractor. Communications on Pure & Applied Analysis, 2011, 10 (2) : 415-433. doi: 10.3934/cpaa.2011.10.415
[16]

Chjan C. Lim, Junping Shi. The role of higher vorticity moments in a variational formulation of Barotropic flows on a rotating sphere. Discrete & Continuous Dynamical Systems - B, 2009, 11 (3) : 717-740. doi: 10.3934/dcdsb.2009.11.717
[17]

Regina Martínez, Carles Simó. Non-integrability of the degenerate cases of the Swinging Atwood's Machine using higher order variational equations. Discrete & Continuous Dynamical Systems - A, 2011, 29 (1) : 1-24. doi: 10.3934/dcds.2011.29.1
[18]

Xinmin Yang, Xiaoqi Yang, Kok Lay Teo. Higher-order symmetric duality in multiobjective programming with invexity. Journal of Industrial & Management Optimization, 2008, 4 (2) : 385-391. doi: 10.3934/jimo.2008.4.385
[19]

Weisheng Niu, Yao Xu. Convergence rates in homogenization of higher-order parabolic systems. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4203-4229. doi: 10.3934/dcds.2018183
[20]

Feliz Minhós, Hugo Carrasco. Solvability of higher-order BVPs in the half-line with unbounded nonlinearities. Conference Publications, 2015, 2015 (special) : 841-850. doi: 10.3934/proc.2015.0841

2018 Impact Factor: 0.525

Tools

Metrics

  • PDF downloads (4)
  • HTML views (0)
  • Cited by (6)

Other articles
by authors

[Back to Top]

Copyright © 2019 American Institute of Mathematical Sciences