- Previous Article
- JGM Home
- This Issue
-
Next Article
On the relation between geometrical quantum mechanics and information geometry
A new multisymplectic unified formalism for second order classical field theories
1. | Departamento de Matemática Aplicada IV, Universitat Politècnica de Catalunya, Campus Norte, Ed. C3. C/ Jordi Girona 1, E-08034 Barcelona, Spain |
2. | Departamento de Matemática Aplicada IV, Universitat Politècnica de Catalunya-BarcelonaTech, Campus Norte, Ed. C-3. C/ Jordi Girona 1, E-08034 Barcelona |
References:
[1] |
R. Abraham and J. E. Marsden, Foundations of Mechanics,, $2^{nd}$ edition, (1978).
|
[2] |
V. Aldaya and J. A. de Azcarraga, Variational principles on $r-th$ order jets of fibre bundles in field theory,, J. Math. Phys., 19 (1978), 1869.
doi: 10.1063/1.523904. |
[3] |
V. Aldaya and J. A. de Azcarraga, Vector Bundles, $r-th$ order Noether invariant and canonical symmetries in Lagrangian field theory,, J. Math. Phys., 19 (1978), 1876.
doi: 10.1063/1.523905. |
[4] |
V. Aldaya and J. A. de Azcarraga, Higher order Hamiltonian formalism in field theory,, J. Phys. A, 13 (1980), 2545.
doi: 10.1088/0305-4470/13/8/004. |
[5] |
U. M. Ascher and R. I. McLachlan, On symplectic and multisymplectic schemes for the KdV equation,, J. Sci. Comput., 25 (2005), 83.
doi: 10.1007/s10915-004-4634-6. |
[6] |
M. Barbero-Liñán, A. Echeverría-Enríquez, D. Martín de Diego, M. C. Muñoz-Lecanda and N. Román-Roy, Skinner-rusk unified formalism for optimal control systems and applications,, J. Math. Phys. A: Math. Theor., 40 (2007), 12071.
doi: 10.1088/1751-8113/40/40/005. |
[7] |
M. Barbero-Liñán, A. Echeverría-Enríquez, D. Martín de Diego, M. C. Muñoz-Lecanda and N. Román-Roy, Unified formalism for non-autonomous mechanical systems,, J. Math. Phys., 49 (2008).
doi: 10.1063/1.2929668. |
[8] |
C. Batlle, J. Gomis, J. M. Pons and N. Román-Roy, Lagrangian and Hamiltonian constraints for second-order singular Lagrangians,, J. Phys. A: Math. Gen., 21 (1988), 2693.
doi: 10.1088/0305-4470/21/12/013. |
[9] |
C. M. Campos, Geometric Methods in Classical Field Theory and Continous Media,, Ph.D. thesis, (2010). Google Scholar |
[10] |
C. M. Campos, Higher-order field theory with constraints,, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 106 (2012), 89.
doi: 10.1007/s13398-011-0025-7. |
[11] |
C. M. Campos, M. de León, D. Martín de Diego and J. Vankerschaver, Unambigous formalism for higher-order lagrangian field theories,, J. Phys A: Math Theor., 42 (2009).
doi: 10.1088/1751-8113/42/47/475207. |
[12] |
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. |
[13] |
J. F. Cariñena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order field theories,, Diff. Geom. Appl., 1 (1991), 345.
doi: 10.1016/0926-2245(91)90013-Y. |
[14] |
L. Colombo, D. Martín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometric approach,, J. Math. Phys., 51 (2010).
doi: 10.1063/1.3456158. |
[15] |
J. Cortés, S. Martínez and F. Cantrijn, Skinner-Rusk approach to time-dependent mechanics,, Physics Letters A, 300 (2002), 250.
doi: 10.1016/S0375-9601(02)00777-6. |
[16] |
M. Crampin and D. J. Saunders, Homogeneity and projective equivalence of differential equation fields,, J. Geom. Mech., 4 (2012), 27.
doi: 10.3934/jgm.2012.4.27. |
[17] |
M. de León, J. Marín-Solano and J. C. Marrero, The constraint algorithm in the jet formalism,, Nuovo Cimento B, (11) 84 (1984), 91. Google Scholar |
[18] |
M. de León, J. Marín-Solano, J. C. Marrero, M. C. Muñoz-Lecanda and N. Román-Roy, Premultisymplectic constraint algorithm for field theories,, Int. J. Geom. Methods Mod. Phys., 2 (2005), 839. Google Scholar |
[19] |
M. de León, J. C. Marrero and D. Martín de Diego, A new geometric setting for classical field theories,, Banach Center Publ., 59 (2003), 189.
doi: 10.4064/bc59-0-10. |
[20] |
M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Math. Studies, (1985). Google Scholar |
[21] |
M. de León and P. R. Rodrigues, Higher-order almost tangent geometry and non-autonomous Lagrangian dynamics,, Rend. Circ. Mat. Palermo, 2 (1987), 157.
|
[22] |
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. |
[23] |
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. |
[24] |
A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector fields and connections: Setting Lagrangian equations in field theories,, J. Math. Phys., 39 (1998), 4578.
doi: 10.1063/1.532525. |
[25] |
A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometry of multisymplectic Hamiltonian first-order field theories,, J. Math. Phys., 41 (2000), 7402.
doi: 10.1063/1.1308075. |
[26] |
M. Ferraris and M. Francaviglia, Applications of the Poincaré-Cartan form in higher order field theories,, in Differential Geometry and its Applications (Brno, (1986), 31.
|
[27] |
M. Francaviglia and D. Krupka, The Hamiltonian formalism in higher order variational problems,, Ann. Inst. H. Poincaré Sect. A (N.S.), 37 (1982), 295.
|
[28] |
P. L. García and J. Muñoz-Masqué, On the geometrical structure of higher order variational calculus,, in Procs. IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, (1982), 127.
|
[29] |
P. L. García and J. Muñoz-Masqué, Higher order regular variational problems,, in Symplectic Geometry and Mathematical Physics (Aix-en-Provence, (1990), 136.
|
[30] |
M. J. Gotay, A multisymplectic approach to the KdV equation,, in Diferential Geometric Methods in Theoretical Physics (eds., (1988), 295.
|
[31] |
K. Grabowska and L. Vitagliano, Tulczyjew triples in higher derivative field theory,, J. Geom. Mech., 7 (2015), 1.
doi: 10.3934/jgm.2015.7.1. |
[32] |
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. |
[33] |
X. Gràcia, J. M. Pons and N. Román-Roy, Higher-order conditions for singular Lagrangian systems,, J. Phys. A: Math. Gen., 25 (1992), 1989.
doi: 10.1088/0305-4470/25/7/037. |
[34] |
D. R. Grigore, On a generalization of the Poincaré-Cartan form in higher-order field theory,, in Variations, (2009), 57.
|
[35] |
M. Horák and I. Kolár, On the higher order Poincaré-Cartan forms,, Czechoslovak Math. J., 33 (1983), 467.
|
[36] |
I. Kolár, A geometrical version of the higher order Hamilton formalism in fibered manifolds,, J. Geom. and Phys., 1 (1984), 127.
doi: 10.1016/0393-0440(84)90007-X. |
[37] |
S. Kouranbaeva and S. Shkoller, A variational approach to second-order multisymplectic field theory,, J. Geom. Phys., 35 (2000), 333.
doi: 10.1016/S0393-0440(00)00012-7. |
[38] |
D. Krupka, On the higher order Hamilton theory in fibered spaces,, in Procs. Conference on Differential Geometry and its Applications, (1984), 167.
|
[39] |
O. Krupkova, Higher-order mechanical systems with constraints,, J. Math. Phys., 41 (2000), 5304.
doi: 10.1063/1.533411. |
[40] |
R. Miron, The Geometry of Higher-order Hamilton Spaces: Applications to Hamiltonian Mechanics,, Fundamental Theories of Physics, (2003).
doi: 10.1007/978-94-010-0070-3. |
[41] |
P. Mukherjee and B. Paul, Gauge invariances of higher derivative Maxwell-Chern-Simons field theories: A new Hamiltonian approach,, Phys. Rev. D, 85 (2012).
doi: 10.1103/PhysRevD.85.045028. |
[42] |
J. Muñoz-Masqué, Canonical Cartan equations for higher order variational problems,, J. Geom. Phys., 1 (1984), 1.
doi: 10.1016/0393-0440(84)90001-9. |
[43] |
J. Muñoz-Masqué, Poincaré-Cartan forms in higher order variational calculus on fibred manifolds,, Rev. Mat. Iberoamericana, 1 (1985), 85.
doi: 10.4171/RMI/20. |
[44] |
P. D. Prieto-Martínez and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for autonomous higher-order dynamical systems,, J. Phys. A Math. Theor., 44 (2011).
doi: 10.1088/1751-8113/44/38/385203. |
[45] |
P. D. Prieto-Martínez, N. Román-Roy, Unified formalism for higher-order non-autonomous dynamical systems,, J. Math. Phys., 53 (2012).
doi: 10.1063/1.3692326. |
[46] |
P. D. Prieto-Martínez and N. Román-Roy, Higher-order mechanics: Variational principles and other topics,, J. Geom. Mech., 5 (2013), 493.
doi: 10.3934/jgm.2013.5.493. |
[47] |
A. M. Rey, N. Román-Roy and M. Salgado, Gunther's formalism k-symplectic formalism in classical field theory: Skinner-Rusk approach and the evolution operator,, J. Math. Phys., 46 (2005).
doi: 10.1063/1.1876872. |
[48] |
A. M. Rey, N. Román-Roy, M. Salgado and S. Vilariño, k-cosymplectic classical field theories: Tulckzyjew and Skinner-Rusk formulations,, Math. Phys. Anal. Geom., 15 (2012), 85.
doi: 10.1007/s11040-012-9104-z. |
[49] |
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. |
[50] |
D. J. Saunders, The Geometry of Jet Bundles,, London Mathematical Society, (1989).
doi: 10.1017/CBO9780511526411. |
[51] |
D. J. Saunders and M. Crampin, On the Legendre map in higher-order field theories,, J. Phys. A: Math. Gen., 23 (1990), 3169.
doi: 10.1088/0305-4470/23/14/016. |
[52] |
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. |
[53] |
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. |
[54] |
P. F. Zhao and M. Z. Qin, Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation,, J. Phys. A, 33 (2000), 3613.
doi: 10.1088/0305-4470/33/18/308. |
show all references
References:
[1] |
R. Abraham and J. E. Marsden, Foundations of Mechanics,, $2^{nd}$ edition, (1978).
|
[2] |
V. Aldaya and J. A. de Azcarraga, Variational principles on $r-th$ order jets of fibre bundles in field theory,, J. Math. Phys., 19 (1978), 1869.
doi: 10.1063/1.523904. |
[3] |
V. Aldaya and J. A. de Azcarraga, Vector Bundles, $r-th$ order Noether invariant and canonical symmetries in Lagrangian field theory,, J. Math. Phys., 19 (1978), 1876.
doi: 10.1063/1.523905. |
[4] |
V. Aldaya and J. A. de Azcarraga, Higher order Hamiltonian formalism in field theory,, J. Phys. A, 13 (1980), 2545.
doi: 10.1088/0305-4470/13/8/004. |
[5] |
U. M. Ascher and R. I. McLachlan, On symplectic and multisymplectic schemes for the KdV equation,, J. Sci. Comput., 25 (2005), 83.
doi: 10.1007/s10915-004-4634-6. |
[6] |
M. Barbero-Liñán, A. Echeverría-Enríquez, D. Martín de Diego, M. C. Muñoz-Lecanda and N. Román-Roy, Skinner-rusk unified formalism for optimal control systems and applications,, J. Math. Phys. A: Math. Theor., 40 (2007), 12071.
doi: 10.1088/1751-8113/40/40/005. |
[7] |
M. Barbero-Liñán, A. Echeverría-Enríquez, D. Martín de Diego, M. C. Muñoz-Lecanda and N. Román-Roy, Unified formalism for non-autonomous mechanical systems,, J. Math. Phys., 49 (2008).
doi: 10.1063/1.2929668. |
[8] |
C. Batlle, J. Gomis, J. M. Pons and N. Román-Roy, Lagrangian and Hamiltonian constraints for second-order singular Lagrangians,, J. Phys. A: Math. Gen., 21 (1988), 2693.
doi: 10.1088/0305-4470/21/12/013. |
[9] |
C. M. Campos, Geometric Methods in Classical Field Theory and Continous Media,, Ph.D. thesis, (2010). Google Scholar |
[10] |
C. M. Campos, Higher-order field theory with constraints,, Rev. R. Acad. Cienc. Exactas Fís. Nat. Ser. A Math. RACSAM, 106 (2012), 89.
doi: 10.1007/s13398-011-0025-7. |
[11] |
C. M. Campos, M. de León, D. Martín de Diego and J. Vankerschaver, Unambigous formalism for higher-order lagrangian field theories,, J. Phys A: Math Theor., 42 (2009).
doi: 10.1088/1751-8113/42/47/475207. |
[12] |
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. |
[13] |
J. F. Cariñena, M. Crampin and L. A. Ibort, On the multisymplectic formalism for first order field theories,, Diff. Geom. Appl., 1 (1991), 345.
doi: 10.1016/0926-2245(91)90013-Y. |
[14] |
L. Colombo, D. Martín de Diego and M. Zuccalli, Optimal control of underactuated mechanical systems: A geometric approach,, J. Math. Phys., 51 (2010).
doi: 10.1063/1.3456158. |
[15] |
J. Cortés, S. Martínez and F. Cantrijn, Skinner-Rusk approach to time-dependent mechanics,, Physics Letters A, 300 (2002), 250.
doi: 10.1016/S0375-9601(02)00777-6. |
[16] |
M. Crampin and D. J. Saunders, Homogeneity and projective equivalence of differential equation fields,, J. Geom. Mech., 4 (2012), 27.
doi: 10.3934/jgm.2012.4.27. |
[17] |
M. de León, J. Marín-Solano and J. C. Marrero, The constraint algorithm in the jet formalism,, Nuovo Cimento B, (11) 84 (1984), 91. Google Scholar |
[18] |
M. de León, J. Marín-Solano, J. C. Marrero, M. C. Muñoz-Lecanda and N. Román-Roy, Premultisymplectic constraint algorithm for field theories,, Int. J. Geom. Methods Mod. Phys., 2 (2005), 839. Google Scholar |
[19] |
M. de León, J. C. Marrero and D. Martín de Diego, A new geometric setting for classical field theories,, Banach Center Publ., 59 (2003), 189.
doi: 10.4064/bc59-0-10. |
[20] |
M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory,, North-Holland Math. Studies, (1985). Google Scholar |
[21] |
M. de León and P. R. Rodrigues, Higher-order almost tangent geometry and non-autonomous Lagrangian dynamics,, Rend. Circ. Mat. Palermo, 2 (1987), 157.
|
[22] |
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. |
[23] |
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. |
[24] |
A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Multivector fields and connections: Setting Lagrangian equations in field theories,, J. Math. Phys., 39 (1998), 4578.
doi: 10.1063/1.532525. |
[25] |
A. Echeverría-Enríquez, M. C. Muñoz-Lecanda and N. Román-Roy, Geometry of multisymplectic Hamiltonian first-order field theories,, J. Math. Phys., 41 (2000), 7402.
doi: 10.1063/1.1308075. |
[26] |
M. Ferraris and M. Francaviglia, Applications of the Poincaré-Cartan form in higher order field theories,, in Differential Geometry and its Applications (Brno, (1986), 31.
|
[27] |
M. Francaviglia and D. Krupka, The Hamiltonian formalism in higher order variational problems,, Ann. Inst. H. Poincaré Sect. A (N.S.), 37 (1982), 295.
|
[28] |
P. L. García and J. Muñoz-Masqué, On the geometrical structure of higher order variational calculus,, in Procs. IUTAM-ISIMM Symposium on Modern Developments in Analytical Mechanics, (1982), 127.
|
[29] |
P. L. García and J. Muñoz-Masqué, Higher order regular variational problems,, in Symplectic Geometry and Mathematical Physics (Aix-en-Provence, (1990), 136.
|
[30] |
M. J. Gotay, A multisymplectic approach to the KdV equation,, in Diferential Geometric Methods in Theoretical Physics (eds., (1988), 295.
|
[31] |
K. Grabowska and L. Vitagliano, Tulczyjew triples in higher derivative field theory,, J. Geom. Mech., 7 (2015), 1.
doi: 10.3934/jgm.2015.7.1. |
[32] |
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. |
[33] |
X. Gràcia, J. M. Pons and N. Román-Roy, Higher-order conditions for singular Lagrangian systems,, J. Phys. A: Math. Gen., 25 (1992), 1989.
doi: 10.1088/0305-4470/25/7/037. |
[34] |
D. R. Grigore, On a generalization of the Poincaré-Cartan form in higher-order field theory,, in Variations, (2009), 57.
|
[35] |
M. Horák and I. Kolár, On the higher order Poincaré-Cartan forms,, Czechoslovak Math. J., 33 (1983), 467.
|
[36] |
I. Kolár, A geometrical version of the higher order Hamilton formalism in fibered manifolds,, J. Geom. and Phys., 1 (1984), 127.
doi: 10.1016/0393-0440(84)90007-X. |
[37] |
S. Kouranbaeva and S. Shkoller, A variational approach to second-order multisymplectic field theory,, J. Geom. Phys., 35 (2000), 333.
doi: 10.1016/S0393-0440(00)00012-7. |
[38] |
D. Krupka, On the higher order Hamilton theory in fibered spaces,, in Procs. Conference on Differential Geometry and its Applications, (1984), 167.
|
[39] |
O. Krupkova, Higher-order mechanical systems with constraints,, J. Math. Phys., 41 (2000), 5304.
doi: 10.1063/1.533411. |
[40] |
R. Miron, The Geometry of Higher-order Hamilton Spaces: Applications to Hamiltonian Mechanics,, Fundamental Theories of Physics, (2003).
doi: 10.1007/978-94-010-0070-3. |
[41] |
P. Mukherjee and B. Paul, Gauge invariances of higher derivative Maxwell-Chern-Simons field theories: A new Hamiltonian approach,, Phys. Rev. D, 85 (2012).
doi: 10.1103/PhysRevD.85.045028. |
[42] |
J. Muñoz-Masqué, Canonical Cartan equations for higher order variational problems,, J. Geom. Phys., 1 (1984), 1.
doi: 10.1016/0393-0440(84)90001-9. |
[43] |
J. Muñoz-Masqué, Poincaré-Cartan forms in higher order variational calculus on fibred manifolds,, Rev. Mat. Iberoamericana, 1 (1985), 85.
doi: 10.4171/RMI/20. |
[44] |
P. D. Prieto-Martínez and N. Román-Roy, Lagrangian-Hamiltonian unified formalism for autonomous higher-order dynamical systems,, J. Phys. A Math. Theor., 44 (2011).
doi: 10.1088/1751-8113/44/38/385203. |
[45] |
P. D. Prieto-Martínez, N. Román-Roy, Unified formalism for higher-order non-autonomous dynamical systems,, J. Math. Phys., 53 (2012).
doi: 10.1063/1.3692326. |
[46] |
P. D. Prieto-Martínez and N. Román-Roy, Higher-order mechanics: Variational principles and other topics,, J. Geom. Mech., 5 (2013), 493.
doi: 10.3934/jgm.2013.5.493. |
[47] |
A. M. Rey, N. Román-Roy and M. Salgado, Gunther's formalism k-symplectic formalism in classical field theory: Skinner-Rusk approach and the evolution operator,, J. Math. Phys., 46 (2005).
doi: 10.1063/1.1876872. |
[48] |
A. M. Rey, N. Román-Roy, M. Salgado and S. Vilariño, k-cosymplectic classical field theories: Tulckzyjew and Skinner-Rusk formulations,, Math. Phys. Anal. Geom., 15 (2012), 85.
doi: 10.1007/s11040-012-9104-z. |
[49] |
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. |
[50] |
D. J. Saunders, The Geometry of Jet Bundles,, London Mathematical Society, (1989).
doi: 10.1017/CBO9780511526411. |
[51] |
D. J. Saunders and M. Crampin, On the Legendre map in higher-order field theories,, J. Phys. A: Math. Gen., 23 (1990), 3169.
doi: 10.1088/0305-4470/23/14/016. |
[52] |
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. |
[53] |
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. |
[54] |
P. F. Zhao and M. Z. Qin, Multisymplectic geometry and multisymplectic Preissmann scheme for the KdV equation,, J. Phys. A, 33 (2000), 3613.
doi: 10.1088/0305-4470/33/18/308. |
[1] |
Narciso Román-Roy, Ángel M. Rey, Modesto Salgado, Silvia Vilariño. On the $k$-symplectic, $k$-cosymplectic and multisymplectic formalisms of classical field theories. Journal of Geometric Mechanics, 2011, 3 (1) : 113-137. doi: 10.3934/jgm.2011.3.113 |
[2] |
Cédric M. Campos, Elisa Guzmán, Juan Carlos Marrero. Classical field theories of first order and Lagrangian submanifolds of premultisymplectic manifolds. Journal of Geometric Mechanics, 2012, 4 (1) : 1-26. doi: 10.3934/jgm.2012.4.1 |
[3] |
Alberto Ibort, Amelia Spivak. Covariant Hamiltonian field theories on manifolds with boundary: Yang-Mills theories. Journal of Geometric Mechanics, 2017, 9 (1) : 47-82. doi: 10.3934/jgm.2017002 |
[4] |
Mamoru Okamoto. Asymptotic behavior of solutions to a higher-order KdV-type equation with critical nonlinearity. Evolution Equations & Control Theory, 2019, 8 (3) : 567-601. doi: 10.3934/eect.2019027 |
[5] |
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 |
[6] |
Clesh Deseskel Elion Ekohela, Daniel Moukoko. On higher-order anisotropic perturbed Caginalp phase field systems. Electronic Research Announcements, 2019, 26: 36-53. doi: 10.3934/era.2019.26.004 |
[7] |
Jibin Li, Weigou Rui, Yao Long, Bin He. Travelling wave solutions for higher-order wave equations of KDV type (III). Mathematical Biosciences & Engineering, 2006, 3 (1) : 125-135. doi: 10.3934/mbe.2006.3.125 |
[8] |
Feng Wang, Fengquan Li, Zhijun Qiao. On the Cauchy problem for a higher-order μ-Camassa-Holm equation. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4163-4187. doi: 10.3934/dcds.2018181 |
[9] |
David F. Parker. Higher-order shallow water equations and the Camassa-Holm equation. Discrete & Continuous Dynamical Systems - B, 2007, 7 (3) : 629-641. doi: 10.3934/dcdsb.2007.7.629 |
[10] |
Min Zhu. On the higher-order b-family equation and Euler equations on the circle. Discrete & Continuous Dynamical Systems - A, 2014, 34 (7) : 3013-3024. doi: 10.3934/dcds.2014.34.3013 |
[11] |
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 |
[12] |
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 |
[13] |
Hongqiu Chen. Well-posedness for a higher-order, nonlinear, dispersive equation on a quarter plane. Discrete & Continuous Dynamical Systems - A, 2018, 38 (1) : 397-429. doi: 10.3934/dcds.2018019 |
[14] |
Shouming Zhou. The Cauchy problem for a generalized $b$-equation with higher-order nonlinearities in critical Besov spaces and weighted $L^p$ spaces. Discrete & Continuous Dynamical Systems - A, 2014, 34 (11) : 4967-4986. doi: 10.3934/dcds.2014.34.4967 |
[15] |
Xingxing Liu. Orbital stability of peakons for a modified Camassa-Holm equation with higher-order nonlinearity. Discrete & Continuous Dynamical Systems - A, 2018, 38 (11) : 5505-5521. doi: 10.3934/dcds.2018242 |
[16] |
Katarzyna Grabowska. Lagrangian and Hamiltonian formalism in Field Theory: A simple model. Journal of Geometric Mechanics, 2010, 2 (4) : 375-395. doi: 10.3934/jgm.2010.2.375 |
[17] |
Shouming Zhou, Chunlai Mu, Liangchen Wang. Well-posedness, blow-up phenomena and global existence for the generalized $b$-equation with higher-order nonlinearities and weak dissipation. Discrete & Continuous Dynamical Systems - A, 2014, 34 (2) : 843-867. doi: 10.3934/dcds.2014.34.843 |
[18] |
Jun Zhou. Global existence and energy decay estimate of solutions for a class of nonlinear higher-order wave equation with general nonlinear dissipation and source term. Discrete & Continuous Dynamical Systems - S, 2017, 10 (5) : 1175-1185. doi: 10.3934/dcdss.2017064 |
[19] |
Mei Yu, Xia Zhang, Binlin Zhang. Property of solutions for elliptic equation involving the higher-order fractional Laplacian in $ \mathbb{R}^n_+ $. Communications on Pure & Applied Analysis, 2020, 19 (7) : 3597-3612. doi: 10.3934/cpaa.2020157 |
[20] |
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 |
2019 Impact Factor: 0.649
Tools
Metrics
Other articles
by authors
[Back to Top]