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On the geometry of the Schmidt-Legendre transformation
1. | Department of Mathematics, Gebze Technical University, 41400 Çayırova, Gebze, Kocaeli, Turkey |
2. | S.N. Bose National Centre for Basic Sciences, JD Block, Sector Ⅲ, Salt Lake, Kolkata - 700098, India |
Tulczyjew's triples are constructed for the Schmidt-Legendre transformations of both second and third-order Lagrangians. Symplectic diffeomorphisms relating the Ostrogradsky-Legendre and the Schmidt-Legendre transformations are derived. Several examples are presented.
References:
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R. Abraham and J. E. Marsden,
Foundations of Mechanics, Reading, Massachusetts, Benjamin/Cummings Publishing Company, 1978. |
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L. Abrunheiro and L. Colombo, Lagrangian Lie subalgebroids generating dynamics for second-order mechanical systems on Lie algebroids,
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Lagrangian and Hamiltonian constraints for second-order singular Lagrangians, Journal of Physics A: Mathematical and General, 21 (1988), 2693-2703.
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Hamiltonian structures for pais-uhlenbeck oscillator, Acta Physica Polonica B, 36 (2205), 2115.
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A. J. Bruce, K. Grabowska and J. Grabowski, Higher order mechanics on graded bundles,
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doi: 10.1088/1751-8113/48/20/205203. |
[11] |
F. Çağatay Uçgun, O. Esen and H. Gümral, Reductions of topologically massive gravity Ⅰ: Hamiltonian analysis of second order degenerate Lagrangians,
Journal of Mathematical Physics, 59 (2018), 013510, 16pp.
doi: 10.1063/1.5021948. |
[12] |
C. M. Campos, M. de León, D. M. de Diego and J. Vankerschaver, Unambiguous formalism for higher order Lagrangian field theories,
Journal of Physics A: Mathematical and Theoretical, 42 (2009), 475207, 24pp.
doi: 10.1088/1751-8113/42/47/475207. |
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Morse families and constrained mechanical systems, Generalized hyperelastic materials. Meccanica, 26 (1991), 161-167.
|
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H. Cendra and S. D. Grillo, Lagrangian systems with higher order constraints,
Journal of Mathematical Physics, 48 (2007), 052904, 35pp.
doi: 10.1063/1.2740470. |
[15] |
T. J. Chen, M. Fasiello, E. A. Lim and A. J. Tolley, Higher derivative theories with constraints: Exorcising Ostrogradski's ghost,
Journal of Cosmology and Astroparticle Physics, 2 (2013), 042, front matter+17 pp. |
[16] |
G. Clément,
Particle-like solutions to topologically massive gravity, Classical and Quantum Gravity, 11 (1994), L115-L120.
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L. Colombo,
Second-order constrained variational problems on Lie algebroids: Applications to optimal control, Journal of Geometric Mechanics, 9 (2017), 1-45.
doi: 10.3934/jgm.2017001. |
[18] |
L. Colombo, M. de León, P. D. Prieto-Martínez and N. Román-Roy,
Unified formalism for the generalized kth-order Hamilton-Jacobi problem, International Journal of Geometric Methods
in Modern Physics, 11 (2014), 1460037, 9pp.
doi: 10.1142/S0219887814600378. |
[19] |
L. Colombo and D. M. de Diego,
Higher-order variational problems on Lie groups and optimal control applications, Journal Geometric Mechanics, 6 (2014), 451-478.
doi: 10.3934/jgm.2014.6.451. |
[20] |
L. Colombo and P. D. Prieto-Martínez, Regularity properties of fiber derivatives associated with higher-order mechanical systems,
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M. Crampin, W. Sarlet and F. Cantrijn, Higher-order differential equations and higher-order
Lagrangian mechanics. In Mathematical Proceedings of the Cambridge Philosophical Society,
99 (1986), 565–587.
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[22] |
A. Deriglazov,
Classical Mechanics, Springer International Publishing, 2017.
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[23] |
N. Deruelle, Y. Sendouda and A. Youssef, Various Hamiltonian formulations of f (R) gravity and their canonical relationships,
Physical Review D, 80 (2009), 084032, 11pp.
doi: 10.1103/PhysRevD.80.084032. |
[24] |
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Topologically massive gauge theories, Annals of Physics, 140 (1982), 372-411.
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S. Deser, R. Jackiw and S. Templeton,
Three-dimensional massive gauge theories, Physical Review Letters, 48 (1982), 975-978.
doi: 10.1103/PhysRevLett.48.975. |
[26] |
P. A. M. Dirac,
Lectures on Quantum Mechanics, Belfer Graduate School of Science, Monograph Series, Yeshiva University, New York, 1967. |
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P. A. M. Dirac,
Generalized hamiltonian dynamics, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 246 (1958), 326-332.
doi: 10.1098/rspa.1958.0141. |
[28] |
C. T. J. Dodson and M. S. Radivoiovici,
Tangent and frame bundles of order two, An. Stiint. Univ. "Al. I. Cuza" Iasi Sect. I a Mat. (N.S.), 28 (1982), 63-71.
|
[29] |
C. T. Dodson, G. Galanis and E. Vassiliou,
Geometry in a Fréchet Context: A Projective Limit Approach, Cambridge University Press, 2016.
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[30] |
O. Esen and H. Gümral,
Tulczyjew's triplet for Lie groups Ⅰ: Trivializations and reductions, Journal of Lie Theory, 24 (2014), 1115-1160.
|
[31] |
O. Esen and H. Gümral,
Tulczyjew's triplet for Lie groups. Ⅱ: Dynamics, Journal of Lie Theory, 27 (2017), 329-356.
|
[32] |
F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X. Vialard,
Invariant higher-order variational problems, Communications in Mathematical Physics, 309 (2012), 413-458.
doi: 10.1007/s00220-011-1313-y. |
[33] |
F. Gay-Balmaz, D. D. Holm and T. S. Ratiu,
Higher order Lagrange-Poincaré and Hamilton-Poincaré reductions, Bulletin of the Brazilian Mathematical Society, 42 (2011), 579-606.
doi: 10.1007/s00574-011-0030-7. |
[34] |
M. J. Gotay and J. M. Nester,
Presymplectic Lagrangian systems. Ⅰ: The constraint algorithm and the equivalence theorem, Ann. Inst. H. Poincaré Sect. A (N.S.), 30 (1979), 129-142.
|
[35] |
M. J. Gotay and J. M. Nester, Generalized constraint algorithm and special presymplectic manifolds, Geometric Methods in Mathematical Physics (Proc. NSF-CBMS Conf., Univ.
Lowell, Lowell, Mass., 1979), Lecture Notes in Math., 775, Springer, Berlin, (1980), 78–104. |
[36] |
M. J. Gotay and J. M. Nester,
Apartheid in the Dirac theory of constraints, Journal of Physics A: Mathematical and General, 17 (1984), 3063-3066.
doi: 10.1088/0305-4470/17/15/023. |
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M. J. Gotay, J. M. Nester and G. Hinds,
Presymplectic manifolds and the Dirac Bergmann theory of constraints, Journal of Mathematical Physics, 19 (1978), 2388-2399.
doi: 10.1063/1.523597. |
[38] |
K. Grabowska and L. Vitagliano,
Tulczyjew triples in higher derivative field theory, Journal of Geometric Mechanics, 7 (2015), 1-33.
doi: 10.3934/jgm.2015.7.1. |
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show all references
References:
[1] |
R. Abraham and J. E. Marsden,
Foundations of Mechanics, Reading, Massachusetts, Benjamin/Cummings Publishing Company, 1978. |
[2] |
L. Abrunheiro and L. Colombo, Lagrangian Lie subalgebroids generating dynamics for second-order mechanical systems on Lie algebroids,
Mediterranean Journal of Mathematics, 15 (2018), Art. 57, 19 pp.
doi: 10.1007/s00009-018-1108-x. |
[3] |
K. Andrzejewski, J. Gonera, P. Machalski and P. Maś lanka, Modified Hamiltonian formalism for higher-derivative theories,
Physical Review D, 82 (2010), 045008.
doi: 10.1103/PhysRevD.82.045008. |
[4] |
V. I. Arnol'd,
Mathematical Methods of Classical Mechanics, Vol. 60, Springer-Verlag, New York, 1989.
doi: 10.1007/978-1-4757-2063-1. |
[5] |
C. Batlle, J. Gomis, J. M. Pons and N. Roman-Roy,
Equivalence between the Lagrangian and Hamiltonian formalism for constrained systems, Journal of Mathematical Physics, 27 (1986), 2953-2962.
doi: 10.1063/1.527274. |
[6] |
C. Batlle, J. Gomis, J. M. Pons and N. Roman-Roy,
Lagrangian and Hamiltonian constraints for second-order singular Lagrangians, Journal of Physics A: Mathematical and General, 21 (1988), 2693-2703.
doi: 10.1088/0305-4470/21/12/013. |
[7] |
S. Benenti,
Hamiltonian Structures and Generating Families, Springer Science & Business Media, 2011.
doi: 10.1007/978-1-4614-1499-5. |
[8] |
A. M. Bloch and P. E. Crouch,
On the equivalence of higher order variational problems and optimal control problems. In Decision and Control, 1996., Proceedings of the 35th IEEE Conference on, IEEE, 2 (1996), 1648-1653.
|
[9] |
K. Bolonek and P. Kosinski,
Hamiltonian structures for pais-uhlenbeck oscillator, Acta Physica Polonica B, 36 (2205), 2115.
|
[10] |
A. J. Bruce, K. Grabowska and J. Grabowski, Higher order mechanics on graded bundles,
Journal of Physics A: Mathematical and Theoretical, 48 (2015), 205203, 32pp.
doi: 10.1088/1751-8113/48/20/205203. |
[11] |
F. Çağatay Uçgun, O. Esen and H. Gümral, Reductions of topologically massive gravity Ⅰ: Hamiltonian analysis of second order degenerate Lagrangians,
Journal of Mathematical Physics, 59 (2018), 013510, 16pp.
doi: 10.1063/1.5021948. |
[12] |
C. M. Campos, M. de León, D. M. de Diego and J. Vankerschaver, Unambiguous formalism for higher order Lagrangian field theories,
Journal of Physics A: Mathematical and Theoretical, 42 (2009), 475207, 24pp.
doi: 10.1088/1751-8113/42/47/475207. |
[13] |
F. Cardin,
Morse families and constrained mechanical systems, Generalized hyperelastic materials. Meccanica, 26 (1991), 161-167.
|
[14] |
H. Cendra and S. D. Grillo, Lagrangian systems with higher order constraints,
Journal of Mathematical Physics, 48 (2007), 052904, 35pp.
doi: 10.1063/1.2740470. |
[15] |
T. J. Chen, M. Fasiello, E. A. Lim and A. J. Tolley, Higher derivative theories with constraints: Exorcising Ostrogradski's ghost,
Journal of Cosmology and Astroparticle Physics, 2 (2013), 042, front matter+17 pp. |
[16] |
G. Clément,
Particle-like solutions to topologically massive gravity, Classical and Quantum Gravity, 11 (1994), L115-L120.
|
[17] |
L. Colombo,
Second-order constrained variational problems on Lie algebroids: Applications to optimal control, Journal of Geometric Mechanics, 9 (2017), 1-45.
doi: 10.3934/jgm.2017001. |
[18] |
L. Colombo, M. de León, P. D. Prieto-Martínez and N. Román-Roy,
Unified formalism for the generalized kth-order Hamilton-Jacobi problem, International Journal of Geometric Methods
in Modern Physics, 11 (2014), 1460037, 9pp.
doi: 10.1142/S0219887814600378. |
[19] |
L. Colombo and D. M. de Diego,
Higher-order variational problems on Lie groups and optimal control applications, Journal Geometric Mechanics, 6 (2014), 451-478.
doi: 10.3934/jgm.2014.6.451. |
[20] |
L. Colombo and P. D. Prieto-Martínez, Regularity properties of fiber derivatives associated with higher-order mechanical systems,
Journal of Mathematical Physics, 57 (2016), 082901, 25pp.
doi: 10.1063/1.4960822. |
[21] |
M. Crampin, W. Sarlet and F. Cantrijn, Higher-order differential equations and higher-order
Lagrangian mechanics. In Mathematical Proceedings of the Cambridge Philosophical Society,
99 (1986), 565–587.
doi: 10.1017/S0305004100064501. |
[22] |
A. Deriglazov,
Classical Mechanics, Springer International Publishing, 2017.
doi: 10.1007/978-3-319-44147-4. |
[23] |
N. Deruelle, Y. Sendouda and A. Youssef, Various Hamiltonian formulations of f (R) gravity and their canonical relationships,
Physical Review D, 80 (2009), 084032, 11pp.
doi: 10.1103/PhysRevD.80.084032. |
[24] |
S. Deser, R. Jackiw and S. Templeton,
Topologically massive gauge theories, Annals of Physics, 140 (1982), 372-411.
doi: 10.1016/0003-4916(82)90164-6. |
[25] |
S. Deser, R. Jackiw and S. Templeton,
Three-dimensional massive gauge theories, Physical Review Letters, 48 (1982), 975-978.
doi: 10.1103/PhysRevLett.48.975. |
[26] |
P. A. M. Dirac,
Lectures on Quantum Mechanics, Belfer Graduate School of Science, Monograph Series, Yeshiva University, New York, 1967. |
[27] |
P. A. M. Dirac,
Generalized hamiltonian dynamics, Proceedings of the Royal Society of London A: Mathematical, Physical and Engineering Sciences, 246 (1958), 326-332.
doi: 10.1098/rspa.1958.0141. |
[28] |
C. T. J. Dodson and M. S. Radivoiovici,
Tangent and frame bundles of order two, An. Stiint. Univ. "Al. I. Cuza" Iasi Sect. I a Mat. (N.S.), 28 (1982), 63-71.
|
[29] |
C. T. Dodson, G. Galanis and E. Vassiliou,
Geometry in a Fréchet Context: A Projective Limit Approach, Cambridge University Press, 2016.
doi: 10.1017/CBO9781316556092. |
[30] |
O. Esen and H. Gümral,
Tulczyjew's triplet for Lie groups Ⅰ: Trivializations and reductions, Journal of Lie Theory, 24 (2014), 1115-1160.
|
[31] |
O. Esen and H. Gümral,
Tulczyjew's triplet for Lie groups. Ⅱ: Dynamics, Journal of Lie Theory, 27 (2017), 329-356.
|
[32] |
F. Gay-Balmaz, D. D. Holm, D. M. Meier, T. S. Ratiu and F.-X. Vialard,
Invariant higher-order variational problems, Communications in Mathematical Physics, 309 (2012), 413-458.
doi: 10.1007/s00220-011-1313-y. |
[33] |
F. Gay-Balmaz, D. D. Holm and T. S. Ratiu,
Higher order Lagrange-Poincaré and Hamilton-Poincaré reductions, Bulletin of the Brazilian Mathematical Society, 42 (2011), 579-606.
doi: 10.1007/s00574-011-0030-7. |
[34] |
M. J. Gotay and J. M. Nester,
Presymplectic Lagrangian systems. Ⅰ: The constraint algorithm and the equivalence theorem, Ann. Inst. H. Poincaré Sect. A (N.S.), 30 (1979), 129-142.
|
[35] |
M. J. Gotay and J. M. Nester, Generalized constraint algorithm and special presymplectic manifolds, Geometric Methods in Mathematical Physics (Proc. NSF-CBMS Conf., Univ.
Lowell, Lowell, Mass., 1979), Lecture Notes in Math., 775, Springer, Berlin, (1980), 78–104. |
[36] |
M. J. Gotay and J. M. Nester,
Apartheid in the Dirac theory of constraints, Journal of Physics A: Mathematical and General, 17 (1984), 3063-3066.
doi: 10.1088/0305-4470/17/15/023. |
[37] |
M. J. Gotay, J. M. Nester and G. Hinds,
Presymplectic manifolds and the Dirac Bergmann theory of constraints, Journal of Mathematical Physics, 19 (1978), 2388-2399.
doi: 10.1063/1.523597. |
[38] |
K. Grabowska and L. Vitagliano,
Tulczyjew triples in higher derivative field theory, Journal of Geometric Mechanics, 7 (2015), 1-33.
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