American Institute of Mathematical Sciences

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September  2018, 10(3): 251-291. doi: 10.3934/jgm.2018010

On the geometry of the Schmidt-Legendre transformation

Oğul Esen 1,, and  Partha Guha 2,

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

* Corresponding author

Received  February 2017 Revised  August 2018 Published  August 2018

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.

Keywords: Ostrogradsky-Legendre transformation, Schmidt-Legendre transformation, second order degenerate Lagrangian systems, Tulczyjew's triple.
Mathematics Subject Classification: Primary: 70H50; Secondary: 70H45.
Citation: Oğul Esen, Partha Guha. On the geometry of the Schmidt-Legendre transformation. Journal of Geometric Mechanics, 2018, 10 (3) : 251-291. doi: 10.3934/jgm.2018010
References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, Reading, Massachusetts, Benjamin/Cummings Publishing Company, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. Benenti, Hamiltonian Structures and Generating Families, Springer Science & Business Media, 2011. doi: 10.1007/978-1-4614-1499-5.  Google Scholar

[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.   Google Scholar

[9]

K. Bolonek and P. Kosinski, Hamiltonian structures for pais-uhlenbeck oscillator, Acta Physica Polonica B, 36 (2205), 2115.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

F. Cardin, Morse families and constrained mechanical systems, Generalized hyperelastic materials. Meccanica, 26 (1991), 161-167.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

G. Clément, Particle-like solutions to topologically massive gravity, Classical and Quantum Gravity, 11 (1994), L115-L120.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

A. Deriglazov, Classical Mechanics, Springer International Publishing, 2017. doi: 10.1007/978-3-319-44147-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science, Monograph Series, Yeshiva University, New York, 1967.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[30]

O. Esen and H. Gümral, Tulczyjew's triplet for Lie groups Ⅰ: Trivializations and reductions, Journal of Lie Theory, 24 (2014), 1115-1160.   Google Scholar

[31]

O. Esen and H. Gümral, Tulczyjew's triplet for Lie groups. Ⅱ: Dynamics, Journal of Lie Theory, 27 (2017), 329-356.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

K. Grabowska and M. Zajac, The Tulczyjew triple in mechanics on a Lie group, Journal of Geometric Mechanics, 8 (2016), 413-435.  doi: 10.3934/jgm.2016014.  Google Scholar

[40]

J. Grabowski, K. Grabowska and P. Urbański, Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings, Journal of Geometric Mechanics, 6 (2014), 503-526.  doi: 10.3934/jgm.2014.6.503.  Google Scholar

[41]

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

[42]

S. W. Hawking and T. Hertog, Living with ghosts, Physical Review D, 65 (2002), 103515, 8pp. doi: 10.1103/PhysRevD.65.103515.  Google Scholar

[43]

M. Jóźwikowski and M. Rotkiewicz, Models for higher algebroids, Journal of Geometric Mechanics, 7 (2015), 317-359.  doi: 10.3934/jgm.2015.7.317.  Google Scholar

[44]

M. Jóźwikowski, Prolongations vs. Tulczyjew triples in Geometric Mechanics, arXiv preprint, arXiv: 1712.09858, (2017). Google Scholar

[45]

U. Kasper, Finding the Hamiltonian for cosmological models in fourth-order gravity theories without resorting to the Ostrogradski or Dirac formalism, General Relativity and Gravitation, 29 (1997), 221-233.  doi: 10.1023/A:1010292128733.  Google Scholar

[46]

B. Lawruk, J. Śniatycki and W. M. Tulczyjew, Special symplectic spaces, Journal of Differential Equations, 17 (1975), 477-497.  doi: 10.1016/0022-0396(75)90057-1.  Google Scholar

[47]

M. de León and D. M. de Diego, Symmetries and constants of the motion for higher order Lagrangian systems, Journal of Mathematical Physics, 36 (1995), 4138-4161.  doi: 10.1063/1.530952.  Google Scholar

[48]

M. de León and E. A. Lacomba, Lagrangian submanifolds and higher-order mechanical systems, Journal of Physics A: Mathematical and General, 22 (1989), 3809-3820.  doi: 10.1088/0305-4470/22/18/019.  Google Scholar

[49]

M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, Journal of Physics A: Mathematical and General, 38 (2005), R241-R308.  doi: 10.1088/0305-4470/38/24/R01.  Google Scholar

[50]

M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory: A Geometrical Approach of Lagrangian and Hamiltonian Formalisms Involving Higher Order Derivatives, North-Holland Publishing Co., Amsterdam, 1985. doi: 10.1088/0266-5611/8/4/006.  Google Scholar

[51]

P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.  Google Scholar

[52]

P. D. Mannheim and A. Davidson, Dirac quantization of the Pais-Uhlenbeck fourth order oscillator, Physical Review A, 71 (2005), 042110, 9pp. doi: 10.1103/PhysRevA.71.042110.  Google Scholar

[53]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Second edition. Texts in Applied Mathematics, 17 Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[54]

E. Martínez, Higher-order variational calculus on Lie algebroids, Journal of Geometric Mechanics, 7 (2015), 81-108.  doi: 10.3934/jgm.2015.7.81.  Google Scholar

[55]

I. Masterov, An alternative Hamiltonian formulation for the Pais-Uhlenbeck oscillator, Nuclear Physics B, 902 (2016), 95-114.  doi: 10.1016/j.nuclphysb.2015.11.011.  Google Scholar

[56]

I. Masterov, The odd-order Pais-Uhlenbeck oscillator, Nuclear Physics B, 907 (2016), 495-508.  doi: 10.1016/j.nuclphysb.2016.04.025.  Google Scholar

[57]

R. Miron, The Geometry of Higher-Order Lagrange Spaces: Applications to Mechanics and Physics, Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-94-017-3338-0.  Google Scholar

[58]

A. Mostafazadeh, A Hamiltonian formulation of the Pais-Uhlenbeck oscillator that yields a stable and unitary quantum system, Physics Letters A, 375 (2010), 93-98.  doi: 10.1016/j.physleta.2010.10.050.  Google Scholar

[59]

M. Nakahara, Geometry, Topology and Physics, Institute of Physics, Bristol, 2003. doi: 10.1201/9781420056945.  Google Scholar

[60]

N. V. Nesterenko, Singular Lagrangians with higher derivatives, Journal of Physics A: Mathematical and General, 22 (1989), 1673-1687.  doi: 10.1088/0305-4470/22/10/021.  Google Scholar

[61]

M. Ostrogradsky, Mem. Ac. St. Petersbourg. Mem. Ac. St. Petersbourg, 14, (19850) 385. Google Scholar

[62]

A. Pais and G. E. Uhlenbeck, On field theories with non-localized action, Physical Review(2), 79 (1950), 145-165.  doi: 10.1103/PhysRev.79.145.  Google Scholar

[63]

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

[64]

J. M. Pons, Ostrogradski's theorem for higher-order singular Lagrangians, Lett. Math. Phys., 17 (1989), 181-189.  doi: 10.1007/BF00401583.  Google Scholar

[65]

M. S. Rashid and S. S. Khalil, Hamiltonian description of higher order Lagrangians, International Journal of Modern Physics A, 11 (1996), 4551-4559.  doi: 10.1142/S0217751X96002108.  Google Scholar

[66]

H. J. Schmidt, Stability and Hamiltonian formulation of higher derivative theories, Physical Review D, 49 (1994), 6354-6366.  doi: 10.1103/PhysRevD.49.6354.  Google Scholar

[67]

H. J. Schmidt, An alternate Hamiltonian formulation of fourth-order theories and its application to cosmology, arXiv preprint, arXiv: 9501019. Google Scholar

[68]

Ö Sarıoğlu and B. Tekin, Topologically massive gravity as a Pais-Uhlenbeck oscillator, Classical and Quantum Gravity, 23 (2006), 7541-7549.  doi: 10.1088/0264-9381/23/24/023.  Google Scholar

[69]

R. Skinner, First order equations of motion for classical mechanics, Journal of Mathematical Physics, 24 (1983), 2581-2588.  doi: 10.1063/1.525653.  Google Scholar

[70]

R. Skinner and R. Rusk, Generalized Hamiltonian dynamics. Ⅰ. Formulation on ${T^*}Q{\rm{ }} \oplus TQ$, Journal of Mathematical Physics, 24 (1983), 2589-2594.  doi: 10.1063/1.525654.  Google Scholar

[71]

R. Skinner and R. Rusk, Generalized Hamiltonian dynamics. Ⅱ. Gauge transformations, Journal of Mathematical Physics, 24 (1983), 2595-2601.  doi: 10.1063/1.525655.  Google Scholar

[72]

J. Śniatycki and W. M. Tulczyjew, Generating forms of Lagrangian submanifolds, Indiana Univ. Math. J., 22 (1972/73), 267-275.  doi: 10.1512/iumj.1973.22.22021.  Google Scholar

[73]

A. Suri, Geometry of the double tangent bundles of Banach manifolds, Journal of Geometry and Physics, 74 (2013), 91-100.  doi: 10.1016/j.geomphys.2013.07.009.  Google Scholar

[74]

A. Suri, Higher order tangent bundles, Mediterr. J. Math. , 14 (2017), Art. 5, 17pp. doi: 10.1007/s00009-016-0812-7.  Google Scholar

[75]

W. M. Tulczyjew, A symplectic formulation of relativistic particle dynamics, Acta Physica Polonica B, 8 (1977), 431-447.   Google Scholar

[76]

W. M. Tulczyjew, The Legendre transformation, Ann. Inst. Henri Poincaré Sec. A: Phys. Théor., 27 (1977), 101-114.   Google Scholar

[77]

W. M. Tulczyjew, A symplectic formulation of particle dynamics, Differential Geometric Methods in Mathematical Physics, Lect. Notes in Math., 570 (1977), 457–463.  Google Scholar

[78]

W. M. Tulczyjew and P. Urbanski, A slow and careful legendre transformation for singular lagrangians, Acta Physica Polonica. Series B, 30 (1999), 2909-2978.   Google Scholar

[79]

A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advan. in Math., 6 (1971), 329-346.  doi: 10.1016/0001-8708(71)90020-X.  Google Scholar

[80]

A. Weinstein, Lectures on Symplectic Manifolds, Exp. lec. from the CBMS, Regional Conference Series in Mathematics, 29 A. M. S., Providence, 1977.  Google Scholar

show all references

References:
[1]

R. Abraham and J. E. Marsden, Foundations of Mechanics, Reading, Massachusetts, Benjamin/Cummings Publishing Company, 1978.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[7]

S. Benenti, Hamiltonian Structures and Generating Families, Springer Science & Business Media, 2011. doi: 10.1007/978-1-4614-1499-5.  Google Scholar

[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.   Google Scholar

[9]

K. Bolonek and P. Kosinski, Hamiltonian structures for pais-uhlenbeck oscillator, Acta Physica Polonica B, 36 (2205), 2115.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[13]

F. Cardin, Morse families and constrained mechanical systems, Generalized hyperelastic materials. Meccanica, 26 (1991), 161-167.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[16]

G. Clément, Particle-like solutions to topologically massive gravity, Classical and Quantum Gravity, 11 (1994), L115-L120.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[22]

A. Deriglazov, Classical Mechanics, Springer International Publishing, 2017. doi: 10.1007/978-3-319-44147-4.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[26]

P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science, Monograph Series, Yeshiva University, New York, 1967.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[30]

O. Esen and H. Gümral, Tulczyjew's triplet for Lie groups Ⅰ: Trivializations and reductions, Journal of Lie Theory, 24 (2014), 1115-1160.   Google Scholar

[31]

O. Esen and H. Gümral, Tulczyjew's triplet for Lie groups. Ⅱ: Dynamics, Journal of Lie Theory, 27 (2017), 329-356.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.   Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[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.  Google Scholar

[39]

K. Grabowska and M. Zajac, The Tulczyjew triple in mechanics on a Lie group, Journal of Geometric Mechanics, 8 (2016), 413-435.  doi: 10.3934/jgm.2016014.  Google Scholar

[40]

J. Grabowski, K. Grabowska and P. Urbański, Geometry of Lagrangian and Hamiltonian formalisms in the dynamics of strings, Journal of Geometric Mechanics, 6 (2014), 503-526.  doi: 10.3934/jgm.2014.6.503.  Google Scholar

[41]

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

[42]

S. W. Hawking and T. Hertog, Living with ghosts, Physical Review D, 65 (2002), 103515, 8pp. doi: 10.1103/PhysRevD.65.103515.  Google Scholar

[43]

M. Jóźwikowski and M. Rotkiewicz, Models for higher algebroids, Journal of Geometric Mechanics, 7 (2015), 317-359.  doi: 10.3934/jgm.2015.7.317.  Google Scholar

[44]

M. Jóźwikowski, Prolongations vs. Tulczyjew triples in Geometric Mechanics, arXiv preprint, arXiv: 1712.09858, (2017). Google Scholar

[45]

U. Kasper, Finding the Hamiltonian for cosmological models in fourth-order gravity theories without resorting to the Ostrogradski or Dirac formalism, General Relativity and Gravitation, 29 (1997), 221-233.  doi: 10.1023/A:1010292128733.  Google Scholar

[46]

B. Lawruk, J. Śniatycki and W. M. Tulczyjew, Special symplectic spaces, Journal of Differential Equations, 17 (1975), 477-497.  doi: 10.1016/0022-0396(75)90057-1.  Google Scholar

[47]

M. de León and D. M. de Diego, Symmetries and constants of the motion for higher order Lagrangian systems, Journal of Mathematical Physics, 36 (1995), 4138-4161.  doi: 10.1063/1.530952.  Google Scholar

[48]

M. de León and E. A. Lacomba, Lagrangian submanifolds and higher-order mechanical systems, Journal of Physics A: Mathematical and General, 22 (1989), 3809-3820.  doi: 10.1088/0305-4470/22/18/019.  Google Scholar

[49]

M. de León, J. C. Marrero and E. Martínez, Lagrangian submanifolds and dynamics on Lie algebroids, Journal of Physics A: Mathematical and General, 38 (2005), R241-R308.  doi: 10.1088/0305-4470/38/24/R01.  Google Scholar

[50]

M. de León and P. R. Rodrigues, Generalized Classical Mechanics and Field Theory: A Geometrical Approach of Lagrangian and Hamiltonian Formalisms Involving Higher Order Derivatives, North-Holland Publishing Co., Amsterdam, 1985. doi: 10.1088/0266-5611/8/4/006.  Google Scholar

[51]

P. Libermann and C.-M. Marle, Symplectic Geometry and Analytical Mechanics, D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.  Google Scholar

[52]

P. D. Mannheim and A. Davidson, Dirac quantization of the Pais-Uhlenbeck fourth order oscillator, Physical Review A, 71 (2005), 042110, 9pp. doi: 10.1103/PhysRevA.71.042110.  Google Scholar

[53]

J. E. Marsden and T. S. Ratiu, Introduction to Mechanics and Symmetry: A Basic Exposition of Classical Mechanical Systems, Second edition. Texts in Applied Mathematics, 17 Springer-Verlag, New York, 1999. doi: 10.1007/978-0-387-21792-5.  Google Scholar

[54]

E. Martínez, Higher-order variational calculus on Lie algebroids, Journal of Geometric Mechanics, 7 (2015), 81-108.  doi: 10.3934/jgm.2015.7.81.  Google Scholar

[55]

I. Masterov, An alternative Hamiltonian formulation for the Pais-Uhlenbeck oscillator, Nuclear Physics B, 902 (2016), 95-114.  doi: 10.1016/j.nuclphysb.2015.11.011.  Google Scholar

[56]

I. Masterov, The odd-order Pais-Uhlenbeck oscillator, Nuclear Physics B, 907 (2016), 495-508.  doi: 10.1016/j.nuclphysb.2016.04.025.  Google Scholar

[57]

R. Miron, The Geometry of Higher-Order Lagrange Spaces: Applications to Mechanics and Physics, Kluwer Academic Publishers Group, Dordrecht, 1997. doi: 10.1007/978-94-017-3338-0.  Google Scholar

[58]

A. Mostafazadeh, A Hamiltonian formulation of the Pais-Uhlenbeck oscillator that yields a stable and unitary quantum system, Physics Letters A, 375 (2010), 93-98.  doi: 10.1016/j.physleta.2010.10.050.  Google Scholar

[59]

M. Nakahara, Geometry, Topology and Physics, Institute of Physics, Bristol, 2003. doi: 10.1201/9781420056945.  Google Scholar

[60]

N. V. Nesterenko, Singular Lagrangians with higher derivatives, Journal of Physics A: Mathematical and General, 22 (1989), 1673-1687.  doi: 10.1088/0305-4470/22/10/021.  Google Scholar

[61]

M. Ostrogradsky, Mem. Ac. St. Petersbourg. Mem. Ac. St. Petersbourg, 14, (19850) 385. Google Scholar

[62]

A. Pais and G. E. Uhlenbeck, On field theories with non-localized action, Physical Review(2), 79 (1950), 145-165.  doi: 10.1103/PhysRev.79.145.  Google Scholar

[63]

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

[64]

J. M. Pons, Ostrogradski's theorem for higher-order singular Lagrangians, Lett. Math. Phys., 17 (1989), 181-189.  doi: 10.1007/BF00401583.  Google Scholar

[65]

M. S. Rashid and S. S. Khalil, Hamiltonian description of higher order Lagrangians, International Journal of Modern Physics A, 11 (1996), 4551-4559.  doi: 10.1142/S0217751X96002108.  Google Scholar

[66]

H. J. Schmidt, Stability and Hamiltonian formulation of higher derivative theories, Physical Review D, 49 (1994), 6354-6366.  doi: 10.1103/PhysRevD.49.6354.  Google Scholar

[67]

H. J. Schmidt, An alternate Hamiltonian formulation of fourth-order theories and its application to cosmology, arXiv preprint, arXiv: 9501019. Google Scholar

[68]

Ö Sarıoğlu and B. Tekin, Topologically massive gravity as a Pais-Uhlenbeck oscillator, Classical and Quantum Gravity, 23 (2006), 7541-7549.  doi: 10.1088/0264-9381/23/24/023.  Google Scholar

[69]

R. Skinner, First order equations of motion for classical mechanics, Journal of Mathematical Physics, 24 (1983), 2581-2588.  doi: 10.1063/1.525653.  Google Scholar

[70]

R. Skinner and R. Rusk, Generalized Hamiltonian dynamics. Ⅰ. Formulation on ${T^*}Q{\rm{ }} \oplus TQ$, Journal of Mathematical Physics, 24 (1983), 2589-2594.  doi: 10.1063/1.525654.  Google Scholar

[71]

R. Skinner and R. Rusk, Generalized Hamiltonian dynamics. Ⅱ. Gauge transformations, Journal of Mathematical Physics, 24 (1983), 2595-2601.  doi: 10.1063/1.525655.  Google Scholar

[72]

J. Śniatycki and W. M. Tulczyjew, Generating forms of Lagrangian submanifolds, Indiana Univ. Math. J., 22 (1972/73), 267-275.  doi: 10.1512/iumj.1973.22.22021.  Google Scholar

[73]

A. Suri, Geometry of the double tangent bundles of Banach manifolds, Journal of Geometry and Physics, 74 (2013), 91-100.  doi: 10.1016/j.geomphys.2013.07.009.  Google Scholar

[74]

A. Suri, Higher order tangent bundles, Mediterr. J. Math. , 14 (2017), Art. 5, 17pp. doi: 10.1007/s00009-016-0812-7.  Google Scholar

[75]

W. M. Tulczyjew, A symplectic formulation of relativistic particle dynamics, Acta Physica Polonica B, 8 (1977), 431-447.   Google Scholar

[76]

W. M. Tulczyjew, The Legendre transformation, Ann. Inst. Henri Poincaré Sec. A: Phys. Théor., 27 (1977), 101-114.   Google Scholar

[77]

W. M. Tulczyjew, A symplectic formulation of particle dynamics, Differential Geometric Methods in Mathematical Physics, Lect. Notes in Math., 570 (1977), 457–463.  Google Scholar

[78]

W. M. Tulczyjew and P. Urbanski, A slow and careful legendre transformation for singular lagrangians, Acta Physica Polonica. Series B, 30 (1999), 2909-2978.   Google Scholar

[79]

A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advan. in Math., 6 (1971), 329-346.  doi: 10.1016/0001-8708(71)90020-X.  Google Scholar

[80]

A. Weinstein, Lectures on Symplectic Manifolds, Exp. lec. from the CBMS, Regional Conference Series in Mathematics, 29 A. M. S., Providence, 1977.  Google Scholar

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