American Institute of Mathematical Sciences

Advanced
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.

[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.  doi: 10.3934/jgm.2015.7.1.

[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.

[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.

[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.

[42]

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

[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.

[44]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[56]

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

[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.

[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.

[59]

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

[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.

[61]

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

[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.

[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.

[64]

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

[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.

[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.

[67]

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

[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.

[69]

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

[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.

[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.

[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.

[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.

[74]

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

[75]

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

[76]

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

[77]

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

[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. 

[79]

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

[80]

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

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.  doi: 10.3934/jgm.2015.7.1.

[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.

[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.

[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.

[42]

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

[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.

[44]

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

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[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.

[56]

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

[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.

[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.

[59]

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

[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.

[61]

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

[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.

[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.

[64]

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

[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.

[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.

[67]

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

[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.

[69]

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

[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.

[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.

[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.

[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.

[74]

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

[75]

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

[76]

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

[77]

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

[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. 

[79]

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

[80]

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

[1]

E. García-Toraño Andrés, Bavo Langerock, Frans Cantrijn. Aspects of reduction and transformation of Lagrangian systems with symmetry. Journal of Geometric Mechanics, 2014, 6 (1) : 1-23. doi: 10.3934/jgm.2014.6.1
[2]

Lijun Yi, Zhongqing Wang. Legendre spectral collocation method for second-order nonlinear ordinary/partial differential equations. Discrete and Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299
[3]

N. Kamran, K. Tenenblat. Periodic systems for the higher-dimensional Laplace transformation. Discrete and Continuous Dynamical Systems, 1998, 4 (2) : 359-378. doi: 10.3934/dcds.1998.4.359
[4]

Katarzyna Grabowska, Marcin Zając. The Tulczyjew triple in mechanics on a Lie group. Journal of Geometric Mechanics, 2016, 8 (4) : 413-435. doi: 10.3934/jgm.2016014
[5]

Richard Hofer, Arne Winterhof. On the arithmetic autocorrelation of the Legendre sequence. Advances in Mathematics of Communications, 2017, 11 (1) : 237-244. doi: 10.3934/amc.2017015
[6]

Sabyasachi Karati, Palash Sarkar. Connecting Legendre with Kummer and Edwards. Advances in Mathematics of Communications, 2019, 13 (1) : 41-66. doi: 10.3934/amc.2019003
[7]

Chun-xiang Guo, Dong Cai, Yu-yang Tan. Outsourcing contract design for the green transformation of manufacturing systems under asymmetric information. Journal of Industrial and Management Optimization, 2021  doi: 10.3934/jimo.2021158
[8]

Nicolás Borda, Javier Fernández, Sergio Grillo. Discrete second order constrained Lagrangian systems: First results. Journal of Geometric Mechanics, 2013, 5 (4) : 381-397. doi: 10.3934/jgm.2013.5.381
[9]

Lyndsey Clark. The $\beta$-transformation with a hole. Discrete and Continuous Dynamical Systems, 2016, 36 (3) : 1249-1269. doi: 10.3934/dcds.2016.36.1249
[10]

Elisabetta Carlini, Francisco J. Silva. A semi-Lagrangian scheme for a degenerate second order mean field game system. Discrete and Continuous Dynamical Systems, 2015, 35 (9) : 4269-4292. doi: 10.3934/dcds.2015.35.4269
[11]

Jie Shen, Li-Lian Wang. Laguerre and composite Legendre-Laguerre Dual-Petrov-Galerkin methods for third-order equations. Discrete and Continuous Dynamical Systems - B, 2006, 6 (6) : 1381-1402. doi: 10.3934/dcdsb.2006.6.1381
[12]

Shan Li, Shi-Mi Yan, Zhong-Qing Wang. Efficient Legendre dual-Petrov-Galerkin methods for odd-order differential equations. Discrete and Continuous Dynamical Systems - B, 2020, 25 (4) : 1543-1563. doi: 10.3934/dcdsb.2019239
[13]

Marc Chamberland, Victor H. Moll. Dynamics of the degree six Landen transformation. Discrete and Continuous Dynamical Systems, 2006, 15 (3) : 905-919. doi: 10.3934/dcds.2006.15.905
[14]

Irena Pawłow, Wojciech M. Zajączkowski. The global solvability of a sixth order Cahn-Hilliard type equation via the Bäcklund transformation. Communications on Pure and Applied Analysis, 2014, 13 (2) : 859-880. doi: 10.3934/cpaa.2014.13.859
[15]

Tomáš Bárta. Exact rate of decay for solutions to damped second order ODE's with a degenerate potential. Evolution Equations and Control Theory, 2018, 7 (4) : 531-543. doi: 10.3934/eect.2018025
[16]

Jean Mawhin. Periodic solutions of second order Lagrangian difference systems with bounded or singular $\phi$-Laplacian and periodic potential. Discrete and Continuous Dynamical Systems - S, 2013, 6 (4) : 1065-1076. doi: 10.3934/dcdss.2013.6.1065
[17]

Sze-Bi Hsu, Bernold Fiedler, Hsiu-Hau Lin. Classification of potential flows under renormalization group transformation. Discrete and Continuous Dynamical Systems - B, 2016, 21 (2) : 437-446. doi: 10.3934/dcdsb.2016.21.437
[18]

Hyukjin Kwean. Kwak transformation and Navier-Stokes equations. Communications on Pure and Applied Analysis, 2004, 3 (3) : 433-446. doi: 10.3934/cpaa.2004.3.433
[19]

Andrey Kochergin. A Besicovitch cylindrical transformation with Hölder function. Electronic Research Announcements, 2015, 22: 87-91. doi: 10.3934/era.2015.22.87
[20]

Xian Chen, Zhi-Ming Ma. A transformation of Markov jump processes and applications in genetic study. Discrete and Continuous Dynamical Systems, 2014, 34 (12) : 5061-5084. doi: 10.3934/dcds.2014.34.5061

2020 Impact Factor: 0.857

Tools

Metrics

  • PDF downloads (363)
  • HTML views (220)
  • Cited by (2)

Other articles
by authors

[Back to Top]

Copyright © 2022 American Institute of Mathematical Sciences | Privacy Policy