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

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

* 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.

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. BatlleJ. GomisJ. 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. BatlleJ. GomisJ. 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. ColomboM. de LeónP. 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. DeserR. 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. DeserR. 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-BalmazD. D. HolmD. M. MeierT. 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-BalmazD. 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. GotayJ. 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. GrabowskiK. 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. GráciaJ. 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. LawrukJ. 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 LeónJ. 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. BatlleJ. GomisJ. 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. BatlleJ. GomisJ. 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. ColomboM. de LeónP. 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. DeserR. 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. DeserR. 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-BalmazD. D. HolmD. M. MeierT. 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-BalmazD. 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. GotayJ. 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. GrabowskiK. 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. GráciaJ. 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. LawrukJ. 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 LeónJ. 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 & Continuous Dynamical Systems - B, 2014, 19 (1) : 299-322. doi: 10.3934/dcdsb.2014.19.299

[3]

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

[4]

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

[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]

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

[7]

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

[8]

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

[9]

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

[10]

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

[11]

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

[12]

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

[13]

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

[14]

Hongguang Xiao, Wen Tan, Dehua Xiang, Lifu Chen, Ning Li. A study of numerical integration based on Legendre polynomial and RLS algorithm. Numerical Algebra, Control & Optimization, 2017, 7 (4) : 457-464. doi: 10.3934/naco.2017028

[15]

Zhong-Qing Wang, Li-Lian Wang. A Legendre-Gauss collocation method for nonlinear delay differential equations. Discrete & Continuous Dynamical Systems - B, 2010, 13 (3) : 685-708. doi: 10.3934/dcdsb.2010.13.685

[16]

Annamaria Canino, Elisa De Giorgio, Berardino Sciunzi. Second order regularity for degenerate nonlinear elliptic equations. Discrete & Continuous Dynamical Systems - A, 2018, 38 (8) : 4231-4242. doi: 10.3934/dcds.2018184

[17]

Sze-Bi Hsu, Bernold Fiedler, Hsiu-Hau Lin. Classification of potential flows under renormalization group transformation. Discrete & 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 & 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 & Continuous Dynamical Systems - A, 2014, 34 (12) : 5061-5084. doi: 10.3934/dcds.2014.34.5061

2017 Impact Factor: 0.561

Metrics

  • PDF downloads (47)
  • HTML views (165)
  • Cited by (0)

Other articles
by authors

[Back to Top]