March  2019, 11(1): 23-44. doi: 10.3934/jgm.2019002

Geometry of Routh reduction

Faculty of Physics, University of Warsaw, ul. Pasteura 5, 02-093 Warszawa, Poland

* Corresponding author

Received  February 2018 Revised  December 2018 Published  January 2019

Fund Project: Research founded by the Polish National Science Centre grant under the contract number DEC-2012/06/A/ST1/00256.

The Routh reduction for Lagrangian systems with cyclic variable is presented as an example of a Lagrangian reduction. It appears that the Routhian, which is a generating object of reduced dynamics, is not a function any more but a section of a bundle of affine values.

Citation: Katarzyna Grabowska, Paweƚ Urbański. Geometry of Routh reduction. Journal of Geometric Mechanics, 2019, 11 (1) : 23-44. doi: 10.3934/jgm.2019002
References:
[1]

L. Adamec, A route to routh the classical setting, Journal of Nonlinear Mathematical Physics, 18 (2011), 87-107.  doi: 10.1142/S1402925111001180.  Google Scholar

[2]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, Berlin, 1997.  Google Scholar

[3]

S. L. Bażański, The Jacobi variational principle revisited, in Classical and Quantum Integrability (Warsaw, 2001), Banach Center Publ., 59 (2003), 99-111. doi: 10.4064/bc59-0-4.  Google Scholar

[4]

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

[5]

M. Crampin and T. Mestdag, Routh procedure for non-Abelian symmetry groups, J. Math. Phys., 49 (2008), 032901, 28 pp. doi: 10.1063/1.2885077.  Google Scholar

[6]

J.-P. Dufour, Introduction aux tissus, (preprint), Séminaire GETODIM, (1991), 55-76.  Google Scholar

[7]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor., 41 (2008), 175204, 25pp. doi: 10.1088/1751-8113/41/17/175204.  Google Scholar

[8]

K. GrabowskaJ. Grabowski and P. Urbański, AV-differential geometry: Poisson and Jacobi structures, J. Geom. Phys., 52 (2004), 398-446.  doi: 10.1016/j.geomphys.2004.04.004.  Google Scholar

[9]

K. GrabowskaJ. Grabowski and P. Urbański, AV-differential geometry: Euler-Lagrange equations, J. Geom. Phys., 57 (2007), 1984-1998.  doi: 10.1016/j.geomphys.2007.04.003.  Google Scholar

[10]

K. GrabowskaJ. Grabowski and P. Urbański, Geometrical Mechanics on algebroids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559-575.  doi: 10.1142/S0219887806001259.  Google Scholar

[11]

K. Grabowska and L. Vitagliano, Tulczyjew triples in higher derivative field theory, J. Geom. Mech., 7 (2015), 1-33.  doi: 10.3934/jgm.2015.7.1.  Google Scholar

[12]

J. Grabowski and M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys., 59 (2009), 1285-1305.  doi: 10.1016/j.geomphys.2009.06.009.  Google Scholar

[13]

J. GrabowskiM. Rotkiewicz and P. Urbanski, Double Affine Bundles, J. Geom. Phys., 60 (2010), 581-598.  doi: 10.1016/j.geomphys.2009.12.008.  Google Scholar

[14]

J. Grabowski and P. Urbanski, Tangent lifts of Poisson and related structures, J. Phys. A: Math. Gen., 28 (1995), 6743-6777.  doi: 10.1088/0305-4470/28/23/024.  Google Scholar

[15]

J. Grabowski and P. Urbanski, Algebroids general differential calculi on vector bundles, J. Geom. Phys, 31 (1999), 111-141.  doi: 10.1016/S0393-0440(99)00007-8.  Google Scholar

[16]

K. Konieczna and P. Urbański, Double vector bundles and duality, Arch. Math. (Brno), 35 (1999), 59-95.   Google Scholar

[17]

B. Langerock, E. García-Toraño Andrés and F. Cantrijn, Routh reduction and the class of magnetic Lagrangian systems, J. Math. Phys., 53 (2012), 062902, 19pp. doi: 10.1063/1.4723841.  Google Scholar

[18]

B. Langerock, F. Cantrijn and J. Vankerschaver, Routhian reduction for quasi-invariant Lagrangians, J. Math. Phys., 51 (2010), 022902, 20pp. doi: 10.1063/1.3277181.  Google Scholar

[19]

P. Liebermann and Ch. M. Marle, Symplectic Geometry and Analytical Mechanics, Reidel Publishing Company, Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.  Google Scholar

[20]

J. E. MarsdenT. S. Ratiu and J. Scheurle, Reduction theory and the Lagrange-Routh equations, Journal of Mathematical Physics, 41 (2000), 3379-3429.  doi: 10.1063/1.533317.  Google Scholar

[21]

T. Mestdag, Finsler geodesics of Lagrangian systems through Routh reduction, Mediterranean Journal of Mathematics, 13 (2016), 825-839.  doi: 10.1007/s00009-014-0505-z.  Google Scholar

[22]

J. Pradines, Fibrés Vectoriels Doubles et Calcul Des Jets Non-Holonomes, Notes polycopiés Amiens, 1977 (in French).  Google Scholar

[23]

E. J. Routh, Stability of a Given State of Motion, Halsted Press, New York, 1877. Google Scholar

[24]

W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sc. Paris, 283 (1976), 15-18.   Google Scholar

[25]

W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sc. Paris, 283 (1976), 675-678.   Google Scholar

[26]

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

[27]

W. M. Tulczyjew, Geometric Formulation of Physical Theories, Bibliopolis, Naples, 1989.  Google Scholar

[28]

W. M. Tulczyjew and P. Urbański, An affine framework for the dynamics of charged particles, Atti Accad. Sci. Torino, Suppl., 126 (1992), 257-265.  Google Scholar

[29]

W. M. Tulczyjew and P. Urbański, A slow and careful Legendre transformation for singular Lagrangians, The Infeld Centennial Meeting (Warsaw, 1998), Acta Phys. Polon. B, 30 (1999), 2909-2978.   Google Scholar

[30]

W. M. TulczyjewP. Urbański and S. Zakrzewski, A pseudocategory of principal bundles, Atti Accad. Sci. Torino, 122 (1988), 66-72.   Google Scholar

[31]

P. Urbański, An affine framework for analytical mechanics, in Classical and quantum integrability (Warsaw, 2001), Banach Center Publ., 59 (2003), 257-279. doi: 10.4064/bc59-0-14.  Google Scholar

[32]

P. Urbański, Double vector bundles in classical mechanics, Rend. Sem. Mat. Univ. Poi. Torino, 54 (1996), 405-421.   Google Scholar

show all references

References:
[1]

L. Adamec, A route to routh the classical setting, Journal of Nonlinear Mathematical Physics, 18 (2011), 87-107.  doi: 10.1142/S1402925111001180.  Google Scholar

[2]

V. I. Arnold, V. V. Kozlov and A. I. Neishtadt, Mathematical Aspects of Classical and Celestial Mechanics, Springer-Verlag, Berlin, 1997.  Google Scholar

[3]

S. L. Bażański, The Jacobi variational principle revisited, in Classical and Quantum Integrability (Warsaw, 2001), Banach Center Publ., 59 (2003), 99-111. doi: 10.4064/bc59-0-4.  Google Scholar

[4]

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

[5]

M. Crampin and T. Mestdag, Routh procedure for non-Abelian symmetry groups, J. Math. Phys., 49 (2008), 032901, 28 pp. doi: 10.1063/1.2885077.  Google Scholar

[6]

J.-P. Dufour, Introduction aux tissus, (preprint), Séminaire GETODIM, (1991), 55-76.  Google Scholar

[7]

K. Grabowska and J. Grabowski, Variational calculus with constraints on general algebroids, J. Phys. A: Math. Theor., 41 (2008), 175204, 25pp. doi: 10.1088/1751-8113/41/17/175204.  Google Scholar

[8]

K. GrabowskaJ. Grabowski and P. Urbański, AV-differential geometry: Poisson and Jacobi structures, J. Geom. Phys., 52 (2004), 398-446.  doi: 10.1016/j.geomphys.2004.04.004.  Google Scholar

[9]

K. GrabowskaJ. Grabowski and P. Urbański, AV-differential geometry: Euler-Lagrange equations, J. Geom. Phys., 57 (2007), 1984-1998.  doi: 10.1016/j.geomphys.2007.04.003.  Google Scholar

[10]

K. GrabowskaJ. Grabowski and P. Urbański, Geometrical Mechanics on algebroids, Int. J. Geom. Meth. Mod. Phys., 3 (2006), 559-575.  doi: 10.1142/S0219887806001259.  Google Scholar

[11]

K. Grabowska and L. Vitagliano, Tulczyjew triples in higher derivative field theory, J. Geom. Mech., 7 (2015), 1-33.  doi: 10.3934/jgm.2015.7.1.  Google Scholar

[12]

J. Grabowski and M. Rotkiewicz, Higher vector bundles and multi-graded symplectic manifolds, J. Geom. Phys., 59 (2009), 1285-1305.  doi: 10.1016/j.geomphys.2009.06.009.  Google Scholar

[13]

J. GrabowskiM. Rotkiewicz and P. Urbanski, Double Affine Bundles, J. Geom. Phys., 60 (2010), 581-598.  doi: 10.1016/j.geomphys.2009.12.008.  Google Scholar

[14]

J. Grabowski and P. Urbanski, Tangent lifts of Poisson and related structures, J. Phys. A: Math. Gen., 28 (1995), 6743-6777.  doi: 10.1088/0305-4470/28/23/024.  Google Scholar

[15]

J. Grabowski and P. Urbanski, Algebroids general differential calculi on vector bundles, J. Geom. Phys, 31 (1999), 111-141.  doi: 10.1016/S0393-0440(99)00007-8.  Google Scholar

[16]

K. Konieczna and P. Urbański, Double vector bundles and duality, Arch. Math. (Brno), 35 (1999), 59-95.   Google Scholar

[17]

B. Langerock, E. García-Toraño Andrés and F. Cantrijn, Routh reduction and the class of magnetic Lagrangian systems, J. Math. Phys., 53 (2012), 062902, 19pp. doi: 10.1063/1.4723841.  Google Scholar

[18]

B. Langerock, F. Cantrijn and J. Vankerschaver, Routhian reduction for quasi-invariant Lagrangians, J. Math. Phys., 51 (2010), 022902, 20pp. doi: 10.1063/1.3277181.  Google Scholar

[19]

P. Liebermann and Ch. M. Marle, Symplectic Geometry and Analytical Mechanics, Reidel Publishing Company, Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.  Google Scholar

[20]

J. E. MarsdenT. S. Ratiu and J. Scheurle, Reduction theory and the Lagrange-Routh equations, Journal of Mathematical Physics, 41 (2000), 3379-3429.  doi: 10.1063/1.533317.  Google Scholar

[21]

T. Mestdag, Finsler geodesics of Lagrangian systems through Routh reduction, Mediterranean Journal of Mathematics, 13 (2016), 825-839.  doi: 10.1007/s00009-014-0505-z.  Google Scholar

[22]

J. Pradines, Fibrés Vectoriels Doubles et Calcul Des Jets Non-Holonomes, Notes polycopiés Amiens, 1977 (in French).  Google Scholar

[23]

E. J. Routh, Stability of a Given State of Motion, Halsted Press, New York, 1877. Google Scholar

[24]

W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sc. Paris, 283 (1976), 15-18.   Google Scholar

[25]

W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sc. Paris, 283 (1976), 675-678.   Google Scholar

[26]

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

[27]

W. M. Tulczyjew, Geometric Formulation of Physical Theories, Bibliopolis, Naples, 1989.  Google Scholar

[28]

W. M. Tulczyjew and P. Urbański, An affine framework for the dynamics of charged particles, Atti Accad. Sci. Torino, Suppl., 126 (1992), 257-265.  Google Scholar

[29]

W. M. Tulczyjew and P. Urbański, A slow and careful Legendre transformation for singular Lagrangians, The Infeld Centennial Meeting (Warsaw, 1998), Acta Phys. Polon. B, 30 (1999), 2909-2978.   Google Scholar

[30]

W. M. TulczyjewP. Urbański and S. Zakrzewski, A pseudocategory of principal bundles, Atti Accad. Sci. Torino, 122 (1988), 66-72.   Google Scholar

[31]

P. Urbański, An affine framework for analytical mechanics, in Classical and quantum integrability (Warsaw, 2001), Banach Center Publ., 59 (2003), 257-279. doi: 10.4064/bc59-0-14.  Google Scholar

[32]

P. Urbański, Double vector bundles in classical mechanics, Rend. Sem. Mat. Univ. Poi. Torino, 54 (1996), 405-421.   Google Scholar

[1]

Rama Ayoub, Aziz Hamdouni, Dina Razafindralandy. A new Hodge operator in discrete exterior calculus. Application to fluid mechanics. Communications on Pure & Applied Analysis, , () : -. doi: 10.3934/cpaa.2021062

[2]

Manuel de León, Víctor M. Jiménez, Manuel Lainz. Contact Hamiltonian and Lagrangian systems with nonholonomic constraints. Journal of Geometric Mechanics, 2021, 13 (1) : 25-53. doi: 10.3934/jgm.2021001

[3]

Davi Obata. Symmetries of vector fields: The diffeomorphism centralizer. Discrete & Continuous Dynamical Systems, 2021  doi: 10.3934/dcds.2021063

[4]

Patrick Henning, Anders M. N. Niklasson. Shadow Lagrangian dynamics for superfluidity. Kinetic & Related Models, 2021, 14 (2) : 303-321. doi: 10.3934/krm.2021006

[5]

Yuri Chekanov, Felix Schlenk. Notes on monotone Lagrangian twist tori. Electronic Research Announcements, 2010, 17: 104-121. doi: 10.3934/era.2010.17.104

[6]

Knut Hüper, Irina Markina, Fátima Silva Leite. A Lagrangian approach to extremal curves on Stiefel manifolds. Journal of Geometric Mechanics, 2021, 13 (1) : 55-72. doi: 10.3934/jgm.2020031

[7]

Hua Shi, Xiang Zhang, Yuyan Zhang. Complex planar Hamiltonian systems: Linearization and dynamics. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3295-3317. doi: 10.3934/dcds.2020406

[8]

Jia Li, Junxiang Xu. On the reducibility of a class of almost periodic Hamiltonian systems. Discrete & Continuous Dynamical Systems - B, 2021, 26 (7) : 3905-3919. doi: 10.3934/dcdsb.2020268

[9]

Huy Dinh, Harbir Antil, Yanlai Chen, Elena Cherkaev, Akil Narayan. Model reduction for fractional elliptic problems using Kato's formula. Mathematical Control & Related Fields, 2021  doi: 10.3934/mcrf.2021004

[10]

Xiaofei Liu, Yong Wang. Weakening convergence conditions of a potential reduction method for tensor complementarity problems. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021080

[11]

Palash Sarkar, Subhadip Singha. Classical reduction of gap SVP to LWE: A concrete security analysis. Advances in Mathematics of Communications, 2021  doi: 10.3934/amc.2021004

[12]

Francisco Braun, Jaume Llibre, Ana Cristina Mereu. Isochronicity for trivial quintic and septic planar polynomial Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2016, 36 (10) : 5245-5255. doi: 10.3934/dcds.2016029

[13]

Montserrat Corbera, Claudia Valls. Reversible polynomial Hamiltonian systems of degree 3 with nilpotent saddles. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 3209-3233. doi: 10.3934/dcdsb.2020225

[14]

Adrian Viorel, Cristian D. Alecsa, Titus O. Pinţa. Asymptotic analysis of a structure-preserving integrator for damped Hamiltonian systems. Discrete & Continuous Dynamical Systems, 2021, 41 (7) : 3319-3341. doi: 10.3934/dcds.2020407

[15]

Peter Benner, Jens Saak, M. Monir Uddin. Balancing based model reduction for structured index-2 unstable descriptor systems with application to flow control. Numerical Algebra, Control & Optimization, 2016, 6 (1) : 1-20. doi: 10.3934/naco.2016.6.1

[16]

Guiyang Zhu. Optimal pricing and ordering policy for defective items under temporary price reduction with inspection errors and price sensitive demand. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021060

[17]

Marian Gidea, Rafael de la Llave, Tere M. Seara. A general mechanism of instability in Hamiltonian systems: Skipping along a normally hyperbolic invariant manifold. Discrete & Continuous Dynamical Systems, 2020, 40 (12) : 6795-6813. doi: 10.3934/dcds.2020166

[18]

Mohsen Abdolhosseinzadeh, Mir Mohammad Alipour. Design of experiment for tuning parameters of an ant colony optimization method for the constrained shortest Hamiltonian path problem in the grid networks. Numerical Algebra, Control & Optimization, 2021, 11 (2) : 321-332. doi: 10.3934/naco.2020028

[19]

Yuzhou Tian, Yulin Zhao. Global phase portraits and bifurcation diagrams for reversible equivariant Hamiltonian systems of linear plus quartic homogeneous polynomials. Discrete & Continuous Dynamical Systems - B, 2021, 26 (6) : 2941-2956. doi: 10.3934/dcdsb.2020214

[20]

Sumon Sarkar, Bibhas C. Giri. Optimal lot-sizing policy for a failure prone production system with investment in process quality improvement and lead time variance reduction. Journal of Industrial & Management Optimization, 2021  doi: 10.3934/jimo.2021048

2019 Impact Factor: 0.649

Metrics

  • PDF downloads (143)
  • HTML views (414)
  • Cited by (0)

Other articles
by authors

[Back to Top]