Lie Cebiroidleri Üzerindeki Lagrange Dinamiğinin Eşlenmesi Problemi Üzerine
Yıl 2021,
, 162 - 171, 20.08.2021
Oğul Esen
,
Hanife Kübra Kaya
,
Serkan Sütlü
Öz
Lie cebiroidleri, bir anlamda tanjant demetini ve Lie cebiri yapısını beraber ihtiva eden ve fakat daha genel olan geometrik inşaalardır. Lagrange dinamiğinin en genel ifadesi Lie cebiroidleri üzerinde mümkündür. Bu makalede, karşılıklı (Lie cebiroidi üzerinde tanımlı) etki içindeki iki Lagrange dinamiğinin beraber davranışı, geometrik ve cebirsel bir yol ile elde edilecektir. Bu bakış açısı ile etkileşim Lie cebiroidlerinin birbirleri üzerine olan lineer temsilleri (etkileri) ifade edilecektir. Bu sayede, belirli uyumluluk şartını sağlayan karşılıklı etki içindeki iki Lie cebiroidinin eşlenmesi, diğer bir ifade ile tek bir Lie cebiroidi olarak yazılması sağlanacaktır. Sonrasında ise eşlenmiş Lie cebiroidi üzerinde Lagrange dinamiği yazılacaktır. Elde edilecek kollektif (eşlenmiş) hareket denklemleri, bireysel davranışların gözlemlenmesinin yanı sıra karşılıklı etki terimlerinin de belirlenmesine olanak verecektir. Çalışmamız esnasında bir çok örnek sunularak teorik tanımların daha net anlatımı yakalanmaya çalışılacaktır.
Destekleyen Kurum
TÜBİTAK(Türkiye Bilimsel ve Teknolojik Araştırma Kurumu)
Teşekkür
Bu çalışma TÜBİTAK, 117F426 numaralı "Eşlenmiş Lagrange ve Hamilton Sistemleri" isimli projenin bir parçasıdır. Destek için TÜBİTAK’a teşekkür ederiz.
Kaynakça
- [1] Abraham, R., Marsden, J. E., Marsden, J. E.
1978. Foundations of mechanics . Reading,
Massachusetts: Benjamin/Cummings Publishing
Company.
- [2] Holm, D. D., Schmah, T., Stoica, C.
2009. Geometric mechanics and symmetry: from
finite to infinite dimensions. Vol. 12. Oxford
University Press.
- [3] Yaremko, Y. 2000. The Tangent Groups of a Lie
Group and Gauge Invariance in Lagrangian
Dynamics. In Proceedings of Institute of
Mathematics of NAS of Ukraine. 30(2), 544-550.
- [4] Marsden, J. E., Ratiu, T. S. 1995. Introduction to
mechanics and symmetry. Physics Today, 48
(12), 65.
- [5] Weinstein, A. 1996. Lagrangian mechanics and
groupoids. Fields Institute Proc. AMS, 7, 207-
231.
- [6] Martínez, E. 2001. Lagrangian mechanics on Lie
algebroids. Acta Applicandae Mathematica, 67
(3), 295-320.
- [7] Martínez, E. 2009. Lie algebroids and Mechanics.
In AIP Conference Proceedings, American
Institute of Physics, 1130(1), 3-33.
- [8] Ratiu, T., Moerbeke, P. V. 1982. The Lagrange
rigid body motion. In Annales de l'institut
Fourier. 32(1), 211-234.
- [9] Holm, D. D., Marsden, J. E., Ratiu, T. S. 1998. The
Euler–Poincaré equations and semidirect
products with applications to continuum
theories. Advances in Mathematics, 137(1), 1-81.
- [10] Esen, O., Sütlü, S. 2017. Lagrangian dynamics on
matched pairs. Journal of Geometry and
Physics, 111, 142-157.
- [11] Esen, O., Sütlü, S. 2021. Discrete dynamical
systems over double cross-product Lie
groupoids. International Journal of Geometric
Methods in Modern Physics. 18(04), 2150057.
- [12] Mackenzie, K., Kirill, M., Mackenzie, K. C.
1987. Lie groupoids and Lie algebroids in
differential geometry. Cambridge university
press.
- [13] Mokri, T. 1997. Matched pairs of Lie algebroids.
Glasgow Mathematical Journal, 39(2), 167-181.
- [14] Pradines, J. 1967. Theorie de Lie pour les
groupoides differentiable. CR Acad. Sci.
Paris, 264, 245-248.
- [15] Crampin, M., Saunders, D. 2016. Cartan
geometries and their symmetries: a Lie
algebroid approach. Springer.
- [16] Mackenzie, K. C., Mackenzie, K. C. 2005. General
theory of Lie groupoids and Lie algebroids.
Cambridge University Press.
- [17] Marrero, J. C., de Diego, D. M., Martínez, E. 2006.
Discrete Lagrangian and Hamiltonian mechanics
on Lie groupoids. Nonlinearity, 19(6), 1313.
- [18] Iglesias, D., Marrero, J. C., Martín de Diego, D.,
Martínez, E., Padrón, E. 2007. Reduction of
symplectic Lie algebroids by a Lie subalgebroid
and a symmetry Lie group. SIGMA. Symmetry,
Integrability and Geometry: Methods and
Applications, 3, 049.
- [19] Cortés, J.,de Leon, M., Marrero, J. C., de Diego, D.
M., Martinez, E. 2006. A survey of Lagrangian
mechanics and control on Lie algebroids and
groupoids. International Journal of Geometric
Methods in Modern Physics, 3(03), 509-558.
- [20] Mackenzie, K. C. 1992. Double Lie algebroids and
second-order geometry, I. Advances in
Mathematics, 94(2), 180-239.
- [21] Brown, R. 1972. Groupoids as
coefficients. Proceedings of the London
Mathematical Society, 3(3), 413-426.
- [22] Majid, S. 1990. Matched pairs of Lie groups
associated to solutions of the Yang-Baxter
equations. Pacific Journal of
Mathematics, 141(2), 311-332.
- [23] Majid, S. 2000. Foundations of quantum group
theory. Cambridge university press.
- [24] Higgins, P. J., Mackenzie, K. 1990. Algebraic
constructions in the category of Lie
algebroids. Journal of Algebra, 129(1), 194-230.
- [25] Lu, J. H. 1997. Lie algebroids associated to
Poisson actions. In Duke Math. J.
- [26] Mackenzie, K. C. 1995. Lie algebroids and Lie
pseudoalgebras. Bulletin of the London
Mathematical Society, 27(2), 97-147.
- [27] Martínez, E. 2015. Higher-order variational
calculus on Lie algebroids. Journal of Geometric
Mechanics, 7(1), 81-108.
- [28] Esen, O., Kudeyt, M., Sütlü, S. 2021. Second order
Lagrangian dynamics on double cross product
groups. Journal of Geometry and Physics, 159,
103934.
- [29] Esen, O., Sütlü, S. 2016. Hamiltonian dynamics
on matched pairs. International Journal of
Geometric Methods in Modern Physics, 13(10),
1650128.
On The Problem of Matched Lagrangian Dynamics on Lie Algebroids
Yıl 2021,
, 162 - 171, 20.08.2021
Oğul Esen
,
Hanife Kübra Kaya
,
Serkan Sütlü
Öz
Lie algebroids are geometric constructions generalizing both tangent bundles and Lie algebras. Lagrangian dynamics is possible on Lie algebroid frameworks in its most general form. In this work, we obtain the joint behaviour of two mutually interacting Lagrangian systems in a geometric and an algebraic way. Here, the interaction is decoded into linear representations (actions) of two Lie algebroids onto each other. By this means, mutally interacting two Lie algebroids those satisfying some certain compatibility condition are matched, in other words, they are recast as trivially intersecting Lie subalgebroids of a single Lie algebroid. Then, Lagrangian dynamics is recast on the matched Lie algebroid. In this framework, the equations involve both the dynamics of constitutive subsystems and the action terms. Along with the theory, we provide several examples.
Kaynakça
- [1] Abraham, R., Marsden, J. E., Marsden, J. E.
1978. Foundations of mechanics . Reading,
Massachusetts: Benjamin/Cummings Publishing
Company.
- [2] Holm, D. D., Schmah, T., Stoica, C.
2009. Geometric mechanics and symmetry: from
finite to infinite dimensions. Vol. 12. Oxford
University Press.
- [3] Yaremko, Y. 2000. The Tangent Groups of a Lie
Group and Gauge Invariance in Lagrangian
Dynamics. In Proceedings of Institute of
Mathematics of NAS of Ukraine. 30(2), 544-550.
- [4] Marsden, J. E., Ratiu, T. S. 1995. Introduction to
mechanics and symmetry. Physics Today, 48
(12), 65.
- [5] Weinstein, A. 1996. Lagrangian mechanics and
groupoids. Fields Institute Proc. AMS, 7, 207-
231.
- [6] Martínez, E. 2001. Lagrangian mechanics on Lie
algebroids. Acta Applicandae Mathematica, 67
(3), 295-320.
- [7] Martínez, E. 2009. Lie algebroids and Mechanics.
In AIP Conference Proceedings, American
Institute of Physics, 1130(1), 3-33.
- [8] Ratiu, T., Moerbeke, P. V. 1982. The Lagrange
rigid body motion. In Annales de l'institut
Fourier. 32(1), 211-234.
- [9] Holm, D. D., Marsden, J. E., Ratiu, T. S. 1998. The
Euler–Poincaré equations and semidirect
products with applications to continuum
theories. Advances in Mathematics, 137(1), 1-81.
- [10] Esen, O., Sütlü, S. 2017. Lagrangian dynamics on
matched pairs. Journal of Geometry and
Physics, 111, 142-157.
- [11] Esen, O., Sütlü, S. 2021. Discrete dynamical
systems over double cross-product Lie
groupoids. International Journal of Geometric
Methods in Modern Physics. 18(04), 2150057.
- [12] Mackenzie, K., Kirill, M., Mackenzie, K. C.
1987. Lie groupoids and Lie algebroids in
differential geometry. Cambridge university
press.
- [13] Mokri, T. 1997. Matched pairs of Lie algebroids.
Glasgow Mathematical Journal, 39(2), 167-181.
- [14] Pradines, J. 1967. Theorie de Lie pour les
groupoides differentiable. CR Acad. Sci.
Paris, 264, 245-248.
- [15] Crampin, M., Saunders, D. 2016. Cartan
geometries and their symmetries: a Lie
algebroid approach. Springer.
- [16] Mackenzie, K. C., Mackenzie, K. C. 2005. General
theory of Lie groupoids and Lie algebroids.
Cambridge University Press.
- [17] Marrero, J. C., de Diego, D. M., Martínez, E. 2006.
Discrete Lagrangian and Hamiltonian mechanics
on Lie groupoids. Nonlinearity, 19(6), 1313.
- [18] Iglesias, D., Marrero, J. C., Martín de Diego, D.,
Martínez, E., Padrón, E. 2007. Reduction of
symplectic Lie algebroids by a Lie subalgebroid
and a symmetry Lie group. SIGMA. Symmetry,
Integrability and Geometry: Methods and
Applications, 3, 049.
- [19] Cortés, J.,de Leon, M., Marrero, J. C., de Diego, D.
M., Martinez, E. 2006. A survey of Lagrangian
mechanics and control on Lie algebroids and
groupoids. International Journal of Geometric
Methods in Modern Physics, 3(03), 509-558.
- [20] Mackenzie, K. C. 1992. Double Lie algebroids and
second-order geometry, I. Advances in
Mathematics, 94(2), 180-239.
- [21] Brown, R. 1972. Groupoids as
coefficients. Proceedings of the London
Mathematical Society, 3(3), 413-426.
- [22] Majid, S. 1990. Matched pairs of Lie groups
associated to solutions of the Yang-Baxter
equations. Pacific Journal of
Mathematics, 141(2), 311-332.
- [23] Majid, S. 2000. Foundations of quantum group
theory. Cambridge university press.
- [24] Higgins, P. J., Mackenzie, K. 1990. Algebraic
constructions in the category of Lie
algebroids. Journal of Algebra, 129(1), 194-230.
- [25] Lu, J. H. 1997. Lie algebroids associated to
Poisson actions. In Duke Math. J.
- [26] Mackenzie, K. C. 1995. Lie algebroids and Lie
pseudoalgebras. Bulletin of the London
Mathematical Society, 27(2), 97-147.
- [27] Martínez, E. 2015. Higher-order variational
calculus on Lie algebroids. Journal of Geometric
Mechanics, 7(1), 81-108.
- [28] Esen, O., Kudeyt, M., Sütlü, S. 2021. Second order
Lagrangian dynamics on double cross product
groups. Journal of Geometry and Physics, 159,
103934.
- [29] Esen, O., Sütlü, S. 2016. Hamiltonian dynamics
on matched pairs. International Journal of
Geometric Methods in Modern Physics, 13(10),
1650128.