October  2017, 37(10): 5163-5190. doi: 10.3934/dcds.2017224

Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability

1. 

Mathematical Institute SANU, Serbian Academy of Sciences and Arts, Kneza Mihaila 36, 11000 Belgrade, Serbia

2. 

Faculty of Sciences, University of Banja Luka, Mladena Stojanovića 2, 51000 Banja Luka, Bosnia and Herzegovina

* Corresponding author

Received  October 2016 Revised  May 2017 Published  June 2017

The aim of the paper is to unify the efforts in the study of integrable billiards within quadrics in flat and curved spaces and to explore further the interplay of symplectic and contact integrability. As a starting point in this direction, we consider virtual billiard dynamics within quadrics in pseudo-Euclidean spaces. In contrast to the usual billiards, the incoming velocity and the velocity after the billiard reflection can be at opposite sides of the tangent plane at the reflection point. In the symmetric case we prove noncommutative integrability of the system and give a geometrical interpretation of integrals, an analog of the classical Chasles and Poncelet theorems and we show that the virtual billiard dynamics provides a natural framework in the study of billiards within quadrics in projective spaces, in particular of billiards within ellipsoids on the sphere ${\mathbb{S}^{n - 1}}$ and the Lobachevsky space $\mathbb H^{n-1}$.

Citation: Božidar Jovanović, Vladimir Jovanović. Virtual billiards in pseudo–euclidean spaces: Discrete hamiltonian and contact integrability. Discrete & Continuous Dynamical Systems - A, 2017, 37 (10) : 5163-5190. doi: 10.3934/dcds.2017224
References:
[1]

V. I. Arnol'd, Matematicheskie Metody Klassicheskoy Mehaniki Moskva, Nauka 1974 (Russian). English translation: V. I. Arnol'd, Mathematical Methods of Classical Mechanics Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. Google Scholar

[2]

M. Audin, Courbes algébriques et systémes intégrables: Géodésiques des quadriques, Expo. Math., 12 (1994), 193-226. Google Scholar

[3]

A. Avila, J. De Simoi and V. Kaloshin, An integrable deformation of an ellipse of small eccentricity is an ellipse, Annals of Mathematics, 184 (2016), 527-558, arXiv: 1412.2853. doi: 10.4007/annals.2016.184.2.5. Google Scholar

[4]

A. Banyaga and P. Molino, Géométrie des formes de contact complétement intégrables de type torique, Séminare Gaston Darboux, Montpellier (1991-92), 1-25 (French). English translation: Complete integrability in contact geometry, Penn State preprint PM 197,1996. Google Scholar

[5]

M. Bialy and A. E. Mironov, Angular billiard and algebraic Birkhoff conjecture, Adv. Math., 313 (2017), 102-126, arXiv: 1601.03196. doi: 10.1016/j.aim.2017.04.001. Google Scholar

[6]

M. Bialy and A. E. Mironov, Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane, Journal of Geometry and Physics, 115 (2017), 150-156, arXiv: 1602.05698. doi: 10.1016/j.geomphys.2016.04.017. Google Scholar

[7]

S. V. Bolotin, Integriruemye bil'yardy na poverhnostyah postoyannoy krivizny, Mat. Zametki, 51 (1992), 20-28, (Russian); English translation: S. V. Bolotin, Integrable billiards on constant curvature surfaces, Math. Notes, Math. Notes, 51 (1992), 117-123. doi: 10.1007/BF02102114. Google Scholar

[8]

V. Dragović, B. Jovanović and M. Radnović, On elliptic billiards in the Lobachevsky space and associated geodesic hierarchies, J. Geom. Phys., 47 (2003), 221-234, arXiv: math-ph/0210019. doi: 10.1016/S0393-0440(02)00219-X. Google Scholar

[9]

V. Dragović and M. Radnović, Geometry of integrable billiards and pencils of quadrics, J. Math. Pures Appl., 85 (2006), 758-790, arXiv: math-ph/0512049. doi: 10.1016/j.matpur.2005.12.002. Google Scholar

[10]

V. Dragović and M. Radnović, Hyperelliptic Jacobians as billiard algebra of pencils of quadrics: beyond Poncelet porisms, Adv. Math., 219 (2008), 1577-1607, arXiv: 0710.3656. doi: 10.1016/j.aim.2008.06.021. Google Scholar

[11]

V. Dragović and M. Radnović, Integriruemye billiardy, kvadriki i mnogomernye porizmy Ponsele, Moskva-Izhevsk, Regulyarnaya i hoiticheskaya dinamika, 2010. (Russian); English translation: V. Dragović and M. Radnović, Poncelet Porisms and Beyond. Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics, Frontiers in Mathematics. Birkhauser/Springer Basel AG, Basel, 2011.Google Scholar

[12]

V. Dragović and M. Radnović, Ellipsoidal billiards in pseudo-euclidean spaces and relativistic quadrics, Adv. Math., 231 (2012), 1173-1201, arXiv: 1108.4552. doi: 10.1016/j.aim.2012.06.004. Google Scholar

[13]

V. Dragović and M. Radnović, Billiard algebra, integrable line congruences, and double reflection nets, Journal of Nonlinear Mathematical Physics 19 (2012), 1250019, 18pp, arXiv: 1112.5860. doi: 10.1142/S1402925112500192. Google Scholar

[14]

V. Dragović and M. Radnović, Bicentennial of the great Poncelet theorem (1813-2013): Current advances, Bulletin AMS, 51 (2014), 373-445, arXiv: 1212.6867. doi: 10.1090/S0273-0979-2014-01437-5. Google Scholar

[15]

Yu. N. Fedorov, Integrable systems, Lax representation and confocal quadrics, Dynamical systems in classical mechanics, Amer. Math. Soc. Transl.(2), 168 (1995), 173-199. doi: 10.1090/trans2/168/07. Google Scholar

[16]

Yu. N. Fedorov, Ellipsoidal'ny billiard s kvadratichnym potentsialom, Funkc. analiz i ego prilozh., 35 (2001), 48-59, 95-96 (Russian); English translation: Yu. N. Fedorov, An ellipsoidal billiard with quadratic potential, Funct. Anal. Appl., 35 (2001), 199-208. doi: 10.1023/A:1012326828456. Google Scholar

[17]

Yu. N. Fedorov and B. Jovanović, Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi-Mumford systems, preprint, arXiv: 1503.07053.Google Scholar

[18]

A. Glutsyuk, On quadrilateral orbits in complex algebraic planar billiards, Moscow Math. J., 14 (2014), 239-289, arXiv: 1309.1843. Google Scholar

[19]

B. Jovanović, Noncommutative integrability and action angle variables in contact geometry, Journal of Symplectic Geometry, 10 (2012), 535-561, arXiv: 1103.3611. doi: 10.4310/JSG.2012.v10.n4.a3. Google Scholar

[20]

B. Jovanović, The Jacobi-Rosochatius problem on an ellipsoid: The Lax representations and billiards, Arch. Rational Mech. Anal. 210 (2013), 101-131, arXiv: 1303.6204. doi: 10.1007/s00205-013-0638-4. Google Scholar

[21]

B. Jovanović, Heisenberg model in pseudo-Euclidean spaces, Regular and Chaotic Dynamics, 19 (2014), 245-250, arXiv: 1405.0905. doi: 10.1134/S1560354714020075. Google Scholar

[22]

B. Jovanović and V. Jovanović, Contact flows and integrable systems, Journal of Geometry and Physics, 87 (2015), 217-232, arXiv: 1212.2918. doi: 10.1016/j.geomphys.2014.07.030. Google Scholar

[23]

B. Jovanović and V. Jovanović, Geodesic and billiard flows on quadrics in pseudo-Euclidean spaces: L-A pairs and Chasles theorem, International Mathematics Research Notices, 15 (2015), 6618-6638, arXiv: 1407.0555. doi: 10.1093/imrn/rnu141. Google Scholar

[24]

B. Khesin and S. Tabachnikov, Pseudo-Riemannian geodesics and billiards, Adv. Math., 221 (2009), 1364-1396, arXiv: math/0608620. doi: 10.1016/j.aim.2009.02.010. Google Scholar

[25]

B. Khesin and S. Tabachnikov, Contact complete integrability, Regular and Chaotic Dynamics, 15 (2010), 504-520, arXiv: 0910.0375. doi: 10.1134/S1560354710040076. Google Scholar

[26]

V. V. Kozlov i D. V. Treshchev, Billiardy, Geneticheskoe vvedenie v dinamiku sistem s udarami, Izd-vo Mosk. un-ta, Moskva, 1991. English translation: V. V. Kozlov and D. V. Treshchev, Billiards. A Genetic Introduction to the Dynamics of Systems with Impacts Transl. Math. Monogr., 89, Amer. Math. Soc., Providence, RI, 1991.Google Scholar

[27]

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

[28]

A. S. Mishchenko and A. T. Fomenko, Obobshchenny metod Liuvillya integrirovaniya gamiltonovyh sistem, Funkc. Analiz i ego Prilozh., 12 (1978), 46-56 (Russian); English translation: A. S. Mishchenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl., 12 (1978), 113-121.Google Scholar

[29]

J. Moser, Geometry of quadric and spectral theory, Chern Symposium 1979, Berlin-Heidelberg-New York, (1980), 147-188. Google Scholar

[30]

J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494. Google Scholar

[31]

N. N. Nekhoroshev, Peremennye deystvie-ugol i ih obobshcheniya Tr. Mosk. Mat. O.-va., 26 (1972), 181-198 (Russian). English translation: N. N. Nekhoroshev, Action-angle variables and their generalization, Trans. Mosc. Math. Soc., 26 (1972), 180-198.Google Scholar

[32]

M. Radnović, Topology of the elliptical billiard with the Hooke's potential, Theoretical and Applied Mehanics, 42 (2015), 1-9, arXiv: 1508.01025.Google Scholar

[33]

Yu. B. Suris, The Problem of Integrable Discretization: Hamiltonian Approach Progress in Mathematics, 219. Birkhäuser Verlag, Basel, 2003. doi: 10.1007/978-3-0348-8016-9. Google Scholar

[34]

S. L. Tabachnikov, Ellipsoids, complete integrability and hyperbolic geometry, Mosc. Math. J., 2 (2002), 183-196. Google Scholar

[35]

S. Tabachnikov, Geometry and Billiards volume 30 of Student Mathematical Library. American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005. doi: 10.1090/stml/030. Google Scholar

[36]

A. P. Veselov, Integriruemye sistemy s diskretnym vremenem i raznostnye operatory, Funkc. Analiz i ego Prilozh. 22 (1988), 1-13 (Russian); English translation: A. P. Veselov, Integrable systems with discrete time, and difference operators, Funct. Anal. Appl. 22 (1988), 83-93. doi: 10.1007/BF01077598. Google Scholar

[37]

A. P. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space, J. Geom. Phys., 7 (1990), 81-107. doi: 10.1016/0393-0440(90)90021-T. Google Scholar

[38]

A. P. Veselov, Integriruemye otobrazheniya, Uspekhi Mat. Nauk, 46 (1991), 3-45 (Russian); English translation: A. P. Veselov, Integrable mappings, Russ. Math. Surv., 46 (1991), 1-51. Google Scholar

[39]

A. P. Veselov, Growth and integrability in the dynamics of mappings, Comm. Math. Phys., 145 (1992), 181-193. doi: 10.1007/BF02099285. Google Scholar

show all references

References:
[1]

V. I. Arnol'd, Matematicheskie Metody Klassicheskoy Mehaniki Moskva, Nauka 1974 (Russian). English translation: V. I. Arnol'd, Mathematical Methods of Classical Mechanics Second edition. Graduate Texts in Mathematics, 60. Springer-Verlag, New York, 1989. Google Scholar

[2]

M. Audin, Courbes algébriques et systémes intégrables: Géodésiques des quadriques, Expo. Math., 12 (1994), 193-226. Google Scholar

[3]

A. Avila, J. De Simoi and V. Kaloshin, An integrable deformation of an ellipse of small eccentricity is an ellipse, Annals of Mathematics, 184 (2016), 527-558, arXiv: 1412.2853. doi: 10.4007/annals.2016.184.2.5. Google Scholar

[4]

A. Banyaga and P. Molino, Géométrie des formes de contact complétement intégrables de type torique, Séminare Gaston Darboux, Montpellier (1991-92), 1-25 (French). English translation: Complete integrability in contact geometry, Penn State preprint PM 197,1996. Google Scholar

[5]

M. Bialy and A. E. Mironov, Angular billiard and algebraic Birkhoff conjecture, Adv. Math., 313 (2017), 102-126, arXiv: 1601.03196. doi: 10.1016/j.aim.2017.04.001. Google Scholar

[6]

M. Bialy and A. E. Mironov, Algebraic Birkhoff conjecture for billiards on Sphere and Hyperbolic plane, Journal of Geometry and Physics, 115 (2017), 150-156, arXiv: 1602.05698. doi: 10.1016/j.geomphys.2016.04.017. Google Scholar

[7]

S. V. Bolotin, Integriruemye bil'yardy na poverhnostyah postoyannoy krivizny, Mat. Zametki, 51 (1992), 20-28, (Russian); English translation: S. V. Bolotin, Integrable billiards on constant curvature surfaces, Math. Notes, Math. Notes, 51 (1992), 117-123. doi: 10.1007/BF02102114. Google Scholar

[8]

V. Dragović, B. Jovanović and M. Radnović, On elliptic billiards in the Lobachevsky space and associated geodesic hierarchies, J. Geom. Phys., 47 (2003), 221-234, arXiv: math-ph/0210019. doi: 10.1016/S0393-0440(02)00219-X. Google Scholar

[9]

V. Dragović and M. Radnović, Geometry of integrable billiards and pencils of quadrics, J. Math. Pures Appl., 85 (2006), 758-790, arXiv: math-ph/0512049. doi: 10.1016/j.matpur.2005.12.002. Google Scholar

[10]

V. Dragović and M. Radnović, Hyperelliptic Jacobians as billiard algebra of pencils of quadrics: beyond Poncelet porisms, Adv. Math., 219 (2008), 1577-1607, arXiv: 0710.3656. doi: 10.1016/j.aim.2008.06.021. Google Scholar

[11]

V. Dragović and M. Radnović, Integriruemye billiardy, kvadriki i mnogomernye porizmy Ponsele, Moskva-Izhevsk, Regulyarnaya i hoiticheskaya dinamika, 2010. (Russian); English translation: V. Dragović and M. Radnović, Poncelet Porisms and Beyond. Integrable Billiards, Hyperelliptic Jacobians and Pencils of Quadrics, Frontiers in Mathematics. Birkhauser/Springer Basel AG, Basel, 2011.Google Scholar

[12]

V. Dragović and M. Radnović, Ellipsoidal billiards in pseudo-euclidean spaces and relativistic quadrics, Adv. Math., 231 (2012), 1173-1201, arXiv: 1108.4552. doi: 10.1016/j.aim.2012.06.004. Google Scholar

[13]

V. Dragović and M. Radnović, Billiard algebra, integrable line congruences, and double reflection nets, Journal of Nonlinear Mathematical Physics 19 (2012), 1250019, 18pp, arXiv: 1112.5860. doi: 10.1142/S1402925112500192. Google Scholar

[14]

V. Dragović and M. Radnović, Bicentennial of the great Poncelet theorem (1813-2013): Current advances, Bulletin AMS, 51 (2014), 373-445, arXiv: 1212.6867. doi: 10.1090/S0273-0979-2014-01437-5. Google Scholar

[15]

Yu. N. Fedorov, Integrable systems, Lax representation and confocal quadrics, Dynamical systems in classical mechanics, Amer. Math. Soc. Transl.(2), 168 (1995), 173-199. doi: 10.1090/trans2/168/07. Google Scholar

[16]

Yu. N. Fedorov, Ellipsoidal'ny billiard s kvadratichnym potentsialom, Funkc. analiz i ego prilozh., 35 (2001), 48-59, 95-96 (Russian); English translation: Yu. N. Fedorov, An ellipsoidal billiard with quadratic potential, Funct. Anal. Appl., 35 (2001), 199-208. doi: 10.1023/A:1012326828456. Google Scholar

[17]

Yu. N. Fedorov and B. Jovanović, Continuous and discrete Neumann systems on Stiefel varieties as matrix generalizations of the Jacobi-Mumford systems, preprint, arXiv: 1503.07053.Google Scholar

[18]

A. Glutsyuk, On quadrilateral orbits in complex algebraic planar billiards, Moscow Math. J., 14 (2014), 239-289, arXiv: 1309.1843. Google Scholar

[19]

B. Jovanović, Noncommutative integrability and action angle variables in contact geometry, Journal of Symplectic Geometry, 10 (2012), 535-561, arXiv: 1103.3611. doi: 10.4310/JSG.2012.v10.n4.a3. Google Scholar

[20]

B. Jovanović, The Jacobi-Rosochatius problem on an ellipsoid: The Lax representations and billiards, Arch. Rational Mech. Anal. 210 (2013), 101-131, arXiv: 1303.6204. doi: 10.1007/s00205-013-0638-4. Google Scholar

[21]

B. Jovanović, Heisenberg model in pseudo-Euclidean spaces, Regular and Chaotic Dynamics, 19 (2014), 245-250, arXiv: 1405.0905. doi: 10.1134/S1560354714020075. Google Scholar

[22]

B. Jovanović and V. Jovanović, Contact flows and integrable systems, Journal of Geometry and Physics, 87 (2015), 217-232, arXiv: 1212.2918. doi: 10.1016/j.geomphys.2014.07.030. Google Scholar

[23]

B. Jovanović and V. Jovanović, Geodesic and billiard flows on quadrics in pseudo-Euclidean spaces: L-A pairs and Chasles theorem, International Mathematics Research Notices, 15 (2015), 6618-6638, arXiv: 1407.0555. doi: 10.1093/imrn/rnu141. Google Scholar

[24]

B. Khesin and S. Tabachnikov, Pseudo-Riemannian geodesics and billiards, Adv. Math., 221 (2009), 1364-1396, arXiv: math/0608620. doi: 10.1016/j.aim.2009.02.010. Google Scholar

[25]

B. Khesin and S. Tabachnikov, Contact complete integrability, Regular and Chaotic Dynamics, 15 (2010), 504-520, arXiv: 0910.0375. doi: 10.1134/S1560354710040076. Google Scholar

[26]

V. V. Kozlov i D. V. Treshchev, Billiardy, Geneticheskoe vvedenie v dinamiku sistem s udarami, Izd-vo Mosk. un-ta, Moskva, 1991. English translation: V. V. Kozlov and D. V. Treshchev, Billiards. A Genetic Introduction to the Dynamics of Systems with Impacts Transl. Math. Monogr., 89, Amer. Math. Soc., Providence, RI, 1991.Google Scholar

[27]

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

[28]

A. S. Mishchenko and A. T. Fomenko, Obobshchenny metod Liuvillya integrirovaniya gamiltonovyh sistem, Funkc. Analiz i ego Prilozh., 12 (1978), 46-56 (Russian); English translation: A. S. Mishchenko and A. T. Fomenko, Generalized Liouville method of integration of Hamiltonian systems, Funct. Anal. Appl., 12 (1978), 113-121.Google Scholar

[29]

J. Moser, Geometry of quadric and spectral theory, Chern Symposium 1979, Berlin-Heidelberg-New York, (1980), 147-188. Google Scholar

[30]

J. Moser and A. Veselov, Discrete versions of some classical integrable systems and factorization of matrix polynomials, Comm. Math. Phys., 139 (1991), 217-243. doi: 10.1007/BF02352494. Google Scholar

[31]

N. N. Nekhoroshev, Peremennye deystvie-ugol i ih obobshcheniya Tr. Mosk. Mat. O.-va., 26 (1972), 181-198 (Russian). English translation: N. N. Nekhoroshev, Action-angle variables and their generalization, Trans. Mosc. Math. Soc., 26 (1972), 180-198.Google Scholar

[32]

M. Radnović, Topology of the elliptical billiard with the Hooke's potential, Theoretical and Applied Mehanics, 42 (2015), 1-9, arXiv: 1508.01025.Google Scholar

[33]

Yu. B. Suris, The Problem of Integrable Discretization: Hamiltonian Approach Progress in Mathematics, 219. Birkhäuser Verlag, Basel, 2003. doi: 10.1007/978-3-0348-8016-9. Google Scholar

[34]

S. L. Tabachnikov, Ellipsoids, complete integrability and hyperbolic geometry, Mosc. Math. J., 2 (2002), 183-196. Google Scholar

[35]

S. Tabachnikov, Geometry and Billiards volume 30 of Student Mathematical Library. American Mathematical Society, Providence, RI; Mathematics Advanced Study Semesters, University Park, PA, 2005. doi: 10.1090/stml/030. Google Scholar

[36]

A. P. Veselov, Integriruemye sistemy s diskretnym vremenem i raznostnye operatory, Funkc. Analiz i ego Prilozh. 22 (1988), 1-13 (Russian); English translation: A. P. Veselov, Integrable systems with discrete time, and difference operators, Funct. Anal. Appl. 22 (1988), 83-93. doi: 10.1007/BF01077598. Google Scholar

[37]

A. P. Veselov, Confocal surfaces and integrable billiards on the sphere and in the Lobachevsky space, J. Geom. Phys., 7 (1990), 81-107. doi: 10.1016/0393-0440(90)90021-T. Google Scholar

[38]

A. P. Veselov, Integriruemye otobrazheniya, Uspekhi Mat. Nauk, 46 (1991), 3-45 (Russian); English translation: A. P. Veselov, Integrable mappings, Russ. Math. Surv., 46 (1991), 1-51. Google Scholar

[39]

A. P. Veselov, Growth and integrability in the dynamics of mappings, Comm. Math. Phys., 145 (1992), 181-193. doi: 10.1007/BF02099285. Google Scholar

Figure 1.  A segment of a virtual billiard trajectory within hyperbola ($a_1>0, a_2<0$) in the Euclidean space $\mathbb E^{2, 0}$. The caustic is an ellipse
Figure 2.  Families of pseudo-confocal quadrics for $a_1>0, a_2<0$ in $\mathbb E^{1, 1}$ (with $\alpha_1=-a_2>\alpha_2=a_1$) and $\mathbb E^{2, 0}$, respectively
Figure 3.  Family of pseudo-confocal quadrics for $a_1>0, a_2<0$ in $\mathbb E^{1, 1}$, where $\alpha_1=a_1>\alpha_2=-a_2$
Figure 4.  The segments of time-like and space-like billiard trajectories for $a_1>0, a_2<0$, $\alpha_1=-a_2>\alpha_2=a_1$ in $\mathbb E^{1, 1}$. The caustics are hyperbolas
Figure 5.  The segment of a space-like billiard trajectory for $a_1>0, a_2<0$, $\alpha_1=a_1>\alpha_2=-a_2$ in $\mathbb E^{1, 1}$. The caustic is an ellipse
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