# American Institute of Mathematical Sciences

March  2014, 6(1): 67-98. doi: 10.3934/jgm.2014.6.67

## Tensor products of Dirac structures and interconnection in Lagrangian mechanics

 1 Department of Mathematics, Imperial College London, South Kensington Campus, London, SW7 2AZ, United Kingdom 2 Applied Mechanics and Aerospace Engineering, Waseda University, Okubo, Shinjuku, Tokyo 169-8555

Received  October 2013 Revised  February 2014 Published  April 2014

Many mechanical systems are large and complex, despite being composed of simple subsystems. In order to understand such large systems it is natural to tear the system into these subsystems. Conversely we must understand how to invert this tearing procedure. In other words, we must understand interconnection of subsystems. Such an understanding has been already shown in the context of Hamiltonian systems on vector spaces via the port-Hamiltonian systems program, in which an interconnection may be achieved through the identification of shared variables, whereupon the notion of composition of Dirac structures allows one to interconnect two systems. In this paper, we seek to extend the program of the port-Hamiltonian systems on vector spaces to the case of Lagrangian systems on manifolds and also extend the notion of composition of Dirac structures appropriately. In particular, we will interconnect Lagrange-Dirac systems by modifying the respective Dirac structures of the involved subsystems. We define the interconnection of Dirac structures via an interaction Dirac structure and a tensor product of Dirac structures. We will show how the dynamics of the interconnected system is formulated as a function of the subsystems, and we will elucidate the associated variational principles. We will then illustrate how this theory extends the theory of port-Hamiltonian systems and the notion of composition of Dirac structures to manifolds with couplings which do not require the identification of shared variables. Lastly, we will show some examples: a mass-spring mechanical systems, an electric circuit, and a nonholonomic mechanical system.
Citation: Henry O. Jacobs, Hiroaki Yoshimura. Tensor products of Dirac structures and interconnection in Lagrangian mechanics. Journal of Geometric Mechanics, 2014, 6 (1) : 67-98. doi: 10.3934/jgm.2014.6.67
##### References:
 [1] E. Afshari, H. S. Bhat, A. Hajimiri and J. E. Marsden, Extremely wideband signal shaping using one- and two-dimensional nonuniform nonlinear transmission lines, Journal of Applied Physics, 99 (2006) 054901. doi: 10.1063/1.2174126. [2] B. M. Maschke and A. J. van der Schaft, Hamiltonian formulation of distributed-parameter systems with boundary energy flow, Journal of Geometry and Physics, 42 (2002), 166-194. doi: 10.1016/S0393-0440(01)00083-3. [3] C. Batlle, I. Massana and E. Simo, Representation of a general composition of dirac structures, In Decision and Control and European Control Conference (CDC-ECC), 2011 50th IEEE Conference on, pages 5199-5204, (2011). doi: 10.1109/CDC.2011.6160588. doi: 10.1109/CDC.2011.6160588. [4] G. Blankenstein, Implicit Hamiltonian Systems: Symmetry and Interconnection, PhD thesis, University of Twente, 2000. [5] A. M. Bloch, Nonholonomic Mechanics and Control, volume 24 of Interdisciplinary Applied Mathematics, Springer Verlag, 2003. doi: 10.1007/b97376. [6] A. M. Bloch and P. E. Crouch, Representations of Dirac structures on vector spaces and nonlinear LC circuits, In Proc. Sympos. Pure Math, volume 64, pages 103-117. American Mathematical Society, 1999. [7] N. Bou-Rabee and J. E. Marsden, Hamilton-Pontryagin integrators on Lie groups: Introduction and structure preserving properties, Foundations of Computational Mathematics, 9 (2009), 197-219. doi: 10.1007/s10208-008-9030-4. [8] R. K. Brayton, Nonlinear reciprocal networks, In Mathematical Aspects of Electrical Network Analysis, H.S. Wilf and F. Harary (eds). SIAM - AMS Proceedings, Volume III, pages 1-15, 1971. [9] H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian Reduction by Stages, Memoirs of the American Mathematical Society. American Mathematical Society, 2001. doi: 10.1090/memo/0722. [10] J. Cervera, A. J. van der Schaft and A. Baños, Interconnection of port-Hamiltonian systems and composition of Dirac structures, Automatica, 43 (2007), 212-225. doi: 10.1016/j.automatica.2006.08.014. [11] D. A. Chang, A. M. Bloch, N. E. Leonard, J. E. Marsden and C. A. Woolsey, The equivalence of controlled Lagrangian and controlled Hamiltonian systems, ESAIM: Control, Optimisation and Calculus of Variations, 8 (2002), 393-422. doi: 10.1051/cocv:2002045. [12] T. J. Courant, Dirac manifolds, Transactions of the American Mathematical Society, 319 (1990), 631-661. doi: 10.1090/S0002-9947-1990-0998124-1. [13] T. J. Courant and A. Weinstein, Beyond Poisson structures, In Action Hamiltoniennes de groupes. Troisième théoréme de Lie (Lyon 1986), volume 27 of Travaux en Cours, pages 39-49. Hermann, 1988. [14] P. A. M. Dirac, Generalized Hamiltonian dynamics, Canadian J. Mathematics, 2 (1950), 129-148. doi: 10.4153/CJM-1950-012-1. [15] I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, (Nonlinear Science: Theory and Applications). Wiley & Sons Ltd., 1993. [16] J. Dufour and A. Wade, On the local structure of Dirac manifolds, Compos. Math., 144 (2008), 774-786. doi: 10.1112/S0010437X07003272. [17] V. Duindam, Port-based Modelling and Control for Efficient Bipedal Walking Robots, PhD thesis, University of Twente, 2006. [18] V. Duindam, A. Macchelli, S. Stramigioli and H. Bruyninckx, editors., Modelling and Control of Complex Physical Systems, Springer, 2009. doi: 10.1007/978-3-642-03196-0. [19] R. Featherstone, Robot Dynamics Algorithms, Kluwer Academic, 1987. [20] G. Golo, V. Talasila, A. van der Schaft and B. Maschke, Hamiltonian discretization of boundary control systems, Automatica, 40 (2004), 757-771. URL http://www.sciencedirect.com/science/article/pii/S0005109804000147. doi: 10.1016/j.automatica.2003.12.017. [21] M. Gualtieri, Generalized complex geometry, Ann. of Math. (2), 174 (2011), 75-123. doi: 10.4007/annals.2011.174.1.3. [22] H. O. Jacobs and J. Vankerschaver, Fluid-structure interaction in the Lagrange-Poincaré formalism, arXiv:1212.1144 [math.DS], Aug 2013. [23] H. O. Jacobs and H. Yoshimura, Interconnection and composition of Dirac structures for Lagrange-Dirac systems, In Decision and Control and European Control Conference (CDC-ECC), 2011 50th IEEE Conference on, pages 928-933, 2011. doi: 10.1109/CDC.2011.6160480. [24] H. O. Jacobs, H. Yoshimura and J. E. Marsden, Interconnection of Lagrange-Dirac dynamical systems for electric circuits, AIP Conference Proceedings, 1281 (2010), 566-569. doi: 10.1063/1.3498539. [25] G. Kron, Diakoptics: The Piecewise Solution of Large-Scale Systems, McDonald, London, 1963. [26] M. Leok and T. Ohsawa, Variational and geometric structures of discrete Dirac mechanics, Foundations of Computational Mathematics, 11 (2011), 529-562. ISSN 1615-3375. doi: 10.1007/s10208-011-9096-2. [27] R. G. Littlejohn, Variational principles of guiding centre motion, Journal of Plasma Physics, 29 (1983), 111-125. doi: 10.1017/S002237780000060X. [28] 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. [29] J. Merker, On the geometric structure of Hamiltonian systems with ports, Journal of Nonlinear Science, 19 (2009), 717-738. doi: 10.1007/s00332-009-9052-3. [30] R. Ortega, J. A. L. Perez, P. J. Nicklasson and H. J. Sira-Ramirez, Passivity-based Control of Euler-Lagrange Systems: Mechanical, Electrical, and Electromechanical Applications, Communications and Control Engineering. Springer-Verlag, 1st edition, 1998. [31] R. Ortega, A. J. van der Schaft, B. M. Maschke and G. Escobar, Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems, Automatica, 38 (2002), 585-596. ISSN 0005-1098. doi: 10.1016/S0005-1098(01)00278-3. [32] H. M. Paynter, Analysis and Design of Engineering Systems, MIT Press, 1961. [33] V. Talasila, J. Clemente-Gallardo and A. J. van der Schaft, Discrete port-Hamiltonian systems, Systems and Control Letters, 55 (2006), 478-486. ISSN 0167-6911. doi: http://dx.doi.org/10.1016/j.sysconle.2005.10.001. doi: 10.1016/j.sysconle.2005.10.001. [34] W. M. Tulczyjew, The Legendre transformation, Annales de l'Institute Henri Poincaré, 27 (1977), 101-114. [35] A. J. van der Schaft, Port-Hamiltonian systems: An introductory survey, In Proceedings of the International Conference of Mathematics, volume 3, pages 1-27. European Mathematical Society, 1996. [36] A. J. van der Schaft and B. M. Maschke, The Hamiltonian formulation of energy conserving physical systems with external ports, Archiv für Elektronik und Übertragungstechnik, 49 (1995), 362-371. [37] J. Vankerschaver, H. Yoshimura and M. Leok, The Hamilton-Pontryagin principle and multi-Dirac structures for classical field theories, Journal of Mathematical Physics, 53 (2012), 072903. doi: 10.1063/1.4731481. [38] A. Weinstein, Symplectic categories, Port. Math., 67 (2010), 261-278. doi: 10.4171/PM/1866. [39] J. L. Wyatt and L. O. Chua, A theory of nonenergic $n$-ports, Circuit Theory and Applications, 5 (1977), 181-208. [40] H. Yoshimura, Dynamics of Flexible Multibody Systems, PhD thesis, Waseda University, 1995. [41] H. Yoshimura and J. E. Marsden, Dirac structures in Lagrangian mechanics part I: Implicit Lagrangian systems, Journal of Geometry and Physics, 57 (2006), 133-156. ISSN 0393-0440. doi: 10.1016/j.geomphys.2006.02.009. [42] H. Yoshimura and J. E. Marsden, Dirac structures in Lagrangian mechanics part II: Variational structures, Journal of Geometry and Physics, 57 (2006), 209-250. ISSN 0393-0440. doi: 10.1016/j.geomphys.2006.02.012. [43] H. Yoshimura and J. E. Marsden, Dirac structures and implicit Lagrangian systems in electric networks, In Proceedings of the 17th International Symposium on the Mathematical Theory of Networks and Systems, pages 1-6, 2006. [44] H. Yoshimura and J. E. Marsden, Dirac Structures and the Legendre transformation for implicit Lagrangian and Hamiltonian systems, In Lagrangian and Hamiltonian Methods for Nonlinear Control 2006, volume 366 of Lecture Notes in Control and Information Sciences, pages 233-247. Springer Berlin/Heidelberg, 2007. doi: 10.1007/978-3-540-73890-9_18. [45] H. Yoshimura and J. E. Marsden, Reduction of Dirac structures and the Hamilton-Pontryagin principle, Reports on Mathematical Physics, 60 (2007), 381-426. ISSN 0034-4877. doi: 10.1016/S0034-4877(08)00004-9. [46] H. Yoshimura and J. E. Marsden, Dirac cotangent bundle reduction, Journal of Geometric Mechanics, 1 (2009), 87-158. doi: 10.3934/jgm.2009.1.87. [47] H. Yoshimura, H. O. Jacobs, and J. E. Marsden, Interconnection of Dirac structures in Lagrange-Dirac dynamical systems, In Proceedings of the 20th International Symposium on the Mathematical Theory of Networks and Systems, 2010.

show all references

##### References:
 [1] E. Afshari, H. S. Bhat, A. Hajimiri and J. E. Marsden, Extremely wideband signal shaping using one- and two-dimensional nonuniform nonlinear transmission lines, Journal of Applied Physics, 99 (2006) 054901. doi: 10.1063/1.2174126. [2] B. M. Maschke and A. J. van der Schaft, Hamiltonian formulation of distributed-parameter systems with boundary energy flow, Journal of Geometry and Physics, 42 (2002), 166-194. doi: 10.1016/S0393-0440(01)00083-3. [3] C. Batlle, I. Massana and E. Simo, Representation of a general composition of dirac structures, In Decision and Control and European Control Conference (CDC-ECC), 2011 50th IEEE Conference on, pages 5199-5204, (2011). doi: 10.1109/CDC.2011.6160588. doi: 10.1109/CDC.2011.6160588. [4] G. Blankenstein, Implicit Hamiltonian Systems: Symmetry and Interconnection, PhD thesis, University of Twente, 2000. [5] A. M. Bloch, Nonholonomic Mechanics and Control, volume 24 of Interdisciplinary Applied Mathematics, Springer Verlag, 2003. doi: 10.1007/b97376. [6] A. M. Bloch and P. E. Crouch, Representations of Dirac structures on vector spaces and nonlinear LC circuits, In Proc. Sympos. Pure Math, volume 64, pages 103-117. American Mathematical Society, 1999. [7] N. Bou-Rabee and J. E. Marsden, Hamilton-Pontryagin integrators on Lie groups: Introduction and structure preserving properties, Foundations of Computational Mathematics, 9 (2009), 197-219. doi: 10.1007/s10208-008-9030-4. [8] R. K. Brayton, Nonlinear reciprocal networks, In Mathematical Aspects of Electrical Network Analysis, H.S. Wilf and F. Harary (eds). SIAM - AMS Proceedings, Volume III, pages 1-15, 1971. [9] H. Cendra, J. E. Marsden and T. S. Ratiu, Lagrangian Reduction by Stages, Memoirs of the American Mathematical Society. American Mathematical Society, 2001. doi: 10.1090/memo/0722. [10] J. Cervera, A. J. van der Schaft and A. Baños, Interconnection of port-Hamiltonian systems and composition of Dirac structures, Automatica, 43 (2007), 212-225. doi: 10.1016/j.automatica.2006.08.014. [11] D. A. Chang, A. M. Bloch, N. E. Leonard, J. E. Marsden and C. A. Woolsey, The equivalence of controlled Lagrangian and controlled Hamiltonian systems, ESAIM: Control, Optimisation and Calculus of Variations, 8 (2002), 393-422. doi: 10.1051/cocv:2002045. [12] T. J. Courant, Dirac manifolds, Transactions of the American Mathematical Society, 319 (1990), 631-661. doi: 10.1090/S0002-9947-1990-0998124-1. [13] T. J. Courant and A. Weinstein, Beyond Poisson structures, In Action Hamiltoniennes de groupes. Troisième théoréme de Lie (Lyon 1986), volume 27 of Travaux en Cours, pages 39-49. Hermann, 1988. [14] P. A. M. Dirac, Generalized Hamiltonian dynamics, Canadian J. Mathematics, 2 (1950), 129-148. doi: 10.4153/CJM-1950-012-1. [15] I. Dorfman, Dirac Structures and Integrability of Nonlinear Evolution Equations, (Nonlinear Science: Theory and Applications). Wiley & Sons Ltd., 1993. [16] J. Dufour and A. Wade, On the local structure of Dirac manifolds, Compos. Math., 144 (2008), 774-786. doi: 10.1112/S0010437X07003272. [17] V. Duindam, Port-based Modelling and Control for Efficient Bipedal Walking Robots, PhD thesis, University of Twente, 2006. [18] V. Duindam, A. Macchelli, S. Stramigioli and H. Bruyninckx, editors., Modelling and Control of Complex Physical Systems, Springer, 2009. doi: 10.1007/978-3-642-03196-0. [19] R. Featherstone, Robot Dynamics Algorithms, Kluwer Academic, 1987. [20] G. Golo, V. Talasila, A. van der Schaft and B. Maschke, Hamiltonian discretization of boundary control systems, Automatica, 40 (2004), 757-771. URL http://www.sciencedirect.com/science/article/pii/S0005109804000147. doi: 10.1016/j.automatica.2003.12.017. [21] M. Gualtieri, Generalized complex geometry, Ann. of Math. (2), 174 (2011), 75-123. doi: 10.4007/annals.2011.174.1.3. [22] H. O. Jacobs and J. Vankerschaver, Fluid-structure interaction in the Lagrange-Poincaré formalism, arXiv:1212.1144 [math.DS], Aug 2013. [23] H. O. Jacobs and H. Yoshimura, Interconnection and composition of Dirac structures for Lagrange-Dirac systems, In Decision and Control and European Control Conference (CDC-ECC), 2011 50th IEEE Conference on, pages 928-933, 2011. doi: 10.1109/CDC.2011.6160480. [24] H. O. Jacobs, H. Yoshimura and J. E. Marsden, Interconnection of Lagrange-Dirac dynamical systems for electric circuits, AIP Conference Proceedings, 1281 (2010), 566-569. doi: 10.1063/1.3498539. [25] G. Kron, Diakoptics: The Piecewise Solution of Large-Scale Systems, McDonald, London, 1963. [26] M. Leok and T. Ohsawa, Variational and geometric structures of discrete Dirac mechanics, Foundations of Computational Mathematics, 11 (2011), 529-562. ISSN 1615-3375. doi: 10.1007/s10208-011-9096-2. [27] R. G. Littlejohn, Variational principles of guiding centre motion, Journal of Plasma Physics, 29 (1983), 111-125. doi: 10.1017/S002237780000060X. [28] 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. [29] J. Merker, On the geometric structure of Hamiltonian systems with ports, Journal of Nonlinear Science, 19 (2009), 717-738. doi: 10.1007/s00332-009-9052-3. [30] R. Ortega, J. A. L. Perez, P. J. Nicklasson and H. J. Sira-Ramirez, Passivity-based Control of Euler-Lagrange Systems: Mechanical, Electrical, and Electromechanical Applications, Communications and Control Engineering. Springer-Verlag, 1st edition, 1998. [31] R. Ortega, A. J. van der Schaft, B. M. Maschke and G. Escobar, Interconnection and damping assignment passivity-based control of port-controlled Hamiltonian systems, Automatica, 38 (2002), 585-596. ISSN 0005-1098. doi: 10.1016/S0005-1098(01)00278-3. [32] H. M. Paynter, Analysis and Design of Engineering Systems, MIT Press, 1961. [33] V. Talasila, J. Clemente-Gallardo and A. J. van der Schaft, Discrete port-Hamiltonian systems, Systems and Control Letters, 55 (2006), 478-486. ISSN 0167-6911. doi: http://dx.doi.org/10.1016/j.sysconle.2005.10.001. doi: 10.1016/j.sysconle.2005.10.001. [34] W. M. Tulczyjew, The Legendre transformation, Annales de l'Institute Henri Poincaré, 27 (1977), 101-114. [35] A. J. van der Schaft, Port-Hamiltonian systems: An introductory survey, In Proceedings of the International Conference of Mathematics, volume 3, pages 1-27. European Mathematical Society, 1996. [36] A. J. van der Schaft and B. M. Maschke, The Hamiltonian formulation of energy conserving physical systems with external ports, Archiv für Elektronik und Übertragungstechnik, 49 (1995), 362-371. [37] J. Vankerschaver, H. Yoshimura and M. Leok, The Hamilton-Pontryagin principle and multi-Dirac structures for classical field theories, Journal of Mathematical Physics, 53 (2012), 072903. doi: 10.1063/1.4731481. [38] A. Weinstein, Symplectic categories, Port. Math., 67 (2010), 261-278. doi: 10.4171/PM/1866. [39] J. L. Wyatt and L. O. Chua, A theory of nonenergic $n$-ports, Circuit Theory and Applications, 5 (1977), 181-208. [40] H. Yoshimura, Dynamics of Flexible Multibody Systems, PhD thesis, Waseda University, 1995. [41] H. Yoshimura and J. E. Marsden, Dirac structures in Lagrangian mechanics part I: Implicit Lagrangian systems, Journal of Geometry and Physics, 57 (2006), 133-156. ISSN 0393-0440. doi: 10.1016/j.geomphys.2006.02.009. [42] H. Yoshimura and J. E. Marsden, Dirac structures in Lagrangian mechanics part II: Variational structures, Journal of Geometry and Physics, 57 (2006), 209-250. ISSN 0393-0440. doi: 10.1016/j.geomphys.2006.02.012. [43] H. Yoshimura and J. E. Marsden, Dirac structures and implicit Lagrangian systems in electric networks, In Proceedings of the 17th International Symposium on the Mathematical Theory of Networks and Systems, pages 1-6, 2006. [44] H. Yoshimura and J. E. Marsden, Dirac Structures and the Legendre transformation for implicit Lagrangian and Hamiltonian systems, In Lagrangian and Hamiltonian Methods for Nonlinear Control 2006, volume 366 of Lecture Notes in Control and Information Sciences, pages 233-247. Springer Berlin/Heidelberg, 2007. doi: 10.1007/978-3-540-73890-9_18. [45] H. Yoshimura and J. E. Marsden, Reduction of Dirac structures and the Hamilton-Pontryagin principle, Reports on Mathematical Physics, 60 (2007), 381-426. ISSN 0034-4877. doi: 10.1016/S0034-4877(08)00004-9. [46] H. Yoshimura and J. E. Marsden, Dirac cotangent bundle reduction, Journal of Geometric Mechanics, 1 (2009), 87-158. doi: 10.3934/jgm.2009.1.87. [47] H. Yoshimura, H. O. Jacobs, and J. E. Marsden, Interconnection of Dirac structures in Lagrange-Dirac dynamical systems, In Proceedings of the 20th International Symposium on the Mathematical Theory of Networks and Systems, 2010.
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