December  2019, 11(4): 487-510. doi: 10.3934/jgm.2019024

Morse families and Dirac systems

1. 

Departamento de Matemática Aplicada, Universidad Politécnica de Madrid, Av. Juan de Herrera 4, 28040 Madrid, Spain

2. 

Departamento de Matemática, Universidad Nacional del Sur, CONICET, Av. Alem 1253, 8000 Bahía Blanca, Argentina

3. 

Instituto de Ciencias Matemáticas (CSIC-UAM-UC3M-UCM), C/Nicolás Cabrera 13-15, 28049 Madrid, Spain

 

Received  April 2018 Revised  April 2019 Published  November 2019

Dirac structures and Morse families are used to obtain a geometric formalism that unifies most of the scenarios in mechanics (constrained calculus, nonholonomic systems, optimal control theory, higher-order mechanics, etc.), as the examples in the paper show. This approach generalizes the previous results on Dirac structures associated with Lagrangian submanifolds. An integrability algorithm in the sense of Mendella, Marmo and Tulczyjew is described for the generalized Dirac dynamical systems under study to determine the set where the implicit differential equations have solutions.

Citation: María Barbero Liñán, Hernán Cendra, Eduardo García Toraño, David Martín de Diego. Morse families and Dirac systems. Journal of Geometric Mechanics, 2019, 11 (4) : 487-510. doi: 10.3934/jgm.2019024
References:
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R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.

[2]

V. I. Arnol'd, V. V. Kozlov and A. I. Neĭshtadt, Dynamical Systems. III, Encyclopaedia of Mathematical Sciences, 3. Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-642-61551-1.

[3]

M. Barbero-Liñán, H. Cendra, E. García-Toraño Andrés and D. Martín de Diego, New insights in the geometry and interconnection of port-Hamiltonian systems, J. Phys. A, 51 (2018), 375201, 30 pp. doi: 10.1088/1751-8121/aad4ba.

[4]

M. Barbero-LiñánD. Iglesias Ponte and D. Martín de Diego, Morse families in optimal control problems, SIAM J. Control Optim., 53 (2015), 414-433.  doi: 10.1137/120903488.

[5]

S. Bates and A. Weinstein, Lectures on the Geometry of Quantization, Berkeley Mathematics Lecture Notes, 8. American Mathematical Society, Providence, RI, Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, 1997. doi: 10.1016/s0898-1221(97)90217-0.

[6]

H. Bursztyn, A brief introduction to Dirac manifolds, Geometric and Topological Methods for Quantum Field Theory, Cambridge Univ. Press, Cambridge, (2013), 4–38.

[7]

H. Bursztyn and O. Radko, Gauge equivalence of Dirac structures and symplectic groupoids, Ann. Inst. Fourier (Grenoble), 53 (2003), 309-337.  doi: 10.5802/aif.1945.

[8]

J. F. Cariñena, Theory of singular Lagrangians, Fortschr. Phys., 38 (1990), 641-679.  doi: 10.1002/prop.2190380902.

[9]

H. CendraM. Etchechoury and S. Ferraro, An extension of the Dirac and Gotay-Nester theories of constraints for Dirac dynamical systems, J. Geom. Mech., 6 (2014), 167-236.  doi: 10.3934/jgm.2014.6.167.

[10]

H. Cendra, L. A. Ibort and J. E. Marsden, Horizontal Lin constraints, Clebsch potentials and variational principles on principal fiber bundles, XV International Colloquium on Group Theoretical Methods in Physics, World Sci. Publ., Teaneck, NJ, (1987), 446–450.

[11]

H. Cendra and J. Marsden, Lin constraints, Clebsch potentials and variational principles, Phys. D, 27 (1987), 63-89.  doi: 10.1016/0167-2789(87)90005-4.

[12]

H. Cendra, T. Ratiu and H. Yoshimura, Dirac-weinstein reduction, preprint, 2017.

[13]

J. CerveraA. J. van der Schaft and A. Baños, On composition of Dirac structures and its implications for control by interconnection, Nonlinear and Adaptive Control, Lect. Notes Control Inf. Sci., Springer, Berlin, 281 (2003), 55-63.  doi: 10.1007/3-540-45802-6_5.

[14]

J. CerveraA. van der Schaft and A. Baños, Interconnection of port-Hamiltonian systems and composition of Dirac structures, Automatica J. IFAC, 43 (2007), 212-225.  doi: 10.1016/j.automatica.2006.08.014.

[15]

J. CortésM. de LeónD. Martín de Diego and S. Martínez, Geometric description of vakonomic and nonholonomic dynamics. Comparison of solutions, SIAM J. Control Optim., 41 (2002), 1389-1412.  doi: 10.1137/S036301290036817X.

[16]

T. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661.  doi: 10.1090/S0002-9947-1990-0998124-1.

[17]

T. Courant and A. Weinstein, Beyond Poisson structures, Action Hamiltoniennes de Groupes, Troisième Théorème de Lie, Travaux en Cours, Hermann, Paris, 27 (1988), 38-49. 

[18]

M. Dalsmo and A. van der Schaft, On representations and integrability of mathematical structures in energy-conserving physical systems, SIAM J. Control Optim, 37 (1999), 54-91.  doi: 10.1137/S0363012996312039.

[19]

M. de LeónJ. Marrero and D. Martín de Diego, Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics, J. Geom. Mech., 2 (2010), 159-198.  doi: 10.3934/jgm.2010.2.159.

[20]

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 Mathematics Studies, 112. Notes on Pure Mathematics, 102. North-Holland Publishing Co., Amsterdam, 1985.

[21]

M. de León and P. Rodrigues, Methods of Differential Geometry in Analytical Mechanics, North-Holland Mathematics Studies, 158. North-Holland Publishing Co., Amsterdam, 1989.

[22]

P. A. M. Dirac, Generalized Hamiltonian dynamics, Canadian J. Math., 2 (1950), 129-148.  doi: 10.4153/CJM-1950-012-1.

[23]

P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science Monographs Series, 2. Belfer Graduate School of Science, New York, Produced and Distributed by Academic Press, Inc., New York, 1967.

[24]

I. Y. Dorfman, Dirac structures of integrable evolution equations, Phys. Lett. A, 125 (1987), 240-246.  doi: 10.1016/0375-9601(87)90201-5.

[25]

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. 

[26]

M. J. GotayJ. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints, J. Math. Phys., 19 (1978), 2388-2399.  doi: 10.1063/1.523597.

[27]

K. Grabowska and J. Grabowski, Dirac algebroids in Lagrangian and Hamiltonian mechanics, J. Geom. Phys., 61 (2011), 2233-2253.  doi: 10.1016/j.geomphys.2011.06.018.

[28]

K. GrabowskaP. Urbański and J. Grabowski, Geometrical mechanics on algebroids, Int. J. Geom. Methods Mod. Phys., 3 (2006), 559-575.  doi: 10.1142/S0219887806001259.

[29]

J. Grabowski, M. de León, J. C. Marrero and D. Martín de Diego, Nonholonomic constraints: A new viewpoint, J. Math. Phys., 50 (2009), 013520, 17 pp. doi: 10.1063/1.3049752.

[30]

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

[31]

V. Guillemin and S. Sternberg, Geometric Asymptotics, Mathematical Surveys, No. 14. American Mathematical Society, Providence, R.I., 1977.

[32]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31. Springer, Heidelberg, 2010.

[33] D. D. Holm, Geometric mechanics. Part I. Dynamics and Symmetry, Second edition, Imperial College Press, London, 2011. 
[34]

D. D. HolmJ. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721.

[35]

L. Hörmander, Fourier integral operators. Ⅰ, Acta Math., 127 (1971), 79-183.  doi: 10.1007/BF02392052.

[36]

D. IglesiasJ. C. MarreroD. Martín de Diego and D. Sosa, Singular Lagrangian systems and variational constrained mechanics on Lie algebroids, Dyn. Syst., 23 (2008), 351-397.  doi: 10.1080/14689360802294220.

[37]

F. Jiménez and H. Yoshimura, Dirac structures in vakonomic mechanics, J. Geom. Phys., 94 (2015), 158-178.  doi: 10.1016/j.geomphys.2014.11.002.

[38]

M. Leok and T. Ohsawa, Variational and geometric structures of discrete Dirac mechanics, Found. Comput. Math., 11 (2011), 529-562.  doi: 10.1007/s10208-011-9096-2.

[39]

P. Libermann and C.-M. Marle. Symplectic Geometry and Analytical Mechanics, Mathematics and its Applications, 35. D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.

[40]

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.

[41]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.

[42]

E. Martínez, Lie algebroids in classical mechanics and optimal control, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), 17.

[43]

G. MendellaG. Marmo and W. Tulczyjew, Integrability of implicit differential equations, J. Phys. A, 28 (1995), 149-163.  doi: 10.1088/0305-4470/28/1/018.

[44]

H. Parks and M. Leok, Variational itegrators for interconnected Lagrange-Dirac systems, J. Nonlinear Sci., 27 (2017), 1399-1434.  doi: 10.1007/s00332-017-9364-7.

[45]

L. S. Pontryagin, V. G. Boltyanskiĭ, R. V. Gamkrelidze and E. F. Mishchenko, Selected Works. Vol. 4. The mathematical Theory of Optimal Processes, Classics of Soviet Mathematics, Gordon & Breach Science Publishers, New York, 1986.

[46]

W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), Ai, A15–A18.

[47]

W. M. Tulczyjew., Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), Av, A675–A678.

[48]

A. van der Schaft and D. Jeltsema, Port-Hamiltonian systems theory: An introductory overview, Foundations and Trends in Systems and Control, 1 (2014), 173-378. 

[49]

A. van der Schaft and B. Maschke, The hamiltonian formulation of energy conserving physical systems with external ports, AEU. Archiv für Elektronik und Übertragungstechnik, 49 (1995), 362-371. 

[50]

A. van der Schaft and B. Maschke, Mathematical modeling of constrained hamiltonian systems, IFAC Proceedings Volumes, 28 (1995), 637-642. 

[51]

A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math., 6 (1971), 329-346.  doi: 10.1016/0001-8708(71)90020-X.

[52]

A. Weinstein, Lectures on Symplectic Manifolds, CBMS Regional Conference Series in Mathematics, 29. American Mathematical Society, Providence, R.I., 1979.

[53]

H. Yoshimura and J. Marsden, Dirac structures in Lagrangian mechanics. Ⅰ. Implicit Lagrangian systems, J. Geom. Phys., 57 (2006), 133-156.  doi: 10.1016/j.geomphys.2006.02.009.

[54]

H. Yoshimura and J. Marsden, Dirac structures in Lagrangian mechanics. Ⅱ. Variational structures, J. Geom. Phys., 57 (2006), 209-250.  doi: 10.1016/j.geomphys.2006.02.012.

show all references

References:
[1]

R. Abraham and J. Marsden, Foundations of Mechanics, Benjamin/Cummings Publishing Co., Inc., Advanced Book Program, Reading, Mass., 1978.

[2]

V. I. Arnol'd, V. V. Kozlov and A. I. Neĭshtadt, Dynamical Systems. III, Encyclopaedia of Mathematical Sciences, 3. Springer-Verlag, Berlin, 1988. doi: 10.1007/978-3-642-61551-1.

[3]

M. Barbero-Liñán, H. Cendra, E. García-Toraño Andrés and D. Martín de Diego, New insights in the geometry and interconnection of port-Hamiltonian systems, J. Phys. A, 51 (2018), 375201, 30 pp. doi: 10.1088/1751-8121/aad4ba.

[4]

M. Barbero-LiñánD. Iglesias Ponte and D. Martín de Diego, Morse families in optimal control problems, SIAM J. Control Optim., 53 (2015), 414-433.  doi: 10.1137/120903488.

[5]

S. Bates and A. Weinstein, Lectures on the Geometry of Quantization, Berkeley Mathematics Lecture Notes, 8. American Mathematical Society, Providence, RI, Berkeley Center for Pure and Applied Mathematics, Berkeley, CA, 1997. doi: 10.1016/s0898-1221(97)90217-0.

[6]

H. Bursztyn, A brief introduction to Dirac manifolds, Geometric and Topological Methods for Quantum Field Theory, Cambridge Univ. Press, Cambridge, (2013), 4–38.

[7]

H. Bursztyn and O. Radko, Gauge equivalence of Dirac structures and symplectic groupoids, Ann. Inst. Fourier (Grenoble), 53 (2003), 309-337.  doi: 10.5802/aif.1945.

[8]

J. F. Cariñena, Theory of singular Lagrangians, Fortschr. Phys., 38 (1990), 641-679.  doi: 10.1002/prop.2190380902.

[9]

H. CendraM. Etchechoury and S. Ferraro, An extension of the Dirac and Gotay-Nester theories of constraints for Dirac dynamical systems, J. Geom. Mech., 6 (2014), 167-236.  doi: 10.3934/jgm.2014.6.167.

[10]

H. Cendra, L. A. Ibort and J. E. Marsden, Horizontal Lin constraints, Clebsch potentials and variational principles on principal fiber bundles, XV International Colloquium on Group Theoretical Methods in Physics, World Sci. Publ., Teaneck, NJ, (1987), 446–450.

[11]

H. Cendra and J. Marsden, Lin constraints, Clebsch potentials and variational principles, Phys. D, 27 (1987), 63-89.  doi: 10.1016/0167-2789(87)90005-4.

[12]

H. Cendra, T. Ratiu and H. Yoshimura, Dirac-weinstein reduction, preprint, 2017.

[13]

J. CerveraA. J. van der Schaft and A. Baños, On composition of Dirac structures and its implications for control by interconnection, Nonlinear and Adaptive Control, Lect. Notes Control Inf. Sci., Springer, Berlin, 281 (2003), 55-63.  doi: 10.1007/3-540-45802-6_5.

[14]

J. CerveraA. van der Schaft and A. Baños, Interconnection of port-Hamiltonian systems and composition of Dirac structures, Automatica J. IFAC, 43 (2007), 212-225.  doi: 10.1016/j.automatica.2006.08.014.

[15]

J. CortésM. de LeónD. Martín de Diego and S. Martínez, Geometric description of vakonomic and nonholonomic dynamics. Comparison of solutions, SIAM J. Control Optim., 41 (2002), 1389-1412.  doi: 10.1137/S036301290036817X.

[16]

T. Courant, Dirac manifolds, Trans. Amer. Math. Soc., 319 (1990), 631-661.  doi: 10.1090/S0002-9947-1990-0998124-1.

[17]

T. Courant and A. Weinstein, Beyond Poisson structures, Action Hamiltoniennes de Groupes, Troisième Théorème de Lie, Travaux en Cours, Hermann, Paris, 27 (1988), 38-49. 

[18]

M. Dalsmo and A. van der Schaft, On representations and integrability of mathematical structures in energy-conserving physical systems, SIAM J. Control Optim, 37 (1999), 54-91.  doi: 10.1137/S0363012996312039.

[19]

M. de LeónJ. Marrero and D. Martín de Diego, Linear almost Poisson structures and Hamilton-Jacobi equation. Applications to nonholonomic mechanics, J. Geom. Mech., 2 (2010), 159-198.  doi: 10.3934/jgm.2010.2.159.

[20]

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 Mathematics Studies, 112. Notes on Pure Mathematics, 102. North-Holland Publishing Co., Amsterdam, 1985.

[21]

M. de León and P. Rodrigues, Methods of Differential Geometry in Analytical Mechanics, North-Holland Mathematics Studies, 158. North-Holland Publishing Co., Amsterdam, 1989.

[22]

P. A. M. Dirac, Generalized Hamiltonian dynamics, Canadian J. Math., 2 (1950), 129-148.  doi: 10.4153/CJM-1950-012-1.

[23]

P. A. M. Dirac, Lectures on Quantum Mechanics, Belfer Graduate School of Science Monographs Series, 2. Belfer Graduate School of Science, New York, Produced and Distributed by Academic Press, Inc., New York, 1967.

[24]

I. Y. Dorfman, Dirac structures of integrable evolution equations, Phys. Lett. A, 125 (1987), 240-246.  doi: 10.1016/0375-9601(87)90201-5.

[25]

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. 

[26]

M. J. GotayJ. M. Nester and G. Hinds, Presymplectic manifolds and the Dirac-Bergmann theory of constraints, J. Math. Phys., 19 (1978), 2388-2399.  doi: 10.1063/1.523597.

[27]

K. Grabowska and J. Grabowski, Dirac algebroids in Lagrangian and Hamiltonian mechanics, J. Geom. Phys., 61 (2011), 2233-2253.  doi: 10.1016/j.geomphys.2011.06.018.

[28]

K. GrabowskaP. Urbański and J. Grabowski, Geometrical mechanics on algebroids, Int. J. Geom. Methods Mod. Phys., 3 (2006), 559-575.  doi: 10.1142/S0219887806001259.

[29]

J. Grabowski, M. de León, J. C. Marrero and D. Martín de Diego, Nonholonomic constraints: A new viewpoint, J. Math. Phys., 50 (2009), 013520, 17 pp. doi: 10.1063/1.3049752.

[30]

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

[31]

V. Guillemin and S. Sternberg, Geometric Asymptotics, Mathematical Surveys, No. 14. American Mathematical Society, Providence, R.I., 1977.

[32]

E. Hairer, C. Lubich and G. Wanner, Geometric Numerical Integration. Structure-Preserving Algorithms for Ordinary Differential Equations, Springer Series in Computational Mathematics, 31. Springer, Heidelberg, 2010.

[33] D. D. Holm, Geometric mechanics. Part I. Dynamics and Symmetry, Second edition, Imperial College Press, London, 2011. 
[34]

D. D. HolmJ. E. Marsden and T. S. Ratiu, The Euler-Poincaré equations and semidirect products with applications to continuum theories, Adv. Math., 137 (1998), 1-81.  doi: 10.1006/aima.1998.1721.

[35]

L. Hörmander, Fourier integral operators. Ⅰ, Acta Math., 127 (1971), 79-183.  doi: 10.1007/BF02392052.

[36]

D. IglesiasJ. C. MarreroD. Martín de Diego and D. Sosa, Singular Lagrangian systems and variational constrained mechanics on Lie algebroids, Dyn. Syst., 23 (2008), 351-397.  doi: 10.1080/14689360802294220.

[37]

F. Jiménez and H. Yoshimura, Dirac structures in vakonomic mechanics, J. Geom. Phys., 94 (2015), 158-178.  doi: 10.1016/j.geomphys.2014.11.002.

[38]

M. Leok and T. Ohsawa, Variational and geometric structures of discrete Dirac mechanics, Found. Comput. Math., 11 (2011), 529-562.  doi: 10.1007/s10208-011-9096-2.

[39]

P. Libermann and C.-M. Marle. Symplectic Geometry and Analytical Mechanics, Mathematics and its Applications, 35. D. Reidel Publishing Co., Dordrecht, 1987. doi: 10.1007/978-94-009-3807-6.

[40]

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.

[41]

J. E. Marsden and M. West, Discrete mechanics and variational integrators, Acta Numer., 10 (2001), 357-514.  doi: 10.1017/S096249290100006X.

[42]

E. Martínez, Lie algebroids in classical mechanics and optimal control, SIGMA Symmetry Integrability Geom. Methods Appl., 3 (2007), 17.

[43]

G. MendellaG. Marmo and W. Tulczyjew, Integrability of implicit differential equations, J. Phys. A, 28 (1995), 149-163.  doi: 10.1088/0305-4470/28/1/018.

[44]

H. Parks and M. Leok, Variational itegrators for interconnected Lagrange-Dirac systems, J. Nonlinear Sci., 27 (2017), 1399-1434.  doi: 10.1007/s00332-017-9364-7.

[45]

L. S. Pontryagin, V. G. Boltyanskiĭ, R. V. Gamkrelidze and E. F. Mishchenko, Selected Works. Vol. 4. The mathematical Theory of Optimal Processes, Classics of Soviet Mathematics, Gordon & Breach Science Publishers, New York, 1986.

[46]

W. M. Tulczyjew, Les sous-variétés lagrangiennes et la dynamique hamiltonienne, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), Ai, A15–A18.

[47]

W. M. Tulczyjew., Les sous-variétés lagrangiennes et la dynamique lagrangienne, C. R. Acad. Sci. Paris Sér. A-B, 283 (1976), Av, A675–A678.

[48]

A. van der Schaft and D. Jeltsema, Port-Hamiltonian systems theory: An introductory overview, Foundations and Trends in Systems and Control, 1 (2014), 173-378. 

[49]

A. van der Schaft and B. Maschke, The hamiltonian formulation of energy conserving physical systems with external ports, AEU. Archiv für Elektronik und Übertragungstechnik, 49 (1995), 362-371. 

[50]

A. van der Schaft and B. Maschke, Mathematical modeling of constrained hamiltonian systems, IFAC Proceedings Volumes, 28 (1995), 637-642. 

[51]

A. Weinstein, Symplectic manifolds and their Lagrangian submanifolds, Advances in Math., 6 (1971), 329-346.  doi: 10.1016/0001-8708(71)90020-X.

[52]

A. Weinstein, Lectures on Symplectic Manifolds, CBMS Regional Conference Series in Mathematics, 29. American Mathematical Society, Providence, R.I., 1979.

[53]

H. Yoshimura and J. Marsden, Dirac structures in Lagrangian mechanics. Ⅰ. Implicit Lagrangian systems, J. Geom. Phys., 57 (2006), 133-156.  doi: 10.1016/j.geomphys.2006.02.009.

[54]

H. Yoshimura and J. Marsden, Dirac structures in Lagrangian mechanics. Ⅱ. Variational structures, J. Geom. Phys., 57 (2006), 209-250.  doi: 10.1016/j.geomphys.2006.02.012.

Figure 1.  The maps $ \Psi_1 $, $ \Psi_2 $ and $ \Psi_3 $
Figure 2.  $ D_M $ and $ D_{\omega_Q} $
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