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Lagrangian reduction of nonholonomic discrete mechanical systems by stages
Symmetry actuated closed-loop Hamiltonian systems
Österreichische Finanzmarktaufsicht (FMA), Otto-Wagner Platz 5, A-1090 Vienna, Austria |
This paper extends the theory of controlled Hamiltonian systems with symmetries due to [
References:
[1] |
V. I. Arnold,
Sur la géométrie différentielle de groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Annales de l'Institut Fourier, 16 (1966), 319-361.
doi: 10.5802/aif.233. |
[2] |
V. Arnold and B. Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, 1998. |
[3] |
A. Arnaudon, A. L. De Castro and D. D. Holm,
Noise and dissipation on coadjoint orbits, J. Nonlinear Sci., 28 (2018), 91-145.
doi: 10.1007/s00332-017-9404-3. |
[4] |
A. Arnaudon, N. Ganaba and D. Holm,
The stochastic energy – Casimir method, Comptes Rendus Mécanique, 346 (2018), 279-290.
doi: 10.1016/j.crme.2018.01.003. |
[5] |
A. Bloch, D. Chang, N. Leonard and J. E. Marsden,
Controlled Lagrangians and the stabilization of mechanical systems II: Potential shaping, IEEE Trans. Automat. Control, 46 (2001), 1556-1571.
doi: 10.1109/9.956051. |
[6] |
A. Bloch, N. Leonard and J. Marsden,
Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem, IEEE Trans. Automat. Control, 45 (2000), 2253-2270.
doi: 10.1109/9.895562. |
[7] |
A. Bloch, N. Leonard and J. Marsden,
Controlled Lagrangians and the stabilization of Euler-Poincaré mechanical systems, Internat. J. Robust Nonlinear Control, 11 (2001), 191-214.
doi: 10.1002/rnc.572. |
[8] |
A. Bloch and N. Leonard, Symmetries, conservation laws, and control, in Geometry, Mechanics and Dynamics, Springer-Verlag, New York, 2002.
doi: 10.1007/b97525. |
[9] |
A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and G. Sánchez de Alvarez,
Stabilization of rigid body dynamics by internal and external torques, Automatica J. IFAC, 28 (1992), 745-756.
doi: 10.1016/0005-1098(92)90034-D. |
[10] |
A. M. Bloch, J. E. Marsden and G. Sánchez de Alvarez, Feedback stabilization of relative equilibria for mechanical systems with symmetry, in Current and Future Directions in Applied Mathematics, Birkhäuser, Boston, MA, 1997. |
[11] |
D. E. 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.
|
[12] |
D. Crisan, F. Flandoli and D. D. Holm,
Solution properties of a 3D stochastic Euler fluid equation, J. Nonlinear Sci., 29 (2019), 813-870.
doi: 10.1007/s00332-018-9506-6. |
[13] |
T. Drivas, D. Holm and J.-M. Leahy,
Lagrangian averaged stochastic advection by Lie transport for fluids, J. Stat. Phys., 179 (2020), 1304-1342.
doi: 10.1007/s10955-020-02493-4. |
[14] |
F. Gay-Balmaz, C. Tronci and C. Vizman,
Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups, J. Geom. Mech., 5 (2013), 39-84.
doi: 10.3934/jgm.2013.5.39. |
[15] |
F. Gay-Balmaz and T. Ratiu,
Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids, J. Symplectic Geom., 6 (2008), 189-237.
doi: 10.4310/JSG.2008.v6.n2.a4. |
[16] |
J. Gibbons, D. D. Holm and B. Kupershmidt,
The Hamiltonian structure of classical chromohydrodynamics, Phys. D, 6 (1982/83), 179-194.
doi: 10.1016/0167-2789(83)90004-0. |
[17] |
S. Hochgerner, Symmetry reduction of Brownian motion and quantum Calogero-Moser models, Stoch. Dyn., 13 (2013), 1250007, 31 pp.
doi: 10.1142/S0219493712500074. |
[18] |
S. Hochgerner and T. Ratiu,
The geometry of non-holonomic diffusion, J. Eur. Math. Soc., 17 (2015), 273-319.
doi: 10.4171/JEMS/504. |
[19] |
S. Hochgerner, A Hamiltonian mean field system for the Navier–Stokes equation, Proc. A, 474 (2018), 20180178, 20 pp.
doi: 10.1098/rspa.2018.0178. |
[20] |
S. Hochgerner, Feedback control of charged ideal fluids, preprint, 2020, arXiv: 1905.04778. Google Scholar |
[21] |
D. Holm, J. Marsden, T. Ratiu and A. Weinstein,
Nonlinear stability of fluid and plasma equilibria, Phys. Rep., 123 (1985), 1-116.
doi: 10.1016/0370-1573(85)90028-6. |
[22] |
D. Holm, Variational principles for stochastic fluid dynamics, Proc. A, 471 (2015), 20140963, 19 pp.
doi: 10.1098/rspa.2014.0963. |
[23] |
P. S. Krishnaprasad,
Lie-Poisson structures, dual-spinspacecraft and asymptotic stability, Nonlinear Anal., 9 (1985), 1011-1035.
doi: 10.1016/0362-546X(85)90083-5. |
[24] |
J.-A. Lazaro-Cami and J.-P. Ortega,
Stochastic Hamiltonian dynamical systems, Rep. Math. Phys., 61 (2008), 65-122.
doi: 10.1016/S0034-4877(08)80003-1. |
[25] |
J. E. Marsden, Park City lectures on mechanics, dynamics, and symmetry, in Symplectic Geometry and Topology, IAS/Park City Mathematics Series, No. 7, American Mathematical Society, Providence, RI, 1999,335–430.
doi: 10.1090/pcms/007/09. |
[26] |
J. E. Marsden, A. Weinstein, T. Ratiu, R. Schmid and R. Spencer, Hamiltonian systems with symmetry, coadjoint orbits and plasma physics, Proc. IUTAM-IS1MM Symposium on Modern Developments in Analytical Mechanics (Torino, 1982), Atti Acad. Sci. Torino Cl. Sci. Fis. Math. Natur., 117, (1983), 289–340. |
[27] |
P. Michor, Topics in Differential Geometry, Graduate Studies in Mathematics, vol. 93, Amer. Math. Soc., Providence, RI, 2008.
doi: 10.1090/gsm/093. |
[28] |
P. Michor, Some geometric evolution equations arising as geodesic equations on groups of diffeomorphism, including the Hamiltonian approach, in Phase space analysis of Partial Differential Equations, Birkhäuser Boston, Boston, 2006,133–215.
doi: 10.1007/978-0-8176-4521-2_11. |
[29] |
M. Puiggalí and A. Bloch,
An extension to the theory of controlled Lagrangians using the Helmholtz conditions, J. Nonlinear Sci., 29 (2019), 345-376.
doi: 10.1007/s00332-018-9490-x. |
[30] |
A. Weinstein,
A universal phase space for particles in Yang-Mills fields, Lett. Math. Phys., 2 (1977/78), 417-420.
doi: 10.1007/BF00400169. |
show all references
References:
[1] |
V. I. Arnold,
Sur la géométrie différentielle de groupes de Lie de dimension infinie et ses applications à l'hydrodynamique des fluides parfaits, Annales de l'Institut Fourier, 16 (1966), 319-361.
doi: 10.5802/aif.233. |
[2] |
V. Arnold and B. Khesin, Topological Methods in Hydrodynamics, Applied Mathematical Sciences, vol. 125, Springer-Verlag, New York, 1998. |
[3] |
A. Arnaudon, A. L. De Castro and D. D. Holm,
Noise and dissipation on coadjoint orbits, J. Nonlinear Sci., 28 (2018), 91-145.
doi: 10.1007/s00332-017-9404-3. |
[4] |
A. Arnaudon, N. Ganaba and D. Holm,
The stochastic energy – Casimir method, Comptes Rendus Mécanique, 346 (2018), 279-290.
doi: 10.1016/j.crme.2018.01.003. |
[5] |
A. Bloch, D. Chang, N. Leonard and J. E. Marsden,
Controlled Lagrangians and the stabilization of mechanical systems II: Potential shaping, IEEE Trans. Automat. Control, 46 (2001), 1556-1571.
doi: 10.1109/9.956051. |
[6] |
A. Bloch, N. Leonard and J. Marsden,
Controlled Lagrangians and the stabilization of mechanical systems I: The first matching theorem, IEEE Trans. Automat. Control, 45 (2000), 2253-2270.
doi: 10.1109/9.895562. |
[7] |
A. Bloch, N. Leonard and J. Marsden,
Controlled Lagrangians and the stabilization of Euler-Poincaré mechanical systems, Internat. J. Robust Nonlinear Control, 11 (2001), 191-214.
doi: 10.1002/rnc.572. |
[8] |
A. Bloch and N. Leonard, Symmetries, conservation laws, and control, in Geometry, Mechanics and Dynamics, Springer-Verlag, New York, 2002.
doi: 10.1007/b97525. |
[9] |
A. M. Bloch, P. S. Krishnaprasad, J. E. Marsden and G. Sánchez de Alvarez,
Stabilization of rigid body dynamics by internal and external torques, Automatica J. IFAC, 28 (1992), 745-756.
doi: 10.1016/0005-1098(92)90034-D. |
[10] |
A. M. Bloch, J. E. Marsden and G. Sánchez de Alvarez, Feedback stabilization of relative equilibria for mechanical systems with symmetry, in Current and Future Directions in Applied Mathematics, Birkhäuser, Boston, MA, 1997. |
[11] |
D. E. 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.
|
[12] |
D. Crisan, F. Flandoli and D. D. Holm,
Solution properties of a 3D stochastic Euler fluid equation, J. Nonlinear Sci., 29 (2019), 813-870.
doi: 10.1007/s00332-018-9506-6. |
[13] |
T. Drivas, D. Holm and J.-M. Leahy,
Lagrangian averaged stochastic advection by Lie transport for fluids, J. Stat. Phys., 179 (2020), 1304-1342.
doi: 10.1007/s10955-020-02493-4. |
[14] |
F. Gay-Balmaz, C. Tronci and C. Vizman,
Geometric dynamics on the automorphism group of principal bundles: Geodesic flows, dual pairs and chromomorphism groups, J. Geom. Mech., 5 (2013), 39-84.
doi: 10.3934/jgm.2013.5.39. |
[15] |
F. Gay-Balmaz and T. Ratiu,
Reduced Lagrangian and Hamiltonian formulations of Euler-Yang-Mills fluids, J. Symplectic Geom., 6 (2008), 189-237.
doi: 10.4310/JSG.2008.v6.n2.a4. |
[16] |
J. Gibbons, D. D. Holm and B. Kupershmidt,
The Hamiltonian structure of classical chromohydrodynamics, Phys. D, 6 (1982/83), 179-194.
doi: 10.1016/0167-2789(83)90004-0. |
[17] |
S. Hochgerner, Symmetry reduction of Brownian motion and quantum Calogero-Moser models, Stoch. Dyn., 13 (2013), 1250007, 31 pp.
doi: 10.1142/S0219493712500074. |
[18] |
S. Hochgerner and T. Ratiu,
The geometry of non-holonomic diffusion, J. Eur. Math. Soc., 17 (2015), 273-319.
doi: 10.4171/JEMS/504. |
[19] |
S. Hochgerner, A Hamiltonian mean field system for the Navier–Stokes equation, Proc. A, 474 (2018), 20180178, 20 pp.
doi: 10.1098/rspa.2018.0178. |
[20] |
S. Hochgerner, Feedback control of charged ideal fluids, preprint, 2020, arXiv: 1905.04778. Google Scholar |
[21] |
D. Holm, J. Marsden, T. Ratiu and A. Weinstein,
Nonlinear stability of fluid and plasma equilibria, Phys. Rep., 123 (1985), 1-116.
doi: 10.1016/0370-1573(85)90028-6. |
[22] |
D. Holm, Variational principles for stochastic fluid dynamics, Proc. A, 471 (2015), 20140963, 19 pp.
doi: 10.1098/rspa.2014.0963. |
[23] |
P. S. Krishnaprasad,
Lie-Poisson structures, dual-spinspacecraft and asymptotic stability, Nonlinear Anal., 9 (1985), 1011-1035.
doi: 10.1016/0362-546X(85)90083-5. |
[24] |
J.-A. Lazaro-Cami and J.-P. Ortega,
Stochastic Hamiltonian dynamical systems, Rep. Math. Phys., 61 (2008), 65-122.
doi: 10.1016/S0034-4877(08)80003-1. |
[25] |
J. E. Marsden, Park City lectures on mechanics, dynamics, and symmetry, in Symplectic Geometry and Topology, IAS/Park City Mathematics Series, No. 7, American Mathematical Society, Providence, RI, 1999,335–430.
doi: 10.1090/pcms/007/09. |
[26] |
J. E. Marsden, A. Weinstein, T. Ratiu, R. Schmid and R. Spencer, Hamiltonian systems with symmetry, coadjoint orbits and plasma physics, Proc. IUTAM-IS1MM Symposium on Modern Developments in Analytical Mechanics (Torino, 1982), Atti Acad. Sci. Torino Cl. Sci. Fis. Math. Natur., 117, (1983), 289–340. |
[27] |
P. Michor, Topics in Differential Geometry, Graduate Studies in Mathematics, vol. 93, Amer. Math. Soc., Providence, RI, 2008.
doi: 10.1090/gsm/093. |
[28] |
P. Michor, Some geometric evolution equations arising as geodesic equations on groups of diffeomorphism, including the Hamiltonian approach, in Phase space analysis of Partial Differential Equations, Birkhäuser Boston, Boston, 2006,133–215.
doi: 10.1007/978-0-8176-4521-2_11. |
[29] |
M. Puiggalí and A. Bloch,
An extension to the theory of controlled Lagrangians using the Helmholtz conditions, J. Nonlinear Sci., 29 (2019), 345-376.
doi: 10.1007/s00332-018-9490-x. |
[30] |
A. Weinstein,
A universal phase space for particles in Yang-Mills fields, Lett. Math. Phys., 2 (1977/78), 417-420.
doi: 10.1007/BF00400169. |
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