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Reduction of degenerate Lagrangian systems. (English) Zbl 0621.58020
Let a degenerate Lagrangian L be given on some velocity phase space (i.e. a tangent bundle). Is it possible to construct another velocity space and a regular Lagrangian which contains the same dynamical information as L ? This important question is referred to as the ”regularization problem” for degenerate Lagrangians. The authors find conditions which guarantee the existence of such a regularization and describe a relevant class of Lagrangians for which the above question admits an affirmative answer. The connection with the canonical approach to the regularization problem of degenerate systems (Dirac’s theory) and the reduction of systems with symmetry (Marsden-Weinstein theory) is also discussed. The paper is completed with a few typical examples and applications. Two appendices are devoted to the proof of those intermediate results which are of interest on their own.
Reviewer: J.Szilasi

MSC:
37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
70H03 Lagrange’s equations
53C80 Applications of global differential geometry to the sciences
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[1] Dirac, P.A.M.; Dirac, P.A.M., Lectures on quantum mechanics, (), 2, 129-148, (1950) · Zbl 0060.45102
[2] Gotay, M.J.; Nester, J.M.; Hinds, G., Presymplectic manifolds and the Dirac-bergmann theory of constraints, J. math. phys., 19, 2388-2399, (1978) · Zbl 0418.58010
[3] Marsden, J.E.; Weinstein, A., Reduction of symplectic manifolds with symmetry, Rep. math. phys., 5, 121-130, (1974) · Zbl 0327.58005
[4] Kunzle, H.P., Degenerate Lagrangian systems, Ann. inst. H. Poincaré, A11, 393-414, (1969) · Zbl 0193.24901
[5] Gotay, M.J.; Nester, J.M., Presymplectic Lagrangian systems I: the constraint algorithm and the equivalence theorem, Ann. inst. H. Poincaré, A30, 129-142, (1979) · Zbl 0414.58015
[6] Gotay, M.J.; Nester, J.M., Presymplectic Lagrangian systems II: the second-order equation problem, Ann. inst. H. Poincaré, A32, 1-13, (1980) · Zbl 0453.58016
[7] Klein, J., Espace variationnels et mécanique, Ann. inst. Fourier (Grenoble), 12, 1-124, (1962) · Zbl 0281.49026
[8] Crampin, M., Tangent bundle geometry for Lagrangian dynamics, J. phys. A: math. gen., 16, 3755-3772, (1983) · Zbl 0536.58004
[9] Grifone, J., Structure presque-tangente et connexions, I, Ann. inst. Fourier (Grenoble), 22, 287-334, (1972) · Zbl 0219.53032
[10] Cariñena, J.F.; Ibort, L.A., Geometric theory of the equivalence of Lagrangians for constrained systems, J. phys. A: math. gen., 18, 3335-3341, (1985) · Zbl 0588.58020
[11] Crampin, M., Defining Euler-Lagrange fields in terms of almost tangent structures, Phys. lett., A95, 466-468, (1983)
[12] Crampin, M., On the differential geometry of the Euler-Lagrange equations, and the inverse problem of Lagrangian dynamics, J. phys. A: math. gen., 14, 2567-2575, (1981) · Zbl 0475.70022
[13] Sarlet, W.; Cantrijn, F.; Crampin, M., A new look at second-order equations and Lagrangian mechanics, J. phys. A: math. gen., 17, 1999-2009, (1984) · Zbl 0542.58011
[14] Crampin, M.; Thompson, G., Affine bundles and integrable almost tangent structures, Math. proc. camb. phil. soc., 98, 61-71, (1985) · Zbl 0572.53031
[15] Brickell, F.; Clark, R.S., Integrable almost tangent structures, J. diff. geom., 9, 557-563, (1974) · Zbl 0302.53018
[16] Guillemin, V.; Sternberg, S., Symplectic techniques in physics, (1984), Cambridge Univ. Press Cambridge · Zbl 0576.58012
[17] Marmo, G., Nijenhuis operators in classical dynamics, (), (preprint) · Zbl 0694.35136
[18] Weinstein, A., Lectures on symplectic manifolds, () · Zbl 0406.53031
[19] Abraham, R.; Marsden, J.E., Foundations of mechanics, (1978), Benjamin/Cummings Publ. Comp Reading (Ma)
[20] Bergmann, P.G.; Goldberg, I., Dirac bracket transformation in phase space, Phys. rev., 98, 531-538, (1955) · Zbl 0065.22803
[21] Hanson, A.J.; Regge, T.; Teitelboim, C., Constrained Hamiltonian systems, Accad. naz. dei lincei, No. 22, (1976), Rome
[22] Sundermeyer, K., Constrained dynamics, ()
[23] Marmo, G.; Mukunda, N.; Samuel, J., Dynamics and symmetry for constrained systems: a geometrical analysis, La riv. del nuovo cim., 6, No. 2, (1983)
[24] Marle, C., Symplectic manifolds, dynamical groups and Hamiltonian mechanics, () · Zbl 0369.53042
[25] Crampin, M., On differentiable manifolds with degenerate metrics, Proc. camb. phil. soc., 64, 307-316, (1968) · Zbl 0159.51002
[26] Kustaanheimo, P., Spinor regularization of the Kepler motion, () · Zbl 0123.40301
[27] Stiefel, E.L.; Scheifele, G., Linear and regular celestial mechanics, () · Zbl 0226.70005
[28] Pirani, F.A.E., Once more the Kepler problem, Nuovo cim., B19, 189-207, (1974)
[29] Kummer, M., On the 3-dimensional lunar problem and other perturbation problems of the Kepler problem, J. math. anal. appl., 93, 142-194, (1983) · Zbl 0569.70013
[30] Asorey, M.; Cariñena, J.F.; Ibort, L.A., Generalized canonical transformations for time-dependent systems, J. math. phys., 24, 2745-2750, (1983) · Zbl 0548.70010
[31] Jakubiec, A., Canonical variables for the Dirac theory, Lett. math. phys., 9, 171-182, (1985) · Zbl 0571.53059
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