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A geometric analysis of dynamical systems with singular Lagrangians. (English) Zbl 1251.37052

The author investigates a singular mechanical system given by the Lagrangian \[ L= \dot q_1\dot q_3- q_2\dot q_3+ q_1 q_3. \] This system has been studied previously using the constraint method of Dirac, but the results were incomplete and conclusions from different authors were not in agreement. The author rectifies this situation by providing a rigorous solution using the Hamiltonian exterior differential systems method. Given the Lagrangian, one obtains the corresponding dynamical distribution in the first jet bundle. She shows that this distribution is not completely integrable and has a non-constant rank. To obtain the dynamics a general integration method (developed by O. Krupková) called the “geometric constraint algorithm” is applied.
The author calculates the Euler-Lagrange and Hamiltonian equations in terms of the corresponding distributions and finds the complete structure of the solutions. It is worth noting that the Hamiltonian and Euler-Lagrange equations are not equivalent and that the dynamics are not representable by a vector field.

MSC:

37J05 Relations of dynamical systems with symplectic geometry and topology (MSC2010)
70G45 Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics
70H03 Lagrange’s equations
70H45 Constrained dynamics, Dirac’s theory of constraints
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References:

[1] Chinea, D., León, M. de, Marrero, J.C.: The constraint algorithm for time-dependent Lagrangians. J. Math. Phys. 35 1994 3410-3447 · Zbl 0810.70014
[2] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Canad. J. Math. II 1950 129-148 · Zbl 0036.14104
[3] El-Zalan, H.A., Muslih, S.I., Elsabaa, F.M.F.: The Hamiltonian-Jacobi analysis of dynamical system with singular higher order Lagrangians. Hadronic Journal 30 2007 209-220 · Zbl 1138.70010
[4] Goldschmidt, H., Sternberg, S.: The Hamilton-Cartan formalism in the calculus of variations. Ann. Inst. Fourier, Grenoble 23 1973 203-267 · Zbl 0243.49011
[5] Gotay, M.J., Nester, J.M.: Presymplectic Lagrangian systems I: the constraint algorithm and the equivalence theorem. Ann. Inst. H. Poincaré Sect. A, 30 1979 129-142 · Zbl 0414.58015
[6] Gotay, M.J., Nester, J.M.: Presymplectic Lagrangian systems II: the second order equation problem. Ann. Inst. H. Poincaré Sect. A, 32 1980 1-13 · Zbl 0453.58016
[7] Gotay, M.J., Nester, J.M., Hinds, G.: Presymplectic manifolds and the Dirac-Bergmann theory of constraints. J. Math. Phys. 19 1978 2388-2399 · Zbl 0418.58010
[8] Krupková, O.: A geometric setting for higher-order Dirac-Bergmann theory of constraints. J. Math. Phys. 35 1994 6557-6576 · Zbl 0823.70016
[9] Krupková, O.: The Geometry of Ordinary Variational Equations. Springer 1997 · Zbl 0936.70001
[10] Saunders, D.J.: The Geometry of Jet Bundles. Cambridge Univ. Press, Cambridge 2nd 2004 · Zbl 0665.58002
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