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singular Lagrangians; Euler-Lagrange form; point symmetry; conservation law; equivalent Lagrangians
Dynamical properties of singular Lagrangian systems differ from those of classical Lagrangians of the form $L=T-V$. Even less is known about symmetries and conservation laws of such Lagrangians and of their corresponding actions. In this article we study symmetries and conservation laws of a concrete singular Lagrangian system interesting in physics. We solve the problem of determining all point symmetries of the Lagrangian and of its Euler-Lagrange form, i.e. of the action. It is known that every point symmetry of a Lagrangian is a point symmetry of its Euler-Lagrange form, and this of course happens also in our case. We are also interested in the converse statement, namely if to every point symmetry $\xi$ of the Euler-Lagrange form $E$ there exists a Lagrangian $\lambda$ for $E$ such that $\xi$ is a point symmetry of $\lambda$. In the case studied the answer is affirmative, moreover we have found that the corresponding Lagrangians are all of order one.
[1] Cariñena, J.F., Férnandez-Núñez, J.: Geometric Theory of Time-Dependent Singular Lagrangians. Fortschr. Phys., 41, 1993, 517-552 MR 1234799
[2] Cawley, R.: Determination of the Hamiltonian in the presence of constraints. Phys. Rev. Lett., 42, 1979, 413-416 DOI 10.1103/PhysRevLett.42.413
[3] Chinea, D., León, M. de, Marrero, J.C.: The constraint algorithm for time-dependent Lagrangians. J. Math. Phys., 35, 1994, 3410-3447 DOI 10.1063/1.530476 | MR 1279313 | Zbl 0810.70014
[4] Dirac, P.A.M.: Generalized Hamiltonian dynamics. Canad. J. Math., II, 1950, 129-148 DOI 10.4153/CJM-1950-012-1 | MR 0043724 | Zbl 0036.14104
[5] 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 MR 2356437
[6] 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 MR 0535369 | Zbl 0414.58015
[7] 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 MR 0574809 | Zbl 0453.58016
[8] Gotay, M.J., Nester, J.M., Hinds, G.: Presymplectic manifolds and the Dirac--Bergmann theory of constraints. J. Math. Phys., 19, 1978, 2388-2399 DOI 10.1063/1.523597 | MR 0506712 | Zbl 0418.58010
[9] Gràcia, X., Pons, J.M.: Guage transformations, Dirac's conjecture and degrees of freedom for constrained systems. Ann. Phys., 187, 1988, 355-368 DOI 10.1016/0003-4916(88)90153-4 | MR 0980754
[10] Gràcia, X., Pons, J.M.: A generalized geometric framework for constrained systems. Differential Geometry and its Applications, 2, 1992, 223-247 DOI 10.1016/0926-2245(92)90012-C | MR 1245325 | Zbl 0763.34001
[11] Havelková, M.: A geometric analysis of dynamical systems with singular Lagrangians. Communications in Mathematics, 19, 2011, 169-178 MR 2897268 | Zbl 1251.37052
[12] Krupka, D.: Some geometric aspects of variational problems in fibered manifolds. Folia Fac. Sci. Nat. UJEP Brunensis, 14, 1973, 1-65
[13] Krupková, O.: A geometric setting for higher-order Dirac-Bergmann theory of constraints. J. Math. Phys., 35, 1994, 6557-6576 DOI 10.1063/1.530691 | MR 1303063 | Zbl 0823.70016
[14] Krupková, O.: The Geometry of Ordinary Variational Equations. 1997, Springer MR 1484970
[15] LI, Z.-P.: Symmetry in a constrained Hamiltonian system with singular higher-order Lagrangian. J. Phys. A: Math. Gen., 24, 1991, 4261-4274 DOI 10.1088/0305-4470/24/18/014 | MR 1127293 | Zbl 0746.70014
[16] LI, Z.-P.: A counterexample to a conjecture of Dirac for a system with singular higher-order Lagrangian. Europhys. Lett., 21, 1993, 141-146 DOI 10.1209/0295-5075/21/2/003
[17] LI, Z.-P.: Symmetry in phase space for a system a singular higher-order Lagrangian. Physical Review E, 50, 1994, 876-887 MR 1381870
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