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Guided by the symmetries of the Euler-Lagrange equations of motion, a studyof the constrained dynamics of singular Lagrangians is presented. We find thatthese equations of motion admit a generalized Lie symmetry, and on theLagrangian phase space the generators of this symmetry lie in the kernel of theLagrangian two-form. Solutions of the energy equation\textemdash calledsecond-order, Euler-Lagrange vector fields (SOELVFs)\textemdash with integralflows that have this symmetry are determined. Importantly, while second-order,Lagrangian vector fields are not such a solution, it is always possible toconstruct from them a SOELVF that is. We find that all SOELVFs are projectableto the Hamiltonian phase space, as are all the dynamical structures in theLagrangian phase space needed for their evolution. In particular, the primaryHamiltonian constraints can be constructed from vectors that lie in the kernelof the Lagrangian two-form, and with this construction, we show that theLagrangian constraint algorithm for the SOELVF is equivalent to the stabilityanalysis of the total Hamiltonian. Importantly, the end result of thisstability analysis gives a Hamiltonian vector field that is the projection ofthe SOELVF obtained from the Lagrangian constraint algorithm. The Lagrangianand Hamiltonian formulations of mechanics for singular Lagrangians are in thisway equivalent.

Constrained Dynamics: Generalized Lie Symmetries, Singular Lagrangians, and the Passage to Hamiltonian Mechanics