Invariance and conservation laws for Lagrangian systems with one degree of freedom

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Abstract
For a general class of linear transformations of both the dependent and the independent variable in Lagrangian systems with one degree of freedom, we investigate the relationship between invariances for the Lagrangian, the equation of motion, and a constant of the motion. Unlike Denman’s work on this subject, we try to show that for every invariance in the equation of motion a Lagrangian can be found with the related invariance. Furthermore, a constant of the motion is considered as being implied by the invariance of the equation of motion if it has the same invariance as the Lagrangian.