Principle of least action; some possible generalizations

In this article we draw the attention to an important variational principle in dynamics: the Maupertuis-Jacobi Least Action Principle (MJLAP). This principle compares varied paths with the same energy h. We give two new proofs of the MJLAP (Sections 3 and 8) as well as a new unified variational principle which contains both Hamilton's Principle (HP) and the MJLAP as particular cases (Sections 4 and 9). The article also shows several new methods for the construction of a Lagrangian for a conservative dynamical system. As an example, we illustrate the theory with the classical Harmonic Oscillator Problem (Section 10). Our method is based on the theory of changes of independent variables in a dynamical system. It indirectly shows how a change of independent variable affects the self-adjointness of a dynamical system (Sections 5, 6, 7). Our new Lagrangians contain an arbitrary constant ..cap alpha.., whose meaning needs to be studied, eventually in relation to the concepts of quantification or gauge transformations. The two important values of the constant ..cap alpha.. are 1 (Hamilton's principle) and 1/2 (Maupertuis-Jacobi Least Action Principle).