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In the one-dimensional stationary case, we construct a mechanical Lagrangiandescribing the quantum motion of a non-relativistic spinless system. ThisLagrangian is written as a difference between a function $T$, which representsthe quantum generalization of the kinetic energy and which depends on thecoordinate $x$ and the temporal derivatives of $x$ up the third order, and theclassical potential $V(x)$. The Hamiltonian is then constructed and thecorresponding canonical equations are deduced. The function $T$ is firstassumed arbitrary. The development of $T$ in a power series together with thedimensional analysis allow us to fix univocally the series coefficients byrequiring that the well-known quantum stationary Hamilton-Jacobi equation bereproduced. As a consequence of this approach, we formulate the law of thequantum motion representing a new version of the quantum Newton's law. We alsoanalytically establish the famous Bohm's relation % $\mu \dot{x} = \partial S_0/\partial x $ % outside of the framework of the hydrodynamical approach andshow that the well-known quantum potential, although it is a part of thekinetic term, it plays really a role of an additional potential as assumed byBohm.Comment: 20 pages, LateX, no figure, some calculations are reported in  appendice

From a Mechanical Lagrangian to the Schrödinger Equation: A Modified Version of the Quantum Newton Law