The Euler-Lagrange and Hamilton-Jacobi actions and the principle of least action

Details The Euler-Lagrange and Hamilton-Jacobi actions and the principle of least action The Euler-Lagrange and Hamilton-Jacobi actions and the principle of least action

2012-03-13

arxiv.org/abs/1203.2736v1

Physics

Quantum Physics

12 pages, submited to Am.J.Phys

Scientific paper

We recall the main properties of the classical action of Euler-Lagrange Scl(x; t; x0), which links the initial position x0 and its position x at time t, and of the Hamilton-Jacobi action, which connects a family of particles of initial action S0(x) to their various positions x at time t. Mathematically, the Euler-Lagrange action can be considered as the elementary solution of the Hamilton-Jacobi equation in a new branch of nonlinear mathematics, the Minplus analysis. Physically, we show that, contrary to the Euler-Lagrange action, the Hamilton-Jacobi action satis?es the principle of least action. It is a clear answer on the interpretation of this principle. Finally, we use the relation-ship between the Hamilton-Jacobi and Euler-Lagrange actions to study the convergence of quantum mechanics, when the Planck constant tends to 0, for a particular class of quantum systems, the statistical semiclassical case. 1

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