Abstract
It is shown that the Hamilton's principle in classical mechanics and the Schrodinger equation in quantum mechanics can both be derived from an application of Gauss' principle of least squares.
Highlights
The essence of Gauss' principle of least squares seems to be the following: Natural laws are many faceted that allow the same law to be manifested in different ways
It is said that Gauss himself seemed to have favored this principle to make it his preferred topic in lecture [1]. Gauss applied this principle in his calculation of the orbit of the planet Ceres and used it to formulate mechanical systems with constraints [1]
We find that a judicious application of the Gauss principle of least squares can lead to the Schrodinger equation
Summary
The essence of Gauss' principle of least squares seems to be the following: Natural laws are many faceted that allow the same law to be manifested in different ways. The equations of motion of individual components are obtained by making the time integral of the Lagrangian stationary, resulting in the Euler-Lagrange equations It appears that the Hamilton's principle is different from and unrelated to the Gauss principle of least squares. (3.3) Since we are assuming the S to always satisfy the Hamilton-Jacobi equation and that a total time derivative of a function does n.ot contribute to the result of a variational principle, (3.2) becomes (3.4). This is exactly the Hamilton principle to obtain possible motions. The Lagrangian appears naturally in this process as the integrand
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