Operational classical mechanics: holonomic systems

• Published in: Journal of physics. A, Mathematical and theoretical Mathematical and theoretical, 2022-10, Vol.55 (40), p.405201
• Main Author: Bermúdez Manjarres, A D
• Format: Article
• Language: English
• Subjects: Hilbert space classical mechanics; holonomic constraints; Koopman–von Neumann
• ISSN: 1751-8113; 1751-8121
• DOI: 10.1088/1751-8121/ac8f75

We construct an operational formulation of classical mechanics without presupposing previous results from analytical mechanics. In doing so, we rediscover several results from analytical mechanics from an entirely new perspective. We start by expressing the position and velocity of point particles as the eigenvalues of self-adjoint operators acting on a suitable Hilbert space. The concept of holonomic constraint is shown to be equivalent to a restriction to a linear subspace of the free Hilbert space. The principal results we obtain are: (1) the Lagrange equations of motion are derived without the use of D’Alembert or Hamilton principles, (2) the constraining forces are obtained without the use of Lagrange multipliers, (3) the passage from a position–velocity to a position–momentum description of the movement is done without the use of a Legendre transformation, (4) the Koopman–von Neumann theory is obtained as a result of our ab initio operational approach, (5) previous work on the Schwinger action principle for classical systems is generalized to include holonomic constraints.