Operational classical mechanics: holonomic systems

Abstract

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.

The Schwinger action principle for classical systems*

“…However, there is a clear difference between the classical transition into chaos and quantum phase transition, despite both being related to divergences of the geometric tensor and the AGP. The primary interest in quantum phase transition is the behavior of the ground state of the quantum Hamiltonian, meanwhile, the Liouvillian (7) does not even have a ground state.…”

“…Quantum mechanics can be expressed in a Hamiltonian way [1] and in the same geometrical fashion as classical mechanics [2,3,4]. Equivalently, classical mechanics can be reformulated as a theory of operators acting on a Hilbert space [5,6,7,8,9]. This is not to say that the two theories are equivalent, of course, but that sometimes mathematical tools developed for one of them can be applied to the other.…”

Projective representation of the Galilei group for classical and quantum–classical systems*

“…Here we will study the classical geometric phases (and adiabatic driving, see below) in the same mathematical language of quantum mechanics. The above will be possible due to the formulation of classical mechanics known as the Koopman-von Neumann Theory (KvN) [8,9,10,11,12]. The KvN theory is an operational version of classical mechanics akin to quantum mechanics.…”

The wave operator representation of quantum and classical dynamics

“…In the case of pure states, this difficulty was resolved with the demonstration that the Wigner function should be interpreted as a phase space probability amplitude. This is in direct analogy with the Koopman-von Neumann (KvN) representation of classical dynamics [19,[21][22][23][24][25][26][27][28][29][30][31][32][33] which explicitly admits a wavefunction on phase space, and which the Wigner function of a pure state corresponds to in the classical limit. The extension of this interpretation to mixed states has to date been lacking however, given that such states must be described by densities and therefore lack a direct correspondence to wavefunctions.…”