Several attempts have been made to approach classical mechanics in ways that are different from Newtonian Mechanics. Over time, we moved to mathematically more complex Lagrangian and Hamiltonian formulations which have given us a deep understanding of the symmetry, mathematical structures, etc. of classical dynamical systems. Meanwhile, the development of Quantum Mechanics presented the need to develop yet another approach to Classical Mechanics. This is because Quantum Mechanics has a mathematical structure that is unique and different from the one used to describe classical physics. The change in the mathematical structure during the transition from the microscopic world (the quantum world) to the macroscopic world (the classical world) proves to be a challenge. The need to study the interplay between Quantum and Classical Mechanics motivated the attempts to express both theories in mathematical formalisms that are similar to each other.
Therefore we need ways to formulate Classical Mechanics in the same mathematical language as that of Quantum Mechanics. Our work aims at studying the various operatorial approaches to classical mechanics and solving classical systems usign the same. We started with understanding the foundations of Koopman-von Neumann(KvN) Mechanics and the key differences in the operatorial language that set it apart from Quantum Mechanics. While the usual Classical Mechanics is formulated in the phase space framework with observables being functions of position, momentum and time, the KvN approach to Classical Mechanics starts with the introduction of a Hilbert Space of complex and square-integrable functions. Further, the modulus square of these wave functions will be equal to the probability density in phase space and the observables are defined to be operators. Even when we use operatorial language, one must note that KvN formalism still deals with classical systems where position and momentum can be measured simultaneously. Further, unlike the quantum mechanical wave function, the modulus and phase of the KvN wave function evolve separately.
We used the KvN formalism to evaluate the free particle and Harmonic Oscillator systems and successfully mapped it back to Newtonian mechanics. Further we looked at the operational formulation of classical mechanics, derived from a unitary, irreducible representation of the Galilei group. We then Utilized the properties of Galilei Group representation in Classical Mechanics to simplify the solutions of Harmonic Oscillator, Cubic Anharmonic Oscillator and Central Potential systems. Here, we found that the results match with those found using the KvN approach. With this, we confirmed that the two approaches are indeed equivalent and in fact the KvN theory is its particular case.
We were able to give a generic approach to solving basic classical mechanical systems using an operatorial approach. Our work could also be extended to solve for other systems by following a similar procedure. But the challanging part of using an operatorial approach is the rigorous mathematics involved,to solve even simple systems. Thus, we cannot view this as a better alternative than any previous classical formalisms, but only as a prescribed framework to look at classical subsystems being part of a larger system involving both classical and quantum systems.