Hamiltonian mechanics is a theory developed as a reformulation of classical mechanics and predicts the same outcomes as non-Hamiltonian classical mechanics. It uses a different mathematical formalism, providing a more abstract understanding of the theory. Historically, it was an important reformulation of classical mechanics, which later contributed to the formulation of statistical mechanics and quantum mechanics.
For simplicity consider a single particle moving in one dimension having a wave function ψ(x,t); in this case total probability is ∫|ψ|2 dx, where the integration is performed from minus infinity to plus infinity. Physical meaning is that the particle must exist somewhere in space. There is also an equation of continuity for the probability current density j(x), which is a vector quantity ∂ρ/∂t + ∇.j=0, which means that if the particle disappears from a specific region, it must appear at a different region. This axiom was questioned in relativistic quantum mechanics where it was assumed that particles can be created or destroyed at the expense of an equivalent amount of energy, as described by the principle of mass energy equivalence.
Thank you for your contributions. This application is done in the thermodynamic. What we are trying to fact check is whether its actually relevant.
Good question. Hold onto the word relevance. Every abstraction of theory provides us a space for conceptualising another phenomenon, which could be removed from the present. For example, that is how Lokta-Volterra frame from biomathematics guided rational choice economics, and now we seek new spaces to mould the reality of institutions. So, what counts is the rigour of thought in a new space and not the immediate practical application.
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