The Hamilton-Jacobi (HJ) equation for a one-dimensional system is obtained by considering the canonical transformation (q,p) → (Q,P ) so that the old Hamiltonian H maps to a trivialized one, that is H˜ = 0. The old and new momenta are expressed in terms of the generating function of such a transformation, the Hamilton’s principal function p = ∂Scl ∂q , P = cnst = − ∂Scl ∂Q |Q=cnst that satisfies the classical HJ equation H ( q,p = ∂S ∂q , t ) + ∂S ∂t = 0.
Hamilton-Jacobi theory is a fundamental subject of classical mechanics and has also an important role in the development of quantum mechanics. Its conceptual framework results from the advantages of transformation theory and, for this reason, relates to the features of the generating function of canonical transformations connecting a specific Hamiltonian H to a new Hamiltonian K chosen to simplify Hamilton's equations of motion. Usually, the choice is between K=0 and a cyclic K depending on the new conjugate momenta only. Here, we investigate more general representations of the new Hamiltonian. Furthermore, it is pointed out that the two common alternatives of K=0 and a cyclic K should be distinguished in more detail. An attempt is made to clearly discern the two regimes for the harmonic oscillator and, not surprisingly, some correspondences to the quantum harmonic oscillator appear. Thanks to this preparatory work, generalized coordinates and momenta associated with the Hamilton's principal function S (i.e., K=0) are used to determine dependences in the Hamilton's characteristic function W (i.e., cyclic K). The procedure leads to the Schrodinger's equation where the Hamilton's characteristic function plays the role of the Schrodinger's wave function, whereas the first integral invariant of Poincare' takes the place of the reduced Planck constant. This finding advances the pedagogical value of the Hamilton-Jacobi theory in view of an introduction to Schrodinger's wave mechanics without the tools of quantum mechanics.
Classical mechanics admits multiple equivalent formulations, from Newton's equations to the variational Lagrange-Hamilton framework and the scalar Hamilton-Jacobi (HJ) theory. In the HJ formulation, classical ensembles evolve through the continuity equation for a real density
ρ=R2 coupled to Hamilton's principal function
S. Here we develop a complementary formulation, the Hamilton-Jacobi-Schrödinger (HJS) theory, by embedding the pair
(R,S) into a single complex field. Starting from a completely general complex ansatz
ψ=f(R,S)eig(R,S), and imposing two minimal structural requirements, we obtain a unique map $ψ= R\, e^{iS/κ}\, $ together with a linear HJS equation whose
∣κ∣→0 limit reproduces the HJ formulation exactly. Remarkably, when
Re(κ)=0, essential features of quantum mechanics, superposition, operator algebra, commutators, the Heisenberg uncertainty principle, Born's rule and unitary evolution, follow naturally as structural consistency conditions. HJS thus provides a unified mathematical viewpoint in which classical and quantum dynamics appear as different limits of a single underlying structure.
We consider the two main theorems in the derivation of the Quantum Hamilton--Jacobi Equation from the Equivalence Postulate (EP) of quantum mechanics. The first one concerns a basic cocycle condition, which holds in any dimension with Euclidean or Minkowski metrics and implies a global conformal symmetry underlying the Quantum Hamilton--Jacobi Equation. In one dimension such a condition fixes the Schwarzian equation. The second theorem concerns energy quantization which follows rigorously from consistency of the equivalence postulate.
We study how the classical Hamilton's principal and characteristic functions are generated from the solutions of the quantum Hamilton–Jacobi equation. While in the classically forbidden regions these quantum quantities directly tend to the classical ones, this is not the case in the allowed regions. There, the limit is reached only if the quantum fluctuations are eliminated by means of coarse-graining averages. Analogously, the classical Hamilton–Jacobi scheme bringing to the motion's equations arises from a similar formal quantum procedure. Quanta 2022; 11: 42–52.
Abstract The Hamilton-Jacobi theory is the apotheosis of Lagrangian and Hamiltonian mechanics: action functions encode all of the possible trajectories of a mechanical system satisfying certain criteria. These action functions are the solutions of a nonlinear, first-order partial differential equation, called the Hamilton-Jacobi equation. The characteristic equations of this differential equation are the extended Hamilton equations. Solution of a class of mechanics problems is thus reduced to the solution of a single partial differential equation. Aside from its use as a problem-solving tool, the Hamilton-Jacobi theory has particular importance because of its close relation to the Schroedinger formulation of quantum mechanics. This chapter discusses the connection between the Hamilton-Jacobi theory and the Schroedinger formulation, the Bohm hidden variable model and Feynman path integral method that are derived from it, Hamilton’s characteristic equations, complete integrals, separation of variables, canonical transformations, general integrals, mono-energetic integrals, relativistic Hamilton-Jacobi equation, quantum Cauchy problem, quantum mechanics, and classical mechanics.