What if we set Hamilton-Jacobi equation as an axiom?

Question

We usually postulate the principle of least action. Then we can get Lagrangian mechanics. After that we can get Hamiltonian mechanics either from postulate or from the equivalent Lagrangian mechanics. Finally we get the Hamilton-Jacobi equation (HJE).

But what if we have HJE as postulate instead? How can we get Hamiltonian mechanics from it?

Answer 1

I) The Hamilton-Jacobi equation (HJE) itself sure ain't enough as an axiom without some kind of context, setting, definitions and identifications of various variables.

II) Let us assume:

1. that Hamilton's principal function $S(q,P,t)$ depends on the old positions $q^i$ and the new momenta $P_j$ and time $t$,

2. the HJE itself,

3. the definition of old momenta $p_i:=\frac{\partial S}{\partial q^i},$

4. the definition of new positions $Q^i:=\frac{\partial S}{\partial P_i}$,

5. that the new phase space variables $(Q^i,P_j)$ are all constants of motion (COM).

III) Then it is a straightforward exercise to derive Hamilton's equations for the old phase space variables $(q^i,p_j)$ provided pertinent rank-conditions are satisfied for the Hessian of $S$. [This result is expected, because (5) is precisely Kamilton's equations and the function $S$ in (1)-(4) is a type-2 generating function of a canonical transformation (CT).]

IV) The Lagrange equations follow next from a Legendre transformation. In turn, the Lagrange equations are Euler-Lagrange (EL) equations from the stationary action principle for $\int \! dt ~L$.