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I) Given the Hamilton-Jacobi equation,$$\frac{\partial F(q,Q,t)}{\partial t}+H\left(q,\frac{\partial F(q,Q,t)}{\partial q},t\right)=0 $$ it is stated that any function $$F(q,Q,t=0)=f(q,Q)\tag{22}$$ would generate a solution to the problem. How does that work?

My question arose after reading the document The Hamilton-Jacobi equation on page 4, equation 22 — this might help to understand my question.

I have little knowledge of PDE's, so I don't know if this is some general property of PDE's or if there is some other reason for it.

Is the argument that I can for example numerically integrate the function from this point using the HJ equation to obtain $F(q,Q,t)$ at all other times?

II) I tried to solve the free particle and realised there are some points that are unclear. Given is $$ H=\frac{p^2}{2m}. $$ Let $$ F(q,Q,t)=W(q,Q)-Qt $$ and $\frac{\partial F}{\partial q}=p$ and $\frac{\partial F}{\partial Q}=P$. The Hamilton-Jacobi equation is $$ H(q,\frac{\partial F}{\partial q},t) + \frac{\partial F}{\partial t} = 0. $$

This reduces to $$ \frac{1}{2m}\left( \frac{\partial W}{\partial q} \right)^2 = Q. $$ Solving for $W(q,Q)$ by integration over $\int^q_{q_0}dq'$ after rearrangement yields $$W(q,Q) = \sqrt{2mQ}(q-q_0) + W(q_0, Q). $$ Now we can write down $F$: $$ F(q,Q,t)=\sqrt{2mQ}(q-q_0) + W(q_0, Q) - Qt. $$

Is this not the general solution for the principal function? Is there still some sort of constant or function which I could add or remove from this solution?

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  • $\begingroup$ Are you asking for a proof of existence of $F$-solutions to the HJ eq? $\endgroup$ – Qmechanic Sep 25 at 7:37
  • $\begingroup$ I am asking why giving $F(q,Q,t=0)=f(q,Q)$ leads to a solution. I also read the statement that the general solution depends due to initial conditions on an arbitrary function $f(q,Q)$. I don't understand why i can pick any function $f(q,Q)$ as initial condition for a solution. Why is that true ? $\endgroup$ – Hans Wurst Sep 25 at 7:57
  • $\begingroup$ A differential equation does not give a unique solution unless we specify initial conditions. Think of $\dot{x}(t) = c$ gives $x(t) = t c + b$. $b$ is an absolutely arbitrary constant that can be fixed by imposing a specific value for the initial condition $x(t=0) = b =2$ for example. $\endgroup$ – Rudyard Sep 25 at 11:14
  • $\begingroup$ As other people have pointed out, $F$ is not an initial condition. $F$ is the generator of the canonical transformamation $p=\partial F/\partial q$. This reduces the $2$nd differential equation to one which is $1$st order in $F$. If the Hamiltonian was independent of $t$, then $F=F_{0}-Et$. $\endgroup$ – Cinaed Simson Sep 26 at 6:55
  • $\begingroup$ Can someone perhaps give an example ? Simply use a free moving particle and show what $f(q,Q)$ could be ? I don't see how it fits in. $\endgroup$ – Hans Wurst Sep 26 at 7:02
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I) We can rewrite the HJE $$\frac{\partial F(q,Q,t)}{\partial t}~=~-H\left(q,\frac{\partial F(q,Q,t)}{\partial q},t\right)$$ with a type 1 generating function $F(q,Q,t)$ into a fixed-point integral equation $$F(q,Q,t_f)~=~F(q,Q,t_i)-\int_{t_i}^{t_f}\! dt~H(q, \frac{\partial F(q,Q,t)}{\partial q}, t).$$ Given an arbitrary initial profile $$F(q,Q,t_i)~=~f(q,Q)\tag{22}$$ we can in principle find a unique local solution $F(q,Q,t)$ to the HJE, cf. the general mathematical theory for 1st-order non-linear PDEs [2,3].

II) OP considers a free 1D particle, and lists a complete solution of type 1. Similarly, a complete solution to Hamilton's principal function (which is a type 2 generating function) is $$S(x,t; \alpha_1, \alpha_2)~=~\alpha_1 x - \frac{\alpha_1^2}{2m}t +\alpha_2. $$ Note however that a complete solution to a PDE is not a general solution [4,5], despite its name!

References:

  1. S. Mathur, Hamilton-Jacobi equation; p. 4 eq. (22).

  2. R.L. Bryant, S. S. Chern, R.B. Gardner, H.L. Goldschmidt & P.A. Griffiths, Exterior Differential Systems, 2011; p. 30-35.

  3. Y. Choquet-Bruhat, C. DeWitt-Morette, Cécile de Witt & M. Dillard-Bleick, Analysis, Manifolds and Physics, Part 1: Basics, 1982; p. 242-258.

  4. H. Goldstein, Classical Mechanics; Section 10.1 first footnote.

  5. L.D. Landau & E.M. Lifshitz, Mechanics, vol. 1 (1976); $\S$47 footnote on p. 148.

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