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I came across the following proposition while reading Goldstein, in chapter 10, page 430, on Hamilton Jacobi theory. Here he performs a canonical transformation and he asserts that,

We can automatically ensure that the new variables are constant in time by requiring that the transformed Hamiltonian, $K$, shall be identically zero, for then the equations of motion are $$\frac{\partial K}{\partial P_i}=\dot{Q_i}=0 , \qquad -\frac{\partial K}{\partial Q_i}=\dot{P_i}=0 .\tag{10.1} $$

I have a few questions,

  • How do we know such a Canonical transformation exists so that $K$ is identically zero?

  • From what I see, Hamilton Jacobi theory helps to solve the initial value problem locally. Am I correct?

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OP is correct: Given certain regularity conditions, we are generically only guaranteed the existence of a local solution to the Hamilton-Jacobi 1st-order non-linear PDE.

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  • $\begingroup$ Thank you for answering. What about my first question? How do we know such a C-T exists? $\endgroup$ – Abhikumbale Mar 20 '18 at 11:59
  • $\begingroup$ The solution $S$ to HJ eq. is a type 2 generating function for the sought-for CT. $\endgroup$ – Qmechanic Mar 20 '18 at 12:02

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