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As I was reading Goldstein, there is the Hamiltonian $H$ such that, $$\delta \int_{t1}^{t2} (p_i\dot q_i - H(q, p ,t)) dt = 0, \tag{9.7}$$ and Kamiltonian $K$, $$\delta \int_{t1}^{t2} (p_i\dot q_i - K(Q, P ,t)) dt = 0. \tag{9.6}$$ This is ok.

Then, $$\lambda(p_i\dot q_i - H) = P_i\dot Q_i - K + \frac{dF}{dt} \tag{9.8}$$ but I am not getting how this last expression does come. Here I am not understating why $\lambda$ and ${dF/dt}$ does come? If you know any mathematical article, paper that would be really helpful.

SK Dash
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anbhadane
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  • Now My question is how we are extremising the second kamiltonion to give the last result? It might me opposite also. $\lambda$ going with RHS also. similar to derivative also might go with RHS. – anbhadane May 17 '20 at 15:37
  • I'm struggling to find Kamiltonian(?) in my version of Goldstein classical mechanics actually, could you point me to the publisher, version and page number? I'll take a read through and get back to you. In the meantime, this answer seemed relevant: https://physics.stackexchange.com/a/43227/253874 – Thormund May 17 '20 at 15:39
  • eq , (9-6) ,(9-7) ,(9-8), p.380 , chapter: Canonical Transformation , Goldstein , second edition , ADDISON-WESLEY PUBLISHING COMPANY. – anbhadane May 17 '20 at 15:44
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    No, My questions are, Are we extremising Kamiltonion action with Hamiltonion one? If we why we are doing so? – anbhadane May 17 '20 at 16:01
  • Extremising both Lagrangians $L = p\dot{q} - H = ...$, will indeed yield the same action $S$. The point of equation (9.8) was to therefore show the non-uniqueness of $H$ (given choice of phase space representation), since $K$ would have yielded the same action, which is as we would expect that physics behaves the same regardless of how we label it. – Thormund May 17 '20 at 16:08

2 Answers2

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As mentioned through the comments, $\frac{dF}{dt}$ is the result of how a Lagrangian $L$ that produces an action $S$ by $$dS = \delta \int L\, dt$$ is only unique up to a derivative, hence the existence of the $\frac{dF}{dt}$ term in eqn (9.8).

The term $\lambda$ has been stated by Goldstein as a scale factor. Improperly, we can imagine this as the jacobian of the transformation $(p,q) \rightarrow (P, Q)$. (Please refer further into Chapter 9.4.)

Also, we note that extremising the action $S$ through both lagrangians $L_K$ and $L_H$ independently must recover the same action, and it is unusual to say that we are extremising (the functional $S$ on) $H$ together with $K$. Nonetheless, the purpose of (9.8) was to therefore show how $H$ would change depending on coordinate representations such that $S$ would be the same.

Thormund
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Eq. (9.8) [which is called an extended canonical transformation (ECT) in my Phys.SE answer here, and which is supposed to be satisfied off-shell] is a sufficient condition for the variational principles (9.6) and (9.7) to be equivalent.

This is because the stationary solution to a variational principle is not changed if the action is modified by an overall non-zero multiplicative factor $\lambda$ or by boundary terms.

On the other hand, the Euler-Lagrange (EL) equations for the variational principles (9.6) and (9.7) are the Kamilton's and Hamilton's equations, respectively. In this way we see that the ECT (9.8) transforms the Hamilton's equations into the Kamilton's equations.

Qmechanic
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