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In Wolfgang Nolting's book Theoretical Physics 2 - Analytical Mechanics, the following theorem is stated:

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While the formal definition of canonical trasformation is given only several pages later in the book, here we are just assuming that both $(q,p)$ and $(Q,P)$ "refer to" an Hamiltonian function.

Now, the author argues that the following (correct) calculation is a proof of both $\{ Q_{i}, Q_{j}\}_{q,p} = 0$ and $\{ Q_{i}, P_{j}\}_{q,p} = \delta_{1j}$:

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This is obtained, the author argues, by comparison. In other words, being $\dot{Q}_{i}$ the initial term, all the final bracket terms must be null except $\{ Q_{i}, P_{i}\}$.

However, I am not convinced about the validity of the conclusion. The comparison between the first and last term does not involve, say, linearly independent vectors, whereby one can simply conclude that all the coefficients of the linear combination are 0. How can this "proof" be fixed?

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The assumption requires (but does not say so explicitly) that Hamilton's equations continue to hold for all $H(q,p)$ and their associated $\tilde H(P,Q)$'s, so the $\dot Q_i$ and $\dot P_i$ can be arbitrary.

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  1. The proof of Theorem 2.4.1 becomes more transparent in the symplectic notation [1] $$z^I~=~(q^i,p_i) \qquad\text{and}\qquad Z^I~=~(Q^i,P_i),\tag{1}$$ where the old and new Hamilton's equations read $$\dot{z}^I~=~ J^{IJ}\frac{\partial H}{\partial z^J} \qquad\text{and}\qquad \dot{Z}^I~=~ J^{IJ}\frac{\partial H}{\partial Z^J},\tag{2O+2N} $$ and where the fundamental Poisson brackets $$\{z^I,z^J\}_z~=~J^{IJ}~=~\{Z^I,Z^J\}_Z\tag{3}$$ are given by the symplectic unit matrix $J^{IJ}$.

  2. By the chain rule (CR) the old Hamilton's equations (2O) become $$\dot{Z}^I~\stackrel{(CR)+(2O)+(5)}{=}~ (MJM^T)^{IJ}\frac{\partial H}{\partial Z^J},\tag{4}$$ where $$ M^I{}_J~:=~\frac{\partial Z^I}{\partial z^J}\tag{5}$$ is a Jacobian matrix.

  3. Assuming that the canonical transformation (CT) works for all Hamiltonians $H$, we can make the gradient $\frac{\partial H}{\partial Z^J}$ (or equivalently the differential $\mathrm{d}H=\frac{\partial H}{\partial Z^J}\mathrm{d}Z^J$) span the full $2n$-dimensional cotangent space (of phase space) by varying the Hamiltonian $H$ appropriately.

  4. With point 3 in mind, and comparing eq. (4) with the new Hamilton's equations (2N), we conclude that the CT is a symplectomorphism $$ \{Z^I,Z^J\}_z~\stackrel{(CR)+(5)}{=}~(MJM^T)^{IJ}~\stackrel{(2N)+(4)}{=}~J^{IJ}, \tag{6}$$ which yields the conclusion of Theorem 2.4.1. $\Box$

References:

  1. H. Goldstein, Classical Mechanics, 2nd eds.; Section 9.3.

  2. H. Goldstein, Classical Mechanics, 3rd eds.; Section 9.4.

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  • $\begingroup$ Provided 2O and 2N are valid for every choice of $H$... $\endgroup$ Commented yesterday
  • $\begingroup$ Thanks @Valter Moretti. I have made that point more explicit in an update. $\endgroup$ Commented yesterday

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