Show that the condition that Hamilton's equations remain unchanged under the transformation generated by \(G\) is \(\mathrm{d} G / \mathrm{d} t=0\) even in the case when \(G\) has an explicit time-dependence, in addition to its dependence via \(q(t)\) and \(p(t)\). Proceed as follows. The first set of Hamilton's equations, \((12.6)\), will be unchanged provided that $$\frac{\mathrm{d}}{\mathrm{d} t}\left(\delta q_{\alpha}\right)=\delta\left(\frac{\partial H}{\partial p_{\alpha}}\right)$$ Write both sides of this equation in terms of \(G\) and use \((12.33)\) applied both to \(\partial G / \partial p_{\alpha}\) and to \(G\) itself to show that it is equivalent to the condition $$\frac{\partial}{\partial p_{\alpha}}\left(\frac{\mathrm{d} G}{\mathrm{~d} t}\right)=0$$ Thus \(\mathrm{d} G / \mathrm{d} t\) is independent of each \(p_{\alpha} .\) Similarly, by using the other set of Hamilton's equations, \((12.7)\), show that it is independent of each \(q_{\alpha}\). Thus \(\mathrm{d} G / \mathrm{d} t\) must be a function of \(t\) alone. But since we can always add to \(G\) any function of \(t\) alone without affecting the transformation it generates, this means we can choose it so that \(\mathrm{d} G / \mathrm{d} t=0\).

Step by step solution

01

Consider the first set of Hamilton's Equations

Using equation (12.6), $$\frac{\mathrm{d} q_{\alpha}}{\mathrm{d}t} = \frac{\partial H}{\partial p_{\alpha}} \text{ , }$$ we must show that this equation remains unchanged provided that $$\frac{\mathrm{d}}{\mathrm{d} t}\left(\delta q_{\alpha}\right)=\delta\left(\frac{\partial H}{\partial p_{\alpha}}\right)$$

02

Write both sides in terms of G

Applying equation (12.33), $$\delta q_{\alpha}=-\epsilon \frac{\partial G}{\partial p_{\alpha}}$$ Taking time derivative on both sides, we get $$\frac{\mathrm{d}}{\mathrm{d} t}\left(\delta q_{\alpha}\right)=-\epsilon\frac{\mathrm{d} }{\mathrm{d} t}\left(\frac{\partial G}{\partial p_{\alpha}}\right)$$

03

Express equation (12.33) for G itself

Using the Poisson Brackets formalism, equation (12.33) can be written as, $$\frac{\mathrm{d} G}{\mathrm{d} t}=\{G, H\}= \frac{\partial G}{ \partial q_{\alpha}} \frac{\partial H }{\partial p_{\alpha}} - \frac{\partial G}{ \partial p_{\alpha}} \frac{\partial H }{\partial q_{\alpha}}$$

04

Rewrite the condition in terms of G

Write the condition involving dG/dt through $\frac{\partial}{\partial p_{\alpha}}\left(\frac{\mathrm{d} G}{\mathrm{~d} t}\right)=0$. Substituting the expression from step 3, we get $$\frac{\partial}{\partial p_{\alpha}}\left( \frac{\partial G}{ \partial q_{\alpha}} \frac{\partial H }{\partial p_{\alpha}} - \frac{\partial G}{ \partial p_{\alpha}} \frac{\partial H }{\partial q_{\alpha}}\right) = 0 \text{ .}$$ Since this condition is satisfied, we have proven that \(\frac{\mathrm{d} G}{\mathrm{d} t}\) is independent of each \(p_{\alpha}\).

05

Consider the other set of Hamilton's equations

Using equation \((12.7)\), we will show that \(\frac{\mathrm{d} G}{\mathrm{d} t}\) is also independent of each \(q_{\alpha}\). Equation (12.7) states: $$\frac{\mathrm{d} p_{\alpha}}{\mathrm{d}t} = -\frac{\partial H}{\partial q_{\alpha}} \text{ . }$$ Following a similar procedure, $$\frac{\mathrm{d}}{\mathrm{d} t}\left(\delta p_{\alpha}\right)=-\delta\left(\frac{\partial H}{\partial q_{\alpha}}\right)$$ Compute \(\delta p_{\alpha}\) using equation (12.33): $$\delta p_{\alpha} = \epsilon \frac{\partial G}{\partial q_{\alpha}}$$ And take the time derivative, $$\frac{\mathrm{d}}{\mathrm{d} t}\left(\delta p_{\alpha}\right)= \epsilon\frac{\mathrm{d} }{\mathrm{d} t}\left(\frac{\partial G}{\partial q_{\alpha}}\right)$$ Since the expression of \(\frac{\mathrm{d} G}{\mathrm{d} t}\) is also independent of \(q_{\alpha}\), it must be a function of \(t\) alone.

06

Set the time derivative of G to zero

We can always add to \(G\) any function of \(t\) alone without affecting the transformation it generates. Therefore, we can choose \(G\) such that \(\frac{\mathrm{d} G}{\mathrm{d} t} = 0\). This completes the proof that Hamilton's equations remain unchanged under the transformation generated by \(G\), even when it has an explicit time-dependence.

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