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Canonical transformation in classical mechanics

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dRic2

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101
Homework Statement
Give the proof of the equation
$$ H' = H + \frac { \partial S } { \partial t }$$
By proceeding as follow. Leave the time t as independent variable and consider a canonical transformation in which the time appears as a parameter. Then obtain the above relation by distinguishing between ##dS## in the canonical integral and ##\delta S## in the definition of canonical transformation. In the first case time is varied, in the second case not.
Homework Equations
Canonical integral:
$$A = \int ( \sum p_i dq_i - Hdt)$$
Definition of canonical transformation:
$$ \sum p_i \delta q_i = \sum P_i \delta Q_i + \delta S$$
I'm stuck from the beginning. I though I understood the difference between ## \delta## and ##d##, but apparently I was wrong, because I don't know how to exploit it here...

Any hint would be greatly appreciated

Thank
Ric
 
Are you sure you have written correct defination of Canonical Transformation?

##p_i\delta q_i -H= P_i\delta Q_i-H' + dS/dt##

In this case ,the equation is easy to derive by using chain rule to S(q,Q,t) and using linear independence of ##\dot q##and ##p##
 

dRic2

Gold Member
560
101
Are you sure you have written correct defination of Canonical
This is the definition found in my book and it makes sense. It seems different from yours though
 
This is the definition found in my book and it makes sense. It seems different from yours though
Ok. Then use the following:

$$\delta S= dS/dt ~dt + dS/dq ~dq + dS/dQ ~dQ$$

Now subtract ##Hdt## on both sides in your defination of Canonical Transformation

Now use the linear independence of q,Q and t
 

dRic2

Gold Member
560
101
Your notation is confusion me. ##\delta S## is not ##dS##. ##\delta## is used to address variations and, since the problem asks to consider time as a parameter, you don't have to vary wrt to time. ##dS## is obviously the total differential.

Even if you meant ##dS## I don't know how to proceed.
 

dRic2

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560
101
Solved it. I was very confused by the notation of the book.
 

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