Generating Function for Lagrangian Invariant System

Physgeek64
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Homework Statement


Given a system with a Lagrangian ##L(q,\dot{q})## and Hamiltonian ##H=H(q,p)## and that the Lagrangian is invariant under the transformation ##q \rightarrow q+ K(q) ## find the generating function, G.

Homework Equations

The Attempt at a Solution


##\delta q = \{ q,G \} = \frac{\partial G}{\partial p} ##
##\delta p = \{ p, G \} = -\frac{\partial G}{\partial q} ##

in this case ##\delta q =K(q)##
Hence
##G= pK + c_1(q) ##
Assuming the Lagrangian is of the form ##L=\frac{1}{2} m q^2 -V(q)##
then ##p=m\dot{q}##
and ##\delta p = m\dot{K} = mK' \dot{q}= pK'##

##\delta p = pK'=-\frac{\partial G}{\partial q} ##
## G= -pK +c_2(p)##

I know this is not right because i have two different expressions that can't be matched, but i can't think of another way to do this.

Many thanks
 
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Physgeek64 said:
Assuming the Lagrangian is of the form
What makes you think you can assume this?

Edit: Also note that { and } are LaTeX delimiters used for grouping. To actually get the brackets of the Poisson bracket you need to use \{ and \}.
 
Orodruin said:
What makes you think you can assume this?

Edit: Also note that { and } are LaTeX delimiters used for grouping. To actually get the brackets of the Poisson bracket you need to use \{ and \}.
Ahh okay then, so I guess i can't make that assumption. In which case i don't know how to proceed, any tips?

Thank you, i will change them :)
 
Consider how the canonical momentum ##P = \partial L/\partial \dot Q## connected to ##Q = q + K(q)## must change relative to ##p##. This should give you ##\delta p##.
 
Orodruin said:
Consider how the canonical momentum ##P = \partial L/\partial \dot Q## connected to ##Q = q + K(q)## must change relative to ##p##. This should give you ##\delta p##.

## \frac{\partial L}{\partial \dot{Q}} = \frac{\partial L}{\partial \dot{q}} \frac{\partial \dot{q}}{\partial \dot{Q}} =\frac{\partial L}{\partial \dot{q}} \frac{1}{1+\dot{K}'} = p \frac{1}{1+\dot{K}'} ##

?? is this along the right lines?

Many thanks
 
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