Generating Function for Lagrangian Invariant System

[Physgeek64]

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

 

[Orodruin]

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 \}.

 

[Physgeek64]

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 :)

 

[Orodruin]

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 $\delta p$.

 

[Physgeek64]

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