- #1
Physgeek64
- 247
- 11
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
Last edited: