[QUE/CM-03007]

The Lagrangian for a system is given by $$ L = {1\over 2}\, m\, (\omega x+\dot{x})^2 + m \omega^2 x \dot{x} t $$ Using equations of motion explicitly verify that $T= {1\over 2} m \dot{x}^2 $ is a constant of motion. Is the  transformation $$ t\rightarrow t^\prime +\epsilon, \qquad x(t) \rightarrow x^\prime (t^\prime)= x(t)$$ a symmetry transformation? If yes find the conserved quantity and compare it with the Hamiltonian and the kinetic energy.