I have the Schrodinger Lagrangian density for a complex scalar field, given by $L=i\phi^{*}\frac{\partial \phi}{\partial t}-\frac{1}{2m}(\frac{\partial \phi^*}{\partial x}\frac{\partial \phi}{\partial x})$ which undergoes a Galilean boost in the x-axis given by $x'=x-vt$ and $t'=t$. The $\phi$ has dependence on both $x$ and $t$.
The solution to the Lagrangian is given by a plain wave as usual. Under this transformation, I know that the Lagrangian is invariant but I am unsure about how I would go about proving this.