How do you derive the FOC's using the lagrangian approach for this equation
Max E(t) Sum of [q(t) - q^2/st] dt
I have no idea what you mean. You say "Optimization" so I think you mean using Lagrange multipliers. But that is to maximize a function under given restraints. You say "max E(t)" but what is E(t)? Did you drop an equals sign? Is E(t)= Sum of [q(t) - q^2/st] dt ? Sum over what index? And what is that "dt" doing there. Did you mean integral rather than sum? If so, you don't want to integrate with respect to t. That would make E a function of s, not t.
But that right hand side looks like an energy formula. Are you finding the Lagrangian of a dynamics problem?