omeg0 = 1.0; beta =  .09; A = 1. ; m = 1;
omeg1 = Sqrt[omeg0^2 - beta^2];
tR = 2  π/omeg0;
tR = Log[10]/beta;
energ0 = m*A^2*omeg0^2/2;
x = A*Exp[-beta*t]*Cos[omeg1*t];
v = -A*
          Exp[-beta*t]*(beta*Cos[omeg1*t] + 
              omeg1*Sin[omeg1*t]);
energy = (m*A^2*omeg0^2/2)*
          Exp[-2*beta*
              t]*(1 + (beta*omeg1/omeg0^2)*
                Sin[2*omeg1*t] + (beta/omeg0)^2*
                Cos[2*omeg1*t]);
energyy = 
        m*(v^2 + (x*omeg0)^2)/2;
DEDT = -2*beta*energ0*
          Exp[-2*beta*
              t]*(1 - (omeg1^2 - beta^2) Cos[2*omeg1*t]/omeg0^2 + 
              2*(beta*omeg1/omeg0^2)*
                Sin[2*omeg1*t]);
DEDTT = -2*beta*v^2;
AVGDEDT = -beta*m*A^2*omeg0^2*
          Exp[-2*beta*t];
Plot[{energy, DEDT, energyy, DEDTT}, {t, 0, tR}, 
        PlotPoints -> 100, PlotLabel -> "Problem 3-11", 
        AspectRatio ->  .45, AxesLabel -> {"t", "E"}, 
        PlotRange -> All];