\( (*\ vector\ field*) 
Clear[x, y]
\)\(v[x_, y_] := y; 
w[x_, y_] := \(-x\); 
\)

\(\( (*initial\ values\ *) 

x[0] = 1; \ y[0] = 0; 
 (*time\ step*) 

    e = 0.01; 
 (*number\ of\ steps*) 
nmax = 2000; \)\)

\(\( (*\ Euler\ integration\ *) 

    Do[x[n + 1] = x[n] + e*v[x[n], y[n]]; 
      y[n + 1] = 
		y[n] + e*w[x[n], y[n]], {n, 0, nmax}]\)\)

ListPlot[Table[{x[n], y[n]}, {n, 0, nmax}]]

\( (*\ a\ better\ method*) 
Clear[x, y]\)\(aa = 
      NDSolve[{\(x'\)[t] == y[t], \(y'\)[t] == \(-x[t]\), x[0] == 1, 
          y[0] == 0}, {x, y}, {t, 0, 170}]; 
x[1] /. aa\ \)

ParametricPlot[{(x[t] /. aa)[[1]], (y[t] /. aa)[[1]]}, {
        t, 0, 100}]