# real plot
plt.figure(2)
plt.clf()
plt.title('Real Space Trajectory')
plt.plot(q[:, 0], q[:, 1])
plt.xlabel('Position, $x$')
plt.ylabel('Position, $y$')
plt.show()
# calculation with the Stoemer-Verlet method
q = np.zeros((N + 1, 2))
p = np.zeros((N + 1, 2))
q[0, 0] = 1.0
p[0, 1] = 1.0
for ii in range(0, N):
q[ii + 1], p[ii + 1] = step.stoemerstep(V, q[ii], p[ii], Dt)
# phase plot
plt.figure(3)
plt.clf()
plt.subplots_adjust(wspace = 0.3)
plt.subplot(121)
plt.title('Phase Plot in $x$')
plt.plot(q[:, 0], p[:, 0])
plt.xlabel('Position, $x$')
plt.ylabel('Momentum, $p_x$')
plt.subplot(122)
plt.title('Phase Plot in $y$')
plt.plot(q[:, 1], p[:, 1])
plt.xlabel('Position, $y$')
plt.ylabel('Momentum, $p_y$')
b = np.zeros(N + 1, dtype = np.float64)
e2norm = np.zeros(N + 1, dtype = np.float64)
energyError = np.zeros(N + 1, dtype = np.float64)
phi = np.zeros(N + 1, dtype = np.float64)
dphidq_analytic = np.zeros(N + 1, dtype = np.float64)
dphidq_numeric = np.zeros(N + 1, dtype = np.float64)
# initial conditions
q[0] = 1.9
v[0] = -0.0001
energyConst = 0.5*v[0] + LJ(q[0])
# calcualte phase space
for ii in range(0, N):
t[ii + 1] = t[ii] + Dt
q[ii + 1], v[ii + 1] = step.stoemerstep(LJ, q[ii], v[ii], Dt, 35.0)
dphidq = -deriv.derivative(LJ, q[ii], np.abs(q[ii + 1] - q[ii]), False)[0]
d2phidq2 = -deriv.derivative(LJdiff, q[ii], np.abs(q[ii + 1] - q[ii]), False)[0]
a[ii] = np.max([1, d2phidq2])
E = 0.5*v[ii]**2 + LJ(q[ii])
b[ii] = 0.5*np.sqrt(dphidq**2 + 2*d2phidq2**2*(E - LJ(q[ii])))
energyError[ii] = E - energyConst
# plot
plt.figure(1)
plt.clf()
ax1 = plt.subplot2grid((3, 2), (0, 0), rowspan = 3)
ax1.plot(q, v)
plt.title('Phase Plot')
plt.xlabel('Generalised Coordainte, $q$')
plt.ylabel('Generalised Velocity, $v(q)$')
