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%% Cell type:markdown id:consistent-cornell tags:
<h1 style="text-align: center; vertical-align: middle;">Numerical Methods in Accelerator Physics</h1>
<h2 style="text-align: center; vertical-align: middle;">Python examples -- Week 1</h2>
%% Cell type:markdown id:terminal-diabetes tags:
<h2>Run this first!</h2>
Imports and modules:
%% Cell type:code id:helpful-slovakia tags:
``` python
from config import *
%matplotlib inline
```
%% Cell type:markdown id:atlantic-finance tags:
<h2>Phase space plots based on Lagragian (slide 21)</h2>
%% Cell type:code id:excited-technical tags:
``` python
m = 1 # point mass
g = 1 # magnitude of the gravitational field
l = 1 # length of the rod
L = 1 # length of the rod
```
%% Cell type:code id:expected-blood tags:
``` python
def plot_arrow(theta, p, dt=1):
# equations of motion:
dtheta = p / (m * L*L) * dt
dp = - m * g * L * np.sin(theta) * dt
# plotting
plt.scatter([theta], [p], s=50, c='r', marker='*', zorder=10)
plt.annotate('',
xytext=(theta, p),
xy=(theta + dtheta, p + dp),
arrowprops=dict(facecolor='black', shrink=0.03))
```
%% Cell type:code id:together-birthday tags:
``` python
plot_arrow(1, 0)
plot_arrow(0, -1)
plot_arrow(-1, 0)
plot_arrow(0, 1)
#plot_arrow(1/1.41, 1/1.41)
# increase momentum:
#for i in np.arange(0.4, 2.5, 0.2):
# plot_arrow(0, i)
# increase angle:
#for i in np.arange(1, np.pi * 1.1, 0.2):
# plot_arrow(i, 0)
set_axes()
```
%% Cell type:markdown id:sunset-dependence tags:
<h2>Hamiltonian of the pendulum (slide 25)</h2>
%% Cell type:code id:practical-keeping tags:
``` python
def hamiltonian(theta, p):
T = p * p / (2 * m * L*L)
U = m * g * L * (1 - np.cos(theta))
return T + U
```
%% Cell type:code id:accessory-perfume tags:
``` python
TH, PP = np.meshgrid(np.linspace(-np.pi * 1.1, np.pi * 1.1, 100),
np.linspace(-3, 3, 100))
HH = hamiltonian(TH, PP)
plt.contour(TH, PP, HH, levels=10)
plt.colorbar(label=r'$\mathcal{H}(\theta, p)$')
set_axes()
```
%% Cell type:code id:random-honor tags:
``` python
TH, PP = np.meshgrid(np.linspace(-np.pi * 1.1, np.pi * 1.1, 100),
np.linspace(-3, 3, 100))
HH = hamiltonian(TH, PP)
plt.contourf(TH, PP, HH, cmap=plt.get_cmap('hot_r'))
plt.colorbar(label=r'$\mathcal{H}(\theta, p)$')
plot_arrow(2, 0)
set_axes()
```
%% Cell type:markdown id:raised-herald tags:
<h2>Explicit Euler Method (slide 28)</h2>
%% Cell type:code id:brilliant-scale tags:
``` python
def solve_euler(theta, p, dt=0.1):
theta_next = theta + dt * p / (m * L*L)
p_next = p - dt * m * g * L * np.sin(theta)
return (theta_next, p_next)
```
%% Cell type:markdown id:cordless-submission tags:
<h3>First example</h3>
%% Cell type:code id:generous-carroll tags:
``` python
theta_ini = -1.1
p_ini = 0
n_steps = 100
```
%% Cell type:code id:infrared-emergency tags:
``` python
results_euler = np.zeros((n_steps, 2), dtype=np.float32)
results_euler[0] = (theta_ini, p_ini)
for k in range(1, n_steps):
results_euler[k] = solve_euler(*results_euler[k - 1])
```
%% Cell type:code id:prescribed-digit tags:
``` python
plt.contour(TH, PP, HH, levels=10, linestyles='--', linewidths=1)
plt.plot(results_euler[:, 0], results_euler[:, 1], c='b')
set_axes()
```
%% Cell type:code id:ethical-property tags:
``` python
plt.plot(
hamiltonian(results_euler[:, 0], results_euler[:, 1]),
c='b', label='Euler integrator')
plt.axhline(hamiltonian(theta_ini, p_ini), c='r', label='theory')
plt.xlabel('Steps $k$')
plt.ylabel(r'$\mathcal{H}(\theta, p)$')
plt.legend();
```
%% Cell type:markdown id:spanish-option tags:
<h3>Second example: even worse!</h3>
%% Cell type:code id:martial-ebony tags:
``` python
theta_ini2 = -3.
p_ini2 = 0
n_steps = 100
```
%% Cell type:code id:elementary-float tags:
``` python
results_euler2 = np.zeros((n_steps, 2), dtype=np.float32)
results_euler2[0] = (theta_ini2, p_ini2)
for k in range(1, n_steps):
results_euler2[k] = solve_euler(*results_euler2[k - 1])
```
%% Cell type:code id:educational-group tags:
``` python
plt.contour(TH, PP, HH, levels=10, linestyles='--', linewidths=1)
plt.plot(results_euler2[:, 0], results_euler2[:, 1], c='b')
set_axes()
```
%% Cell type:code id:structured-discount tags:
``` python
h_sep = hamiltonian(-np.pi, 0) # Value of Hamiltonian at separatrix
h_sep
```
%% Cell type:code id:decreased-deficit tags:
``` python
plt.plot(
hamiltonian(results_euler2[:, 0], results_euler2[:, 1]),
c='b', label='Euler integrator')
plt.axhline(hamiltonian(theta_ini2, p_ini2), c='r', label='theory')
plt.axhline(h_sep, c='k', label='$\mathcal{H}_{sep}$')
plt.xlabel('Steps $k$')
plt.ylabel(r'$\mathcal{H}(\theta, p)$')
plt.legend(bbox_to_anchor=(1.05, 1))
plt.ylim(top=2.1);
```
%% Cell type:code id:identified-opportunity tags:
``` python
```
......
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