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import matplotlib.pyplot as plt

import numpy as np

class billiard_circle():

def init(self,x_0,y_0,vx_0,vy_0,N,dt,alpha):

self.x_0 = x_0

self.y_0 = y_0

self.vx_0 = vx_0

self.vy_0 = vy_0

self.N = N

self.dt = dt

self.alpha=alpha

def motion_calculate(self):

self.x = []

self.y = []

self.vx = []

self.vy = []

self.t = [0]

self.x.append(self.x_0)

self.y.append(self.y_0)

self.vx.append(self.vx_0)

self.vy.append(self.vy_0)

for i in range(1,self.N):

self.x.append(self.x[i - 1] + self.vx[i - 1]*self.dt)

self.y.append(self.y[i - 1] + self.vy[i - 1]*self.dt)

self.vx.append(self.vx[i - 1])

self.vy.append(self.vy[i - 1])

if (np.sqrt( self.x[i]2+(self.y[i]-self.alpha)2 ) > 1.0) and self.y[i]>self.alpha:

self.x[i],self.y[i] = self.correct('np.sqrt(x2+(y-self.alpha)2) < 1.0',self.x[i - 1], self.y[i - 1], self.vx[i - 1], self.vy[i - 1])

self.vx[i],self.vy[i] = self.reflect1(self.x[i],self.y[i],self.vx[i - 1], self.vy[i - 1])

elif (np.sqrt( self.x[i]2+(self.y[i]+self.alpha)2 ) > 1.0) and self.y[i]<-self.alpha:

self.x[i],self.y[i] = self.correct('np.sqrt(x2+(y+self.alpha)2) < 1.0',self.x[i - 1], self.y[i - 1], self.vx[i - 1], self.vy[i - 1])

self.vx[i],self.vy[i] = self.reflect2(self.x[i],self.y[i],self.vx[i - 1], self.vy[i - 1])

elif (self.x[i] < -1.0) and self.y[i]>-self.alpha and self.y[i]

self.x[i],self.y[i] = self.correct('x>-1.0',self.x[i - 1], self.y[i - 1], self.vx[i - 1], self.vy[i - 1])

self.vx[i] = - self.vx[i]

elif (self.x[i] > 1.0) and self.y[i]>-self.alpha and self.y[i]

self.x[i],self.y[i] = self.correct('x<1.0',self.x[i - 1], self.y[i - 1], self.vx[i - 1], self.vy[i - 1])

self.vx[i] = - self.vx[i]

self.t.append(self.t[i - 1] + self.dt)

return self.x, self.y

def correct(self,condition,x,y,vx,vy):

vx_c = vx/100.0

vy_c = vy/100.0

while eval(condition):

x = x + vx_c*self.dt

y = y + vy_c*self.dt

return x-vx_cself.dt,y-vy_cself.dt

def reflect1(self,x,y,vx,vy):

module = np.sqrt(x2+(y-self.alpha)2) ### normalization

x = x/module

y = (y-self.alpha)/module+self.alpha

v = np.sqrt(vx2+vy2)

cos1 = (vxx+vy(y-self.alpha))/v

cos2 = (vx(y-self.alpha)-vyx)/v

vt = -v*cos1

vc = v*cos2

vx_n = vtx+vc(y-self.alpha)

vy_n = vt(y-self.alpha)-vcx

return vx_n,vy_n

def reflect2(self,x,y,vx,vy):

module = np.sqrt(x2+(y+self.alpha)2) ### normalization

x = x/module

y = (y+self.alpha)/module-self.alpha

v = np.sqrt(vx2+vy2)

cos1 = (vxx+vy(y+self.alpha))/v

cos2 = (vx(y+self.alpha)-vyx)/v

vt = -v*cos1

vc = v*cos2

vx_n = vtx+vc(y+self.alpha)

vy_n = vt(y+self.alpha)-vcx

return vx_n,vy_n

def plot(self):

plt.figure(figsize = (8,8))

plt.xlim(-1,1)

plt.ylim(-1,1)

plt.xlabel('x')

plt.ylabel('y')

plt.title('Stadium billiard $\alpha$=0.01')

self.plot_boundary()

plt.plot(self.x,self.y,'y')

plt.savefig('chapter3_3.31.png',dpi = 144)

plt.show()

def plot_boundary(self):

theta = 0

x = []

y = []

while theta < np.pi:

x.append(np.cos(theta))

y.append(np.sin(theta)+0.01)

theta+= 0.01

plt.plot(x,y,'g.')

while theta > np.pi and theta< 2*np.pi:

x.append(np.cos(theta))

y.append(np.sin(theta)-0.01)

theta+= 0.01

plt.plot(x,y,'g.')

def phase_plot(self):

record_x = []

record_vx = []

for i in range(len(self.x)):

if (abs(self.y[i] - 0)<0.001):

record_vx.append(self.vx[i])

record_x.append(self.x[i])

return record_vx, record_x

plt.xlabel('x')

plt.ylabel(r'$v_x$')

plt.scatter(record_x,record_vx,s=1)

plt.savefig('chapter3_3.31_phasey=0.png', dpi= 144)

plt.show()

sub1=plt.subplot(221)

A=billiard_circle(0.2,0,1,0.6,500000,0.01,0)

A.motion_calculate()

vx,x=A.phase_plot()

sub1.scatter(x, vx,s=1)

plt.xlabel('x')

plt.ylabel('$v_x$')

sub1.set_title('$\alpha=0$')

plt.show()

sub2=plt.subplot(222)

A=billiard_circle(0.2,0,1,0.6,500000,0.01,0.001)

A.motion_calculate()

vx,x=A.phase_plot()

sub2.scatter(x, vx,s=1)

plt.xlabel('x')

plt.ylabel('$v_x$')

sub2.set_title('$\alpha=0.001$')

plt.show()

sub3=plt.subplot(223)

A=billiard_circle(0.2,0,1,0.6,500000,0.01,0.01)

A.motion_calculate()

vx,x=A.phase_plot()

sub3.scatter(x, vx,s=1)

plt.xlabel('x')

plt.ylabel('$v_x$')

sub3.set_title('$\alpha=0.01$')

plt.show()

sub4=plt.subplot(224)

A=billiard_circle(0.2,0,1,0.6,500000,0.01,0.1)

A.motion_calculate()

vx,x=A.phase_plot()

sub4.scatter(x, vx,s=1)

plt.xlabel('x')

plt.ylabel('$v_x$')

sub4.set_title('$\alpha=0.1$')

plt.show()