def _external_force(x,t,pot):
dim= len(x)
if dim == 3:
#x is rectangular so calculate R and phi
R= nu.sqrt(x[0]**2.+x[1]**2.)
phi= nu.arccos(x[0]/R)
sinphi= x[1]/R
cosphi= x[0]/R
if x[1] < 0.: phi= 2.*nu.pi-phi
#calculate forces
Rforce= evaluateRforces(R,x[2],pot,phi=phi,t=t)
phiforce= evaluatephiforces(R,x[2],pot,phi=phi,t=t)
return nu.array([cosphi*Rforce-1./R*sinphi*phiforce,
sinphi*Rforce+1./R*cosphi*phiforce,
evaluatezforces(R,x[2],pot,phi=phi,t=t)])
elif dim == 2:
#x is rectangular so calculate R and phi
R= nu.sqrt(x[0]**2.+x[1]**2.)
phi= nu.arccos(x[0]/R)
sinphi= x[1]/R
cosphi= x[0]/R
if x[1] < 0.: phi= 2.*nu.pi-phi
#calculate forces
Rforce= evaluateplanarRforces(R,pot,phi=phi,t=t)
phiforce= evaluateplanarphiforces(R,pot,phi=phi,t=t)
return nu.array([cosphi*Rforce-1./R*sinphi*phiforce,
sinphi*Rforce+1./R*cosphi*phiforce])
elif dim == 1:
return evaluatelinearForces(x,pot,t=t)
def _rectForce(x,pot,t=0.):
"""
NAME:
_rectForce
PURPOSE:
returns the force in the rectangular frame
INPUT:
x - current position
t - current time
pot - (list of) Potential instance(s)
OUTPUT:
force
HISTORY:
2011-02-02 - Written - Bovy (NYU)
"""
#x is rectangular so calculate R and phi
R= nu.sqrt(x[0]**2.+x[1]**2.)
phi= nu.arccos(x[0]/R)
sinphi= x[1]/R
cosphi= x[0]/R
if x[1] < 0.: phi= 2.*nu.pi-phi
#calculate forces
Rforce= evaluateplanarRforces(R,pot,phi=phi,t=t)
phiforce= evaluateplanarphiforces(R,pot,phi=phi,t=t)
return nu.array([cosphi*Rforce-1./R*sinphi*phiforce,
sinphi*Rforce+1./R*cosphi*phiforce])
def _rectForce(x, pot, t=0.):
"""
NAME:
_rectForce
PURPOSE:
returns the force in the rectangular frame
INPUT:
x - current position
t - current time
pot - (list of) Potential instance(s)
OUTPUT:
force
HISTORY:
2011-02-02 - Written - Bovy (NYU)
"""
#x is rectangular so calculate R and phi
R = nu.sqrt(x[0]**2. + x[1]**2.)
phi = nu.arccos(x[0] / R)
sinphi = x[1] / R
cosphi = x[0] / R
if x[1] < 0.: phi = 2. * nu.pi - phi
#calculate forces
Rforce = evaluateplanarRforces(R, pot, phi=phi, t=t)
phiforce = evaluateplanarphiforces(R, pot, phi=phi, t=t)
return nu.array([
cosphi * Rforce - 1. / R * sinphi * phiforce,
sinphi * Rforce + 1. / R * cosphi * phiforce
])
def _EOM_dxdv(x,t,pot):
"""
NAME:
_EOM_dxdv
PURPOSE:
implements the EOM, i.e., the right-hand side of the differential
equation, for integrating phase space differences, rectangular
INPUT:
x - current phase-space position
t - current time
pot - (list of) Potential instance(s)
OUTPUT:
dy/dt
HISTORY:
2011-10-18 - Written - Bovy (NYU)
"""
#x is rectangular so calculate R and phi
R= nu.sqrt(x[0]**2.+x[1]**2.)
phi= nu.arccos(x[0]/R)
sinphi= x[1]/R
cosphi= x[0]/R
if x[1] < 0.: phi= 2.*nu.pi-phi
#calculate forces
Rforce= evaluateplanarRforces(R,pot,phi=phi,t=t)
phiforce= evaluateplanarphiforces(R,pot,phi=phi,t=t)
R2deriv= evaluateplanarPotentials(R,pot,phi=phi,t=t,dR=2)
phi2deriv= evaluateplanarPotentials(R,pot,phi=phi,t=t,dphi=2)
Rphideriv= evaluateplanarPotentials(R,pot,phi=phi,t=t,dR=1,dphi=1)
#Calculate derivatives and derivatives+time derivatives
dFxdx= -cosphi**2.*R2deriv\
+2.*cosphi*sinphi/R**2.*phiforce\
+sinphi**2./R*Rforce\
+2.*sinphi*cosphi/R*Rphideriv\
-sinphi**2./R**2.*phi2deriv
dFxdy= -sinphi*cosphi*R2deriv\
+(sinphi**2.-cosphi**2.)/R**2.*phiforce\
-cosphi*sinphi/R*Rforce\
-(cosphi**2.-sinphi**2.)/R*Rphideriv\
+cosphi*sinphi/R**2.*phi2deriv
dFydx= -cosphi*sinphi*R2deriv\
+(sinphi**2.-cosphi**2.)/R**2.*phiforce\
+(sinphi**2.-cosphi**2.)/R*Rphideriv\
-sinphi*cosphi/R*Rforce\
+sinphi*cosphi/R**2.*phi2deriv
dFydy= -sinphi**2.*R2deriv\
-2.*sinphi*cosphi/R**2.*phiforce\
-2.*sinphi*cosphi/R*Rphideriv\
+cosphi**2./R*Rforce\
-cosphi**2./R**2.*phi2deriv
return nu.array([x[2],x[3],
cosphi*Rforce-1./R*sinphi*phiforce,
sinphi*Rforce+1./R*cosphi*phiforce,
x[6],x[7],
dFxdx*x[4]+dFxdy*x[5],
dFydx*x[4]+dFydy*x[5]])
def _EOM_dxdv(x, t, pot):
"""
NAME:
_EOM_dxdv
PURPOSE:
implements the EOM, i.e., the right-hand side of the differential
equation, for integrating phase space differences, rectangular
INPUT:
x - current phase-space position
t - current time
pot - (list of) Potential instance(s)
OUTPUT:
dy/dt
HISTORY:
2011-10-18 - Written - Bovy (NYU)
"""
#x is rectangular so calculate R and phi
R = nu.sqrt(x[0]**2. + x[1]**2.)
phi = nu.arccos(x[0] / R)
sinphi = x[1] / R
cosphi = x[0] / R
if x[1] < 0.: phi = 2. * nu.pi - phi
#calculate forces
Rforce = evaluateplanarRforces(R, pot, phi=phi, t=t)
phiforce = evaluateplanarphiforces(R, pot, phi=phi, t=t)
R2deriv = evaluateplanarPotentials(R, pot, phi=phi, t=t, dR=2)
phi2deriv = evaluateplanarPotentials(R, pot, phi=phi, t=t, dphi=2)
Rphideriv = evaluateplanarPotentials(R, pot, phi=phi, t=t, dR=1, dphi=1)
#Calculate derivatives and derivatives+time derivatives
dFxdx= -cosphi**2.*R2deriv\
+2.*cosphi*sinphi/R**2.*phiforce\
+sinphi**2./R*Rforce\
+2.*sinphi*cosphi/R*Rphideriv\
-sinphi**2./R**2.*phi2deriv
dFxdy= -sinphi*cosphi*R2deriv\
+(sinphi**2.-cosphi**2.)/R**2.*phiforce\
-cosphi*sinphi/R*Rforce\
-(cosphi**2.-sinphi**2.)/R*Rphideriv\
+cosphi*sinphi/R**2.*phi2deriv
dFydx= -cosphi*sinphi*R2deriv\
+(sinphi**2.-cosphi**2.)/R**2.*phiforce\
+(sinphi**2.-cosphi**2.)/R*Rphideriv\
-sinphi*cosphi/R*Rforce\
+sinphi*cosphi/R**2.*phi2deriv
dFydy= -sinphi**2.*R2deriv\
-2.*sinphi*cosphi/R**2.*phiforce\
-2.*sinphi*cosphi/R*Rphideriv\
+cosphi**2./R*Rforce\
-cosphi**2./R**2.*phi2deriv
return nu.array([
x[2], x[3], cosphi * Rforce - 1. / R * sinphi * phiforce,
sinphi * Rforce + 1. / R * cosphi * phiforce, x[6], x[7],
dFxdx * x[4] + dFxdy * x[5], dFydx * x[4] + dFydy * x[5]
])
def _EOM(y,t,pot):
"""
NAME:
_EOM
PURPOSE:
implements the EOM, i.e., the right-hand side of the differential
equation
INPUT:
y - current phase-space position
t - current time
pot - (list of) Potential instance(s)
l2 - angular momentum squared
OUTPUT:
dy/dt
HISTORY:
2010-07-20 - Written - Bovy (NYU)
"""
l2= (y[0]**2.*y[3])**2.
return [y[1],
l2/y[0]**3.+evaluateplanarRforces(y[0],pot,phi=y[2],t=t),
y[3],
1./y[0]**2.*(evaluateplanarphiforces(y[0],pot,phi=y[2],t=t)-
2.*y[0]*y[1]*y[3])]
def _EOM(y, t, pot):
"""
NAME:
_EOM
PURPOSE:
implements the EOM, i.e., the right-hand side of the differential
equation
INPUT:
y - current phase-space position
t - current time
pot - (list of) Potential instance(s)
l2 - angular momentum squared
OUTPUT:
dy/dt
HISTORY:
2010-07-20 - Written - Bovy (NYU)
"""
l2 = (y[0]**2. * y[3])**2.
return [
y[1], l2 / y[0]**3. + evaluateplanarRforces(y[0], pot, phi=y[2], t=t),
y[3],
1. / y[0]**2. * (evaluateplanarphiforces(y[0], pot, phi=y[2], t=t) -
2. * y[0] * y[1] * y[3])
]