def Er(self,R,z,vR,vz,E,Lz,sinh2u0,u0):
"""
NAME:
Er
PURPOSE:
calculate the 'radial energy'
INPUT:
R, z, vR, vz - coordinates
E - energy
Lz - angular momentum
sinh2u0, u0 - sinh^2 and u0
OUTPUT:
Er
HISTORY:
2012-11-29 - Written - Bovy (IAS)
"""
u,v= bovy_coords.Rz_to_uv(R,z,self._delta)
pu= (vR*numpy.cosh(u)*numpy.sin(v)
+vz*numpy.sinh(u)*numpy.cos(v)) #no delta, bc we will divide it out
out= (pu**2./2.+Lz**2./2./self._delta**2.*(1./numpy.sinh(u)**2.-1./sinh2u0)
-E*(numpy.sinh(u)**2.-sinh2u0)
+(numpy.sinh(u)**2.+1.)*actionAngleStaeckel.potentialStaeckel(u,numpy.pi/2.,self._pot,self._delta)
-(sinh2u0+1.)*actionAngleStaeckel.potentialStaeckel(u0,numpy.pi/2.,self._pot,self._delta))
# +(numpy.sinh(u)**2.+numpy.sin(v)**2.)*actionAngleStaeckel.potentialStaeckel(u,v,self._pot,self._delta)
# -(sinh2u0+numpy.sin(v)**2.)*actionAngleStaeckel.potentialStaeckel(u0,v,self._pot,self._delta))
return out
def _u0Eq(logu,delta,pot,E,Lz22):
"""The equation that needs to be minimized to find u0"""
u= numpy.exp(logu)
sinh2u= numpy.sinh(u)**2.
cosh2u= numpy.cosh(u)**2.
dU= cosh2u*actionAngleStaeckel.potentialStaeckel(u,numpy.pi/2.,pot,delta)
return -(E*sinh2u-dU-Lz22/delta**2./sinh2u)
def vatu0(self,E,Lz,u0,R,retv2=False):
"""
NAME:
vatu0
PURPOSE:
calculate the velocity at u0
INPUT:
E - energy
Lz - angular momentum
u0 - u0
R - radius corresponding to u0,pi/2.
retv2= (False), if True return v^2
OUTPUT:
velocity
HISTORY:
2012-11-29 - Written - Bovy (IAS)
"""
v2= (2.*(E-actionAngleStaeckel.potentialStaeckel(u0,numpy.pi/2.,
self._pot,
self._delta))
-Lz**2./R**2.)
if retv2: return v2
if isinstance(E,float) and v2 < 0. and v2 > -10.**-7.:
return 0. #rounding errors
elif isinstance(E,float):
return numpy.sqrt(v2)
elif isinstance(v2,numpy.ndarray):
v2[(v2 < 0.)*(v2 > -10.**-7.)]= 0.
return numpy.sqrt(v2)
