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(pot, R, phi=phi, t=t)
    phiforce = _evaluateplanarphiforces(pot, R, phi=phi, t=t)
    return nu.array([
        cosphi * Rforce - 1. / R * sinphi * phiforce,
        sinphi * Rforce + 1. / R * cosphi * phiforce
    ])
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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(pot,R,phi=phi,t=t)
    phiforce= _evaluateplanarphiforces(pot,R,phi=phi,t=t)
    return nu.array([cosphi*Rforce-1./R*sinphi*phiforce,
                     sinphi*Rforce+1./R*cosphi*phiforce])
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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(pot,R,phi=phi,t=t)
    phiforce= _evaluateplanarphiforces(pot,R,phi=phi,t=t)
    R2deriv= _evaluateplanarPotentials(pot,R,phi=phi,t=t,dR=2)
    phi2deriv= _evaluateplanarPotentials(pot,R,phi=phi,t=t,dphi=2)
    Rphideriv= _evaluateplanarPotentials(pot,R,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 _REOM(y, t, pot, l2):
    """
    NAME:
       _REOM
    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)
    """
    return [y[1], l2 / y[0]**3. + _evaluateplanarRforces(pot, y[0], t=t)]
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def _REOM(y,t,pot,l2):
    """
    NAME:
       _REOM
    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)
    """
    return [y[1],
            l2/y[0]**3.+_evaluateplanarRforces(pot,y[0],t=t)]