For example,
var('t R_u c')
map(function,('R_b', 'psi', 'm_z'))
r = R_b(t)*sin(psi(t))
z = R_b(t)*cos(psi(t))+m_z(t)
Dr, Dz = r.diff(t), z.diff(t)
v=vector([Dr,Dz]) * vector([cos(psi(t)),sin(psi(t))])
w=v.simplify_trig()
w.substitute_function(m_z,lambda t:-sqrt(R_b(t)^2-R_u^2))
2*sin(psi(t))*cos(psi(t))*D[0](R_b)(t) -
sin(psi(t))*R_b(t)*D[0](R_b)(t)/sqrt(-R_u^2 + R_b(t)^2) -
(2*sin(psi(t))^2*R_b(t) - R_b(t))*D[0](psi)(t)
Alec Mihailovs
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