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Hi there. I am trying to evaluate the tangent vector at a specific value of the affine parameter. I have tried everything from https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/differentiable/integrated_curve.html but it raises an error i can't find a solution to.

v0 = initial_vector(r0, al=1,b=1, ph0=0, inward=True)
geod = M.integrated_geodesic(g, (s, 0, 500), v0, across_charts=True)
sol = geod.solve_across_charts(step=0.01,method='odeint',parameters_values={a:1,q:1},
                               solution_key='sol',verbose=True) 

interp = geod.interpolate(solution_key='sol',
                          interpolation_key='interp 1', verbose=True)
geodesics = geod

sage:Performing numerical integration with method 'ode'.
Integration will take place on the whole manifold domain.
Integration successful.
Performing cubic spline interpolation by default...
Interpolation completed and associated with the key 'interp 1' (if this key already referred to a former interpolation, such an interpolation was erased).

where using :

v = geod.tangent_vector_eval_at(200, verbose=True)

raises :

Evaluating tangent vector components from the interpolation associated with the key 'interp 1' by default...

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-60-276d62a6acbd> in <module>
----> 1 v2 = geod.tangent_vector_eval_at(Integer(200), verbose=True)

    /opt/sagemath-9.2/local/lib/python3.7/site-packages/sage/manifolds/differentiable/integrated_curve.py in tangent_vector_eval_at(self, t, interpolation_key, verbose)
       2275             # partial test, in case future interpolation objects do not
       2276             # contain lists of instances of the Spline class
    -> 2277             raise TypeError("unexpected type of interpolation object")
       2278 
       2279         interpolated_coordinates=[coordinate_curve_spline(t)

    TypeError: unexpected type of interpolation object

Hi there. I am trying to evaluate the tangent vector at a specific value of the affine parameter. I have tried everything from https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/differentiable/integrated_curve.html but it raises an error i can't find a solution to.

v0 = initial_vector(r0, al=1,b=1, ph0=0, inward=True)
geod = M.integrated_geodesic(g, (s, 0, 500), v0, across_charts=True)
sol = geod.solve_across_charts(step=0.01,method='odeint',parameters_values={a:1,q:1},
                               solution_key='sol',verbose=True) 

interp = geod.interpolate(solution_key='sol',
                          interpolation_key='interp 1', verbose=True)
geodesics = geod

sage:Performing numerical integration with method 'ode'.
Integration will take place on the whole manifold domain.
Integration successful.
Performing cubic spline interpolation by default...
Interpolation completed and associated with the key 'interp 1' (if this key already referred to a former interpolation, such an interpolation was erased).

where using :

v = geod.tangent_vector_eval_at(200, verbose=True)

raises :

Evaluating tangent vector components from the interpolation associated with the key 'interp 1' by default...

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-60-276d62a6acbd> in <module>
----> 1 v2 = geod.tangent_vector_eval_at(Integer(200), verbose=True)

    /opt/sagemath-9.2/local/lib/python3.7/site-packages/sage/manifolds/differentiable/integrated_curve.py in tangent_vector_eval_at(self, t, interpolation_key, verbose)
       2275             # partial test, in case future interpolation objects do not
       2276             # contain lists of instances of the Spline class
    -> 2277             raise TypeError("unexpected type of interpolation object")
       2278 
       2279         interpolated_coordinates=[coordinate_curve_spline(t)

    TypeError: unexpected type of interpolation object

Hi there. I am trying to evaluate the tangent vector at a specific value of the affine parameter. I have tried everything from https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/differentiable/integrated_curve.html but it raises an error i can't find a solution to.

v0 = initial_vector(r0, al=1,b=1, ph0=0, inward=True)
geod = M.integrated_geodesic(g, (s, 0, 500), v0, across_charts=True)
sol = geod.solve_across_charts(step=0.01,method='odeint',parameters_values={a:1,q:1},
                               solution_key='sol',verbose=True) 

interp = geod.interpolate(solution_key='sol',
                          interpolation_key='interp 1', verbose=True)
geodesics = geod

sage:Performing numerical integration with method 'ode'.
Integration will take place on the whole manifold domain.
Integration successful.
Performing cubic spline interpolation by default...
Interpolation completed and associated with the key 'interp 1' (if this key already referred to a former interpolation, such an interpolation was erased).

where using :

v = geod.tangent_vector_eval_at(200, verbose=True)

raises :

Evaluating tangent vector components from the interpolation associated with the key 'interp 1' by default...

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-60-276d62a6acbd> in <module>
----> 1 v2 = geod.tangent_vector_eval_at(Integer(200), verbose=True)

    /opt/sagemath-9.2/local/lib/python3.7/site-packages/sage/manifolds/differentiable/integrated_curve.py in tangent_vector_eval_at(self, t, interpolation_key, verbose)
       2275             # partial test, in case future interpolation objects do not
       2276             # contain lists of instances of the Spline class
    -> 2277             raise TypeError("unexpected type of interpolation object")
       2278 
       2279         interpolated_coordinates=[coordinate_curve_spline(t)

    TypeError: unexpected type of interpolation object

Hi there. I am trying to evaluate the tangent vector at a specific value of the affine parameter. I have tried everything from https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/differentiable/integrated_curve.html but it raises an error i can't find a solution to.

M = Manifold(4, 'M', latex_name=r'\mathcal{M}', structure='Lorentzian')
X.<t,r,th,ph>=M.chart(r"t r:(2.001,+inf) th:(0,pi):\theta ph:(0,2*pi):periodic\phi")

def initial_vector(r0,al,b,  ph0=0, E=1, inward=False):
    t0,th0=0,pi/2
    L = -b*E
    vt0=(E*g33(m,r0,th0,a)+L*g03(m,r0,th0,a))/(D(m,r0,th0,a))
    vr0=sqrt(((1/D(m,r0,th0,a))*((L^2)*g00(m,r0,th0,a)+2*E*L*g03(m,r0,th0,a)+(E^2)*g33(m,r0,th0,a))/g11(m,r0,th0,a))

    if inward:
        vr0 = - vr0
    vth0 = al/(100*r0)
    vph0 = -(1/D(m,r0,th0,a))*(E*g03(m,r0,th0,a)+L*(g00(m,r0,th0,a)))
    p0 = M((t0, r0, th0, ph0), name='p_0')
    return M.tangent_space(p0)((vt0, vr0, vth0, vph0), name='v_0')


v0 = initial_vector(r0, al=1,b=1, ph0=0, inward=True)
geod = M.integrated_geodesic(g, (s, 0, 500), v0, across_charts=True)
sol = geod.solve_across_charts(step=0.01,method='odeint',parameters_values={a:1,q:1},
                               solution_key='sol',verbose=True) 

interp = geod.interpolate(solution_key='sol',
                          interpolation_key='interp 1', verbose=True)
geodesics = geod

sage:Performing numerical integration with method 'ode'.
Integration will take place on the whole manifold domain.
Integration successful.
Performing cubic spline interpolation by default...
Interpolation completed and associated with the key 'interp 1' (if this key already referred to a former interpolation, such an interpolation was erased).

where using :

v = geod.tangent_vector_eval_at(200, verbose=True)

raises :

Evaluating tangent vector components from the interpolation associated with the key 'interp 1' by default...

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-60-276d62a6acbd> in <module>
----> 1 v2 = geod.tangent_vector_eval_at(Integer(200), verbose=True)

    /opt/sagemath-9.2/local/lib/python3.7/site-packages/sage/manifolds/differentiable/integrated_curve.py in tangent_vector_eval_at(self, t, interpolation_key, verbose)
       2275             # partial test, in case future interpolation objects do not
       2276             # contain lists of instances of the Spline class
    -> 2277             raise TypeError("unexpected type of interpolation object")
       2278 
       2279         interpolated_coordinates=[coordinate_curve_spline(t)

    TypeError: unexpected type of interpolation object

Hi there. I am trying to evaluate the tangent vector at a specific value of the affine parameter. I have tried everything from https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/differentiable/integrated_curve.html but it raises an error i can't find a solution to.

M = Manifold(4, 'M', latex_name=r'\mathcal{M}', structure='Lorentzian')
X.<t,r,th,ph>=M.chart(r"t r:(2.001,+inf) th:(0,pi):\theta ph:(0,2*pi):periodic\phi")

def initial_vector(r0,al,b,  ph0=0, E=1, inward=False):
    t0,th0=0,pi/2
    L = -b*E
    vt0=(E*g33(m,r0,th0,a)+L*g03(m,r0,th0,a))/(D(m,r0,th0,a))
    vr0=sqrt(((1/D(m,r0,th0,a))*((L^2)*g00(m,r0,th0,a)+2*E*L*g03(m,r0,th0,a)+(E^2)*g33(m,r0,th0,a))/g11(m,r0,th0,a))

    if inward:
        vr0 = - vr0
    vth0 = al/(100*r0)
    vph0 = -(1/D(m,r0,th0,a))*(E*g03(m,r0,th0,a)+L*(g00(m,r0,th0,a)))
    p0 = M((t0, r0, th0, ph0), name='p_0')
    return M.tangent_space(p0)((vt0, vr0, vth0, vph0), name='v_0')


v0 = initial_vector(r0, al=1,b=1, ph0=0, inward=True)
geod = M.integrated_geodesic(g, (s, 0, 500), v0, across_charts=True)
sol = geod.solve_across_charts(step=0.01,method='odeint',parameters_values={a:1,q:1},
                               solution_key='sol',verbose=True) 

interp = geod.interpolate(solution_key='sol',
                          interpolation_key='interp 1', verbose=True)
geodesics = geod

sage:Performing numerical integration with method 'ode'.
Integration will take place on the whole manifold domain.
Integration successful.
Performing cubic spline interpolation by default...
Interpolation completed and associated with the key 'interp 1' (if this key already referred to a former interpolation, such an interpolation was erased).

where using :

v = geod.tangent_vector_eval_at(200, verbose=True)

raises :

Evaluating tangent vector components from the interpolation associated with the key 'interp 1' by default...

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-60-276d62a6acbd> in <module>
----> 1 v2 = geod.tangent_vector_eval_at(Integer(200), verbose=True)

    /opt/sagemath-9.2/local/lib/python3.7/site-packages/sage/manifolds/differentiable/integrated_curve.py in tangent_vector_eval_at(self, t, interpolation_key, verbose)
       2275             # partial test, in case future interpolation objects do not
       2276             # contain lists of instances of the Spline class
    -> 2277             raise TypeError("unexpected type of interpolation object")
       2278 
       2279         interpolated_coordinates=[coordinate_curve_spline(t)

    TypeError: unexpected type of interpolation object

Hi there. I am trying to evaluate the tangent vector at a specific value of the affine parameter. I have tried everything from https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/differentiable/integrated_curve.html but it raises an error i can't find a solution to.

edit: Just to be precise and avoid taking up anyone's time. I am actually trying to create a poincare section of my orbit so i think i could use the numerical solution by calculating the velocity where the sol meets the criteria. Maybe there's an easier way (?). Couldnt think of anything else that can help me find the poincare section after and not during the integration.

M = Manifold(4, 'M', latex_name=r'\mathcal{M}', structure='Lorentzian')
X.<t,r,th,ph>=M.chart(r"t r:(2.001,+inf) X.<t,r,th,ph> = M.chart(r't r:(2.0001,+inf) th:(0,pi):\theta ph:(0,2*pi):periodic\phi")

def initial_vector(r0,al,b,  ph:(0,2*pi):periodic\phi') 
X.coord_range()

var('m,b,a', domain='real')
g = M.metric()
m=1

g[0,0] = -(1-2*m*r/(r^2 + (a*cos(th))^2))
g[0,3] = -2*a*m*r*sin(th)^2/(r^2 + (a*cos(th))^2)
g[1,1] = (r^2 + (a*cos(th))^2)/(r^2 -2*m*r + a^2)
g[2,2] = (r^2 + (a*cos(th))^2)
g[3,3] = (r^2+a^2+2*m*r*(a*sin(th))^2/(r^2 + (a*cos(th))^2))*sin(th)^2
g.display()

def g00(m,r,th,a):return g[0,0](m,r,th,a)
def g03(m,r,th,a):return g[0,3](m,r,th,a)
def g11(m,r,th,a):return g[1,1](m,r,th,a)
def g22(m,r,th,a):return g[2,2](m,r,th,a)
def g33(m,r,th,a):return g[3,3](m,r,th,a)
def D(m,r,th,a): return (g03(m,r,th,a))^2-g00(m,r,th,a)*g33(m,r,th,a)

E.<x,y,z> = EuclideanSpace()
phi = M.diff_map(E, [r*sin(th)*cos(ph), r*sin(th)*sin(ph), r*cos(th)])
phi.display()

def initial_vector(r0, b,al, ph0=0, E=1, inward=False):
    t0,th0=0,pi/2
    L = -b*E
    vt0=(E*g33(m,r0,th0,a)+L*g03(m,r0,th0,a))/(D(m,r0,th0,a))
    vr0=sqrt(((1/D(m,r0,th0,a))*((L^2)*g00(m,r0,th0,a)+2*E*L*g03(m,r0,th0,a)+(E^2)*g33(m,r0,th0,a))/g11(m,r0,th0,a))

vr0=sqrt(((1/D(m,r0,th0,a))*((L^2)*g00(m,r0,th0,a)+2*E*L*g03(m,r0,th0,a)+(E^2)*g33(m,r0,th0,a)))/g11(m,r0,th0,a))
    if inward:
        vr0 = - vr0
    vth0 = al/(100*r0)
    vph0 = -(1/D(m,r0,th0,a))*(E*g03(m,r0,th0,a)+L*(g00(m,r0,th0,a)))
    p0 = M((t0, r0, th0, ph0), name='p_0')
    return M.tangent_space(p0)((vt0, vr0, vth0, vph0), name='v_0')

r0 = 100 # απόσταση πηγής
th0=pi/2
s = var('s')

v0 = initial_vector(r0, al=1,b=1, ph0=0, inward=True)
geod = M.integrated_geodesic(g, (s, 0, 500), v0, across_charts=True)
sol = geod.solve_across_charts(step=0.01,method='odeint',parameters_values={a:1,q:1},
geod.solve_across_charts(step=0.01,method='odeint',parameters_values={a:1},
                               solution_key='sol',verbose=True) 

interp = geod.interpolate(solution_key='sol',
                          interpolation_key='interp 1', verbose=True)
geodesics = geod

sage:Performing numerical integration with method 'ode'.
Integration will take place on the whole manifold domain.
Integration successful.
Performing cubic spline interpolation by default...
Interpolation completed and associated with the key 'interp 1' (if this key already referred to a former interpolation, such an interpolation was erased).

where using :using:

v = geod.tangent_vector_eval_at(200, verbose=True)

raises :

Evaluating tangent vector components from the interpolation associated with the key 'interp 1' by default...

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-60-276d62a6acbd> <ipython-input-24-6fbce2379e9a> in <module>
----> 1 v2 v = geod.tangent_vector_eval_at(Integer(200), verbose=True)

 /opt/sagemath-9.2/local/lib/python3.7/site-packages/sage/manifolds/differentiable/integrated_curve.py in tangent_vector_eval_at(self, t, interpolation_key, verbose)
    2275             # partial test, in case future interpolation objects do not
    2276             # contain lists of instances of the Spline class
 -> 2277             raise TypeError("unexpected type of interpolation object")
    2278 
    2279         interpolated_coordinates=[coordinate_curve_spline(t)

 TypeError: unexpected type of interpolation object

Hi there. I am trying to evaluate the tangent vector at a specific value of the affine parameter. I have tried everything from https://doc.sagemath.org/html/en/reference/manifolds/sage/manifolds/differentiable/integrated_curve.html but it raises an error i can't find a solution to.

edit: Just to be precise and avoid taking up anyone's time. I am actually trying to create a poincare section of my orbit so i think i could use the numerical solution by calculating the velocity where the sol meets the criteria. Maybe there's an easier way (?). Couldnt think of anything else that can help me find the poincare section after and not during the integration.

M = Manifold(4, 'M', latex_name=r'\mathcal{M}', structure='Lorentzian')
X.<t,r,th,ph> = M.chart(r't r:(2.0001,+inf) th:(0,pi):\theta ph:(0,2*pi):periodic\phi') 
X.coord_range()

var('m,b,a', domain='real')
g = M.metric()
m=1

g[0,0] = -(1-2*m*r/(r^2 + (a*cos(th))^2))
g[0,3] = -2*a*m*r*sin(th)^2/(r^2 + (a*cos(th))^2)
g[1,1] = (r^2 + (a*cos(th))^2)/(r^2 -2*m*r + a^2)
g[2,2] = (r^2 + (a*cos(th))^2)
g[3,3] = (r^2+a^2+2*m*r*(a*sin(th))^2/(r^2 + (a*cos(th))^2))*sin(th)^2
g.display()

def g00(m,r,th,a):return g[0,0](m,r,th,a)
def g03(m,r,th,a):return g[0,3](m,r,th,a)
def g11(m,r,th,a):return g[1,1](m,r,th,a)
def g22(m,r,th,a):return g[2,2](m,r,th,a)
def g33(m,r,th,a):return g[3,3](m,r,th,a)
def D(m,r,th,a): return (g03(m,r,th,a))^2-g00(m,r,th,a)*g33(m,r,th,a)

E.<x,y,z> = EuclideanSpace()
phi = M.diff_map(E, [r*sin(th)*cos(ph), r*sin(th)*sin(ph), r*cos(th)])
phi.display()

def initial_vector(r0, b,al, ph0=0, E=1, inward=False):
    t0,th0=0,pi/2
    L = -b*E
    vt0=(E*g33(m,r0,th0,a)+L*g03(m,r0,th0,a))/(D(m,r0,th0,a))
    vr0=sqrt(((1/D(m,r0,th0,a))*((L^2)*g00(m,r0,th0,a)+2*E*L*g03(m,r0,th0,a)+(E^2)*g33(m,r0,th0,a)))/g11(m,r0,th0,a))
    if inward:
        vr0 = - vr0
    vth0 = al/(100*r0)
    vph0 = -(1/D(m,r0,th0,a))*(E*g03(m,r0,th0,a)+L*(g00(m,r0,th0,a)))
    p0 = M((t0, r0, th0, ph0), name='p_0')
    return M.tangent_space(p0)((vt0, vr0, vth0, vph0), name='v_0')

r0 = 100 # απόσταση πηγής
 th0=pi/2
s = var('s')

v0 = initial_vector(r0, al=1,b=1, ph0=0, inward=True)
geod = M.integrated_geodesic(g, (s, 0, 500), v0, across_charts=True)
sol = geod.solve_across_charts(step=0.01,method='odeint',parameters_values={a:1},
                               solution_key='sol',verbose=True) 

interp = geod.interpolate(solution_key='sol',
                          interpolation_key='interp 1', verbose=True)

where using:

v = geod.tangent_vector_eval_at(200, verbose=True)

raises :

Evaluating tangent vector components from the interpolation associated with the key 'interp 1' by default...

---------------------------------------------------------------------------
TypeError                                 Traceback (most recent call last)
<ipython-input-24-6fbce2379e9a> in <module>
----> 1 v = geod.tangent_vector_eval_at(Integer(200), verbose=True)

/opt/sagemath-9.2/local/lib/python3.7/site-packages/sage/manifolds/differentiable/integrated_curve.py in tangent_vector_eval_at(self, t, interpolation_key, verbose)
   2275             # partial test, in case future interpolation objects do not
   2276             # contain lists of instances of the Spline class
-> 2277             raise TypeError("unexpected type of interpolation object")
   2278 
   2279         interpolated_coordinates=[coordinate_curve_spline(t)

TypeError: unexpected type of interpolation object