In [1]:

    

from __future__ import division
import sympy
from sympy import *
from sympy import Rational as frac
import simpletensors
from simpletensors import Vector, TensorProduct, SymmetricTensorProduct, Tensor
init_printing()


var('vartheta, varphi')
var('nu, m, delta, c, t')

# These are related scalar functions of time
var('r, v, Omega', cls=Function)
r = r(t)
v = v(t)
Omega = Omega(t)

# These get redefined momentarily, but have to exist first
var('nHat, lambdaHat, ellHat', cls=Function)
# And now we define them as vector functions of time
nHat = Vector('nHat', r'\hat{n}', [cos(Omega*t),sin(Omega*t),0,])(t)
lambdaHat = Vector('lambdaHat', r'\hat{\lambda}', [-sin(Omega*t),cos(Omega*t),0,])(t)
ellHat = Vector('ellHat', r'\hat{\ell}', [0,0,1,])(t)

# These are the spin functions -- first, the individual components as regular sympy.Function objects; then the vectors themselves
var('S_n, S_lambda, S_ell', cls=Function)
var('Sigma_n, Sigma_lambda, Sigma_ell', cls=Function)
SigmaVec = Vector('SigmaVec', r'\vec{\Sigma}', [Sigma_n(t), Sigma_lambda(t), Sigma_ell(t)])(t)
SVec = Vector('S', r'\vec{S}', [S_n(t), S_lambda(t), S_ell(t)])(t)


    

In [3]:

    

nHat


    

    
Out[3]:





$$\hat{n}$$

In [4]:

    

diff(nHat, t)


    

    
Out[4]:





$$\partial_t \hat{n}$$

In [5]:

    

diff(lambdaHat, t)


    

    
Out[5]:





$$\partial_t \hat{\lambda}$$

In [6]:

    

diff(lambdaHat, t).components


    

    
Out[6]:





$$\begin{bmatrix}- \left(t \frac{d}{d t} \Omega{\left (t \right )} + \Omega{\left (t \right )}\right) \cos{\left (t \Omega{\left (t \right )} \right )}, & - \left(t \frac{d}{d t} \Omega{\left (t \right )} + \Omega{\left (t \right )}\right) \sin{\left (t \Omega{\left (t \right )} \right )}, & 0\end{bmatrix}$$

In [7]:

    

diff(lambdaHat, t).subs(t,0).components


    

    
Out[7]:





$$\begin{bmatrix}- \Omega{\left (0 \right )}, & 0, & 0\end{bmatrix}$$

In [8]:

    

diff(lambdaHat, t, 2).components


    

    
Out[8]:





$$\begin{bmatrix}\left(t \frac{d}{d t} \Omega{\left (t \right )} + \Omega{\left (t \right )}\right)^{2} \sin{\left (t \Omega{\left (t \right )} \right )} - \left(t \frac{d^{2}}{d t^{2}}  \Omega{\left (t \right )} + 2 \frac{d}{d t} \Omega{\left (t \right )}\right) \cos{\left (t \Omega{\left (t \right )} \right )}, & - \left(t \frac{d}{d t} \Omega{\left (t \right )} + \Omega{\left (t \right )}\right)^{2} \cos{\left (t \Omega{\left (t \right )} \right )} - \left(t \frac{d^{2}}{d t^{2}}  \Omega{\left (t \right )} + 2 \frac{d}{d t} \Omega{\left (t \right )}\right) \sin{\left (t \Omega{\left (t \right )} \right )}, & 0\end{bmatrix}$$

In [9]:

    

diff(lambdaHat, t, 2).subs(t,0).components


    

    
Out[9]:





$$\begin{bmatrix}- 2 \left. \frac{d}{d t} \Omega{\left (t \right )} \right|_{\substack{ t=0 }}, & - \Omega^{2}{\left (0 \right )}, & 0\end{bmatrix}$$

In [10]:

    

diff(ellHat, t)


    

    
Out[10]:





$$0$$

In [11]:

    

diff(nHat, t, 2)


    

    
Out[11]:





$$\partial_t^{2} \hat{n}$$

In [12]:

    

diff(nHat,t, 3)


    

    
Out[12]:





$$\partial_t^{3} \hat{n}$$

In [13]:

    

diff(nHat,t, 4)


    

    
Out[13]:





$$\partial_t^{4} \hat{n}$$

In [14]:

    

diff(SigmaVec,t, 0)


    

    
Out[14]:





$$\vec{\Sigma}$$

In [15]:

    

SigmaVec.fdiff()


    

    
Out[15]:





$$\partial_t \vec{\Sigma}$$

In [16]:

    

diff(SigmaVec,t, 1)


    

    
Out[16]:





$$\partial_t \vec{\Sigma}$$

In [17]:

    

diff(SigmaVec,t, 2)


    

    
Out[17]:





$$\partial_t^{2} \vec{\Sigma}$$

In [18]:

    

diff(SigmaVec,t, 2) | nHat


    

    
Out[18]:





$$\sin{\left (t \Omega{\left (t \right )} \right )} \frac{d^{2}}{d t^{2}}  \Sigma_{\lambda}{\left (t \right )} + \cos{\left (t \Omega{\left (t \right )} \right )} \frac{d^{2}}{d t^{2}}  \Sigma_{n}{\left (t \right )}$$

In [19]:

    

T1 = TensorProduct(SigmaVec, SigmaVec, ellHat, coefficient=1)
T2 = TensorProduct(SigmaVec, nHat, lambdaHat, coefficient=1)
tmp = Tensor(T1,T2)
display(T1, T2, tmp)


    

    






$$\left(\vec{\Sigma} \otimes \vec{\Sigma} \otimes \hat{\ell}\right)$$






    






$$\left(\vec{\Sigma} \otimes \hat{n} \otimes \hat{\lambda}\right)$$






    






$$\begin{align*}&\left[ \left(\vec{\Sigma} \otimes \vec{\Sigma} \otimes \hat{\ell}\right) \right. \nonumber \\&\quad \left. + \left(\vec{\Sigma} \otimes \hat{n} \otimes \hat{\lambda}\right) \right]\end{align*}$$

In [20]:

    

diff(tmp, t, 1)


    

    
Out[20]:





$$\begin{align*}&\left[ \left(\partial_t \vec{\Sigma} \otimes \vec{\Sigma} \otimes \hat{\ell}\right) \right. \nonumber \\&\quad \left. + \left(\vec{\Sigma} \otimes \partial_t \vec{\Sigma} \otimes \hat{\ell}\right) \right. \nonumber \\&\quad \left. + \left(\partial_t \vec{\Sigma} \otimes \hat{n} \otimes \hat{\lambda}\right) \right. \nonumber \\&\quad \left. + \left(\vec{\Sigma} \otimes \partial_t \hat{n} \otimes \hat{\lambda}\right) \right. \nonumber \\&\quad \left. + \left(\vec{\Sigma} \otimes \hat{n} \otimes \partial_t \hat{\lambda}\right) \right]\end{align*}$$

In [21]:

    

T1+T2


    

    
Out[21]:





$$\begin{align*}&\left[ \left(\vec{\Sigma} \otimes \vec{\Sigma} \otimes \hat{\ell}\right) \right. \nonumber \\&\quad \left. + \left(\vec{\Sigma} \otimes \hat{n} \otimes \hat{\lambda}\right) \right]\end{align*}$$

In [22]:

    

T2*ellHat


    

    
Out[22]:





$$\left(\vec{\Sigma} \otimes \hat{n} \otimes \hat{\lambda} \otimes \hat{\ell}\right)$$

In [23]:

    

ellHat*T2


    

    
Out[23]:





$$\left(\hat{\ell} \otimes \vec{\Sigma} \otimes \hat{n} \otimes \hat{\lambda}\right)$$

In [24]:

    

T1.trace(0,1)


    

    
Out[24]:





$$\left\{ \left[ \Sigma_{\ell}^{2}{\left (t \right )} + \Sigma_{\lambda}^{2}{\left (t \right )} + \Sigma_{n}^{2}{\left (t \right )} \right]\, \hat{\ell} \right\}$$

In [25]:

    

T2*ellHat


    

    
Out[25]:





$$\left(\vec{\Sigma} \otimes \hat{n} \otimes \hat{\lambda} \otimes \hat{\ell}\right)$$

In [26]:

    

for k in range(1,4):
    display((T2*ellHat).trace(0,k))


    

    






$$\left\{ \left[ \Sigma_{\lambda}{\left (t \right )} \sin{\left (t \Omega{\left (t \right )} \right )} + \Sigma_{n}{\left (t \right )} \cos{\left (t \Omega{\left (t \right )} \right )} \right]\, \hat{\lambda} \otimes \hat{\ell} \right\}$$






    






$$\left\{ \left[ \Sigma_{\lambda}{\left (t \right )} \cos{\left (t \Omega{\left (t \right )} \right )} - \Sigma_{n}{\left (t \right )} \sin{\left (t \Omega{\left (t \right )} \right )} \right]\, \hat{n} \otimes \hat{\ell} \right\}$$






    






$$\left\{ \left[ \Sigma_{\ell}{\left (t \right )} \right]\, \hat{n} \otimes \hat{\lambda} \right\}$$

In [27]:

    

for k in range(1,4):
    display((T2*ellHat).trace(0,k).subs(t,0))


    

    






$$\left\{ \left[ \Sigma_{\lambda}{\left (t \right )} \sin{\left (t \Omega{\left (t \right )} \right )} + \Sigma_{n}{\left (t \right )} \cos{\left (t \Omega{\left (t \right )} \right )} \right]\, \hat{\lambda} \otimes \hat{\ell} \right\}$$






    






$$\left\{ \left[ \Sigma_{\lambda}{\left (t \right )} \cos{\left (t \Omega{\left (t \right )} \right )} - \Sigma_{n}{\left (t \right )} \sin{\left (t \Omega{\left (t \right )} \right )} \right]\, \hat{n} \otimes \hat{\ell} \right\}$$






    






$$\left\{ \left[ \Sigma_{\ell}{\left (t \right )} \right]\, \hat{n} \otimes \hat{\lambda} \right\}$$

In [28]:

    

T1.trace(0,1) * T2


    

    
Out[28]:





$$\left\{ \left[ \Sigma_{\ell}^{2}{\left (t \right )} + \Sigma_{\lambda}^{2}{\left (t \right )} + \Sigma_{n}^{2}{\left (t \right )} \right]\, \hat{\ell} \otimes \vec{\Sigma} \otimes \hat{n} \otimes \hat{\lambda} \right\}$$

In [29]:

    

display(T1, T2)


    

    






$$\left(\vec{\Sigma} \otimes \vec{\Sigma} \otimes \hat{\ell}\right)$$






    






$$\left(\vec{\Sigma} \otimes \hat{n} \otimes \hat{\lambda}\right)$$

In [30]:

    

T3 = SymmetricTensorProduct(SigmaVec, SigmaVec, ellHat, coefficient=1)
display(T3)
T3.trace(0,1)


    

    






$$\left(\vec{\Sigma} \otimes_{\mathrm{s}} \vec{\Sigma} \otimes_{\mathrm{s}} \hat{\ell}\right)$$






    
Out[30]:





$$\begin{align*}&\left\{ \left\{ \left[ \frac{1}{3} \Sigma_{\ell}^{2}{\left (t \right )} + \frac{1}{3} \Sigma_{\lambda}^{2}{\left (t \right )} + \frac{1}{3} \Sigma_{n}^{2}{\left (t \right )} \right]\, \hat{\ell} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{2}{3} \Sigma_{\ell}{\left (t \right )} \right]\, \vec{\Sigma} \right\} \right\}\end{align*}$$

In [31]:

    

diff(T3, t, 1)


    

    
Out[31]:





$$\begin{align*}&\left\{ \left[ \left(2\right)\, \partial_t \vec{\Sigma} \otimes_{\mathrm{s}} \vec{\Sigma} \otimes_{\mathrm{s}} \hat{\ell} \right] \right\}\end{align*}$$

In [32]:

    

T3.symmetric


    

    
Out[32]:





True

In [33]:

    

T3*ellHat


    

    
Out[33]:





$$\left(\vec{\Sigma} \otimes_{\mathrm{s}} \vec{\Sigma} \otimes_{\mathrm{s}} \hat{\ell} \otimes_{\mathrm{s}} \hat{\ell}\right)$$

In [34]:

    

ellHat*T3


    

    
Out[34]:





$$\left(\hat{\ell} \otimes_{\mathrm{s}} \vec{\Sigma} \otimes_{\mathrm{s}} \vec{\Sigma} \otimes_{\mathrm{s}} \hat{\ell}\right)$$

In [35]:

    

T1+T3


    

    
Out[35]:





$$\begin{align*}&\left[ \left(\vec{\Sigma} \otimes \vec{\Sigma} \otimes \hat{\ell}\right) \right. \nonumber \\&\quad \left. + \left(\vec{\Sigma} \otimes_{\mathrm{s}} \vec{\Sigma} \otimes_{\mathrm{s}} \hat{\ell}\right) \right]\end{align*}$$

In [6]:

    

T1 = SymmetricTensorProduct(SigmaVec, SigmaVec, ellHat, nHat, coefficient=1)
display(T1)
display(T1.trace())


    

    






$$\left(\vec{\Sigma} \otimes_{\mathrm{s}} \vec{\Sigma} \otimes_{\mathrm{s}} \hat{\ell} \otimes_{\mathrm{s}} \hat{n}\right)$$






    






$$\begin{align*}&\left( \left\{ \left[ \frac{1}{12} \Sigma_{\ell}^{2}{\left (t \right )} + \frac{1}{12} \Sigma_{\lambda}^{2}{\left (t \right )} + \frac{1}{12} \Sigma_{n}^{2}{\left (t \right )} \right]\, \hat{\ell} \otimes_{\mathrm{s}} \hat{n} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{6} \Sigma_{\ell}{\left (t \right )} \right]\, \vec{\Sigma} \otimes_{\mathrm{s}} \hat{n} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{6} \Sigma_{\lambda}{\left (t \right )} \sin{\left (t \Omega{\left (t \right )} \right )} + \frac{1}{6} \Sigma_{n}{\left (t \right )} \cos{\left (t \Omega{\left (t \right )} \right )} \right]\, \vec{\Sigma} \otimes_{\mathrm{s}} \hat{\ell} \right\} \right)\end{align*}$$

In [37]:

    

T1*T2


    

    
Out[37]:





$$\left(\vec{\Sigma} \otimes_{\mathrm{s}} \vec{\Sigma} \otimes_{\mathrm{s}} \hat{\ell} \otimes_{\mathrm{s}} \hat{n} \otimes_{\mathrm{s}} \vec{\Sigma} \otimes_{\mathrm{s}} \hat{n} \otimes_{\mathrm{s}} \hat{\lambda}\right)$$

In [38]:

    

type(_)


    

    
Out[38]:





simpletensors.TensorProductFunction_73

In [39]:

    

import simpletensors
isinstance(__, simpletensors.TensorProductFunction)


    

    
Out[39]:





True

In [40]:

    

SymmetricTensorProduct(nHat, nHat, nHat).trace()


    

    
Out[40]:





$$\begin{align*}&\left[ \left(\hat{n}\right) \right]\end{align*}$$

In [7]:

    

diff(T1.trace(), t, 1)


    

    
Out[7]:





$$\begin{align*}&\left( \left\{ \left[ \frac{1}{6} \Sigma_{\ell}{\left (t \right )} \frac{d}{d t} \Sigma_{\ell}{\left (t \right )} + \frac{1}{6} \Sigma_{\lambda}{\left (t \right )} \frac{d}{d t} \Sigma_{\lambda}{\left (t \right )} + \frac{1}{6} \Sigma_{n}{\left (t \right )} \frac{d}{d t} \Sigma_{n}{\left (t \right )} \right]\, \hat{\ell} \otimes_{\mathrm{s}} \hat{n} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{12} \Sigma_{\ell}^{2}{\left (t \right )} + \frac{1}{12} \Sigma_{\lambda}^{2}{\left (t \right )} + \frac{1}{12} \Sigma_{n}^{2}{\left (t \right )} \right]\, \hat{\ell} \otimes_{\mathrm{s}} \partial_t \hat{n} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{6} \frac{d}{d t} \Sigma_{\ell}{\left (t \right )} \right]\, \vec{\Sigma} \otimes_{\mathrm{s}} \hat{n} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{6} \Sigma_{\ell}{\left (t \right )} \right]\, \partial_t \vec{\Sigma} \otimes_{\mathrm{s}} \hat{n} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{6} \Sigma_{\ell}{\left (t \right )} \right]\, \vec{\Sigma} \otimes_{\mathrm{s}} \partial_t \hat{n} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{6} \left(t \frac{d}{d t} \Omega{\left (t \right )} + \Omega{\left (t \right )}\right) \Sigma_{\lambda}{\left (t \right )} \cos{\left (t \Omega{\left (t \right )} \right )} - \frac{1}{6} \left(t \frac{d}{d t} \Omega{\left (t \right )} + \Omega{\left (t \right )}\right) \Sigma_{n}{\left (t \right )} \sin{\left (t \Omega{\left (t \right )} \right )} + \frac{1}{6} \sin{\left (t \Omega{\left (t \right )} \right )} \frac{d}{d t} \Sigma_{\lambda}{\left (t \right )} + \frac{1}{6} \cos{\left (t \Omega{\left (t \right )} \right )} \frac{d}{d t} \Sigma_{n}{\left (t \right )} \right]\, \vec{\Sigma} \otimes_{\mathrm{s}} \hat{\ell} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{6} \Sigma_{\lambda}{\left (t \right )} \sin{\left (t \Omega{\left (t \right )} \right )} + \frac{1}{6} \Sigma_{n}{\left (t \right )} \cos{\left (t \Omega{\left (t \right )} \right )} \right]\, \partial_t \vec{\Sigma} \otimes_{\mathrm{s}} \hat{\ell} \right\} \right)\end{align*}$$

In [8]:

    

diff(T1.trace(), t, 2)


    

    
Out[8]:





$$\begin{align*}&\left( \left\{ \left[ \frac{1}{6} \Sigma_{\ell}{\left (t \right )} \frac{d^{2}}{d t^{2}}  \Sigma_{\ell}{\left (t \right )} + \frac{1}{6} \Sigma_{\lambda}{\left (t \right )} \frac{d^{2}}{d t^{2}}  \Sigma_{\lambda}{\left (t \right )} + \frac{1}{6} \Sigma_{n}{\left (t \right )} \frac{d^{2}}{d t^{2}}  \Sigma_{n}{\left (t \right )} + \frac{1}{6} \left(\frac{d}{d t} \Sigma_{\ell}{\left (t \right )}\right)^{2} + \frac{1}{6} \left(\frac{d}{d t} \Sigma_{\lambda}{\left (t \right )}\right)^{2} + \frac{1}{6} \left(\frac{d}{d t} \Sigma_{n}{\left (t \right )}\right)^{2} \right]\, \hat{\ell} \otimes_{\mathrm{s}} \hat{n} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{3} \Sigma_{\ell}{\left (t \right )} \frac{d}{d t} \Sigma_{\ell}{\left (t \right )} + \frac{1}{3} \Sigma_{\lambda}{\left (t \right )} \frac{d}{d t} \Sigma_{\lambda}{\left (t \right )} + \frac{1}{3} \Sigma_{n}{\left (t \right )} \frac{d}{d t} \Sigma_{n}{\left (t \right )} \right]\, \hat{\ell} \otimes_{\mathrm{s}} \partial_t \hat{n} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{12} \Sigma_{\ell}^{2}{\left (t \right )} + \frac{1}{12} \Sigma_{\lambda}^{2}{\left (t \right )} + \frac{1}{12} \Sigma_{n}^{2}{\left (t \right )} \right]\, \hat{\ell} \otimes_{\mathrm{s}} \partial_t^{2} \hat{n} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{6} \frac{d^{2}}{d t^{2}}  \Sigma_{\ell}{\left (t \right )} \right]\, \vec{\Sigma} \otimes_{\mathrm{s}} \hat{n} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{3} \frac{d}{d t} \Sigma_{\ell}{\left (t \right )} \right]\, \partial_t \vec{\Sigma} \otimes_{\mathrm{s}} \hat{n} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{3} \frac{d}{d t} \Sigma_{\ell}{\left (t \right )} \right]\, \vec{\Sigma} \otimes_{\mathrm{s}} \partial_t \hat{n} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{6} \Sigma_{\ell}{\left (t \right )} \right]\, \partial_t^{2} \vec{\Sigma} \otimes_{\mathrm{s}} \hat{n} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{3} \Sigma_{\ell}{\left (t \right )} \right]\, \partial_t \vec{\Sigma} \otimes_{\mathrm{s}} \partial_t \hat{n} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{6} \Sigma_{\ell}{\left (t \right )} \right]\, \vec{\Sigma} \otimes_{\mathrm{s}} \partial_t^{2} \hat{n} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ - \frac{1}{6} \left(t \frac{d}{d t} \Omega{\left (t \right )} + \Omega{\left (t \right )}\right)^{2} \Sigma_{\lambda}{\left (t \right )} \sin{\left (t \Omega{\left (t \right )} \right )} - \frac{1}{6} \left(t \frac{d}{d t} \Omega{\left (t \right )} + \Omega{\left (t \right )}\right)^{2} \Sigma_{n}{\left (t \right )} \cos{\left (t \Omega{\left (t \right )} \right )} - \frac{1}{3} \left(t \frac{d}{d t} \Omega{\left (t \right )} + \Omega{\left (t \right )}\right) \sin{\left (t \Omega{\left (t \right )} \right )} \frac{d}{d t} \Sigma_{n}{\left (t \right )} + \frac{1}{3} \left(t \frac{d}{d t} \Omega{\left (t \right )} + \Omega{\left (t \right )}\right) \cos{\left (t \Omega{\left (t \right )} \right )} \frac{d}{d t} \Sigma_{\lambda}{\left (t \right )} + \frac{1}{6} \left(t \frac{d^{2}}{d t^{2}}  \Omega{\left (t \right )} + 2 \frac{d}{d t} \Omega{\left (t \right )}\right) \Sigma_{\lambda}{\left (t \right )} \cos{\left (t \Omega{\left (t \right )} \right )} - \frac{1}{6} \left(t \frac{d^{2}}{d t^{2}}  \Omega{\left (t \right )} + 2 \frac{d}{d t} \Omega{\left (t \right )}\right) \Sigma_{n}{\left (t \right )} \sin{\left (t \Omega{\left (t \right )} \right )} + \frac{1}{6} \sin{\left (t \Omega{\left (t \right )} \right )} \frac{d^{2}}{d t^{2}}  \Sigma_{\lambda}{\left (t \right )} + \frac{1}{6} \cos{\left (t \Omega{\left (t \right )} \right )} \frac{d^{2}}{d t^{2}}  \Sigma_{n}{\left (t \right )} \right]\, \vec{\Sigma} \otimes_{\mathrm{s}} \hat{\ell} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{3} \left(t \frac{d}{d t} \Omega{\left (t \right )} + \Omega{\left (t \right )}\right) \Sigma_{\lambda}{\left (t \right )} \cos{\left (t \Omega{\left (t \right )} \right )} - \frac{1}{3} \left(t \frac{d}{d t} \Omega{\left (t \right )} + \Omega{\left (t \right )}\right) \Sigma_{n}{\left (t \right )} \sin{\left (t \Omega{\left (t \right )} \right )} + \frac{1}{3} \sin{\left (t \Omega{\left (t \right )} \right )} \frac{d}{d t} \Sigma_{\lambda}{\left (t \right )} + \frac{1}{3} \cos{\left (t \Omega{\left (t \right )} \right )} \frac{d}{d t} \Sigma_{n}{\left (t \right )} \right]\, \partial_t \vec{\Sigma} \otimes_{\mathrm{s}} \hat{\ell} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{6} \Sigma_{\lambda}{\left (t \right )} \sin{\left (t \Omega{\left (t \right )} \right )} + \frac{1}{6} \Sigma_{n}{\left (t \right )} \cos{\left (t \Omega{\left (t \right )} \right )} \right]\, \partial_t^{2} \vec{\Sigma} \otimes_{\mathrm{s}} \hat{\ell} \right\} \right)\end{align*}$$

In [9]:

    

diff(T1.trace(), t, 2).subs(t,0)


    

    
Out[9]:





$$\begin{align*}&\left( \left\{ \left[ \frac{1}{6} \Sigma_{\ell}{\left (0 \right )} \left. \frac{d^{2}}{d t^{2}}  \Sigma_{\ell}{\left (t \right )} \right|_{\substack{ t=0 }} + \frac{1}{6} \Sigma_{\lambda}{\left (0 \right )} \left. \frac{d^{2}}{d t^{2}}  \Sigma_{\lambda}{\left (t \right )} \right|_{\substack{ t=0 }} + \frac{1}{6} \Sigma_{n}{\left (0 \right )} \left. \frac{d^{2}}{d t^{2}}  \Sigma_{n}{\left (t \right )} \right|_{\substack{ t=0 }} + \frac{1}{6} \left(\left. \frac{d}{d t} \Sigma_{\ell}{\left (t \right )} \right|_{\substack{ t=0 }}\right)^{2} + \frac{1}{6} \left(\left. \frac{d}{d t} \Sigma_{\lambda}{\left (t \right )} \right|_{\substack{ t=0 }}\right)^{2} + \frac{1}{6} \left(\left. \frac{d}{d t} \Sigma_{n}{\left (t \right )} \right|_{\substack{ t=0 }}\right)^{2} \right]\, \left.\hat{\ell}\right|_{t=0} \otimes_{\mathrm{s}} \left.\hat{n}\right|_{t=0} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{3} \Sigma_{\ell}{\left (0 \right )} \left. \frac{d}{d t} \Sigma_{\ell}{\left (t \right )} \right|_{\substack{ t=0 }} + \frac{1}{3} \Sigma_{\lambda}{\left (0 \right )} \left. \frac{d}{d t} \Sigma_{\lambda}{\left (t \right )} \right|_{\substack{ t=0 }} + \frac{1}{3} \Sigma_{n}{\left (0 \right )} \left. \frac{d}{d t} \Sigma_{n}{\left (t \right )} \right|_{\substack{ t=0 }} \right]\, \left.\hat{\ell}\right|_{t=0} \otimes_{\mathrm{s}} \left.\partial_t \hat{n}\right|_{t=0} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{12} \Sigma_{\ell}^{2}{\left (0 \right )} + \frac{1}{12} \Sigma_{\lambda}^{2}{\left (0 \right )} + \frac{1}{12} \Sigma_{n}^{2}{\left (0 \right )} \right]\, \left.\hat{\ell}\right|_{t=0} \otimes_{\mathrm{s}} \left.\partial_t^{2} \hat{n}\right|_{t=0} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{6} \left. \frac{d^{2}}{d t^{2}}  \Sigma_{\ell}{\left (t \right )} \right|_{\substack{ t=0 }} \right]\, \left.\vec{\Sigma}\right|_{t=0} \otimes_{\mathrm{s}} \left.\hat{n}\right|_{t=0} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{3} \left. \frac{d}{d t} \Sigma_{\ell}{\left (t \right )} \right|_{\substack{ t=0 }} \right]\, \left.\partial_t \vec{\Sigma}\right|_{t=0} \otimes_{\mathrm{s}} \left.\hat{n}\right|_{t=0} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{3} \left. \frac{d}{d t} \Sigma_{\ell}{\left (t \right )} \right|_{\substack{ t=0 }} \right]\, \left.\vec{\Sigma}\right|_{t=0} \otimes_{\mathrm{s}} \left.\partial_t \hat{n}\right|_{t=0} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{6} \Sigma_{\ell}{\left (0 \right )} \right]\, \left.\partial_t^{2} \vec{\Sigma}\right|_{t=0} \otimes_{\mathrm{s}} \left.\hat{n}\right|_{t=0} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{3} \Sigma_{\ell}{\left (0 \right )} \right]\, \left.\partial_t \vec{\Sigma}\right|_{t=0} \otimes_{\mathrm{s}} \left.\partial_t \hat{n}\right|_{t=0} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{6} \Sigma_{\ell}{\left (0 \right )} \right]\, \left.\vec{\Sigma}\right|_{t=0} \otimes_{\mathrm{s}} \left.\partial_t^{2} \hat{n}\right|_{t=0} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ - \frac{1}{6} \Omega^{2}{\left (0 \right )} \Sigma_{n}{\left (0 \right )} + \frac{1}{3} \Omega{\left (0 \right )} \left. \frac{d}{d t} \Sigma_{\lambda}{\left (t \right )} \right|_{\substack{ t=0 }} + \frac{1}{3} \Sigma_{\lambda}{\left (0 \right )} \left. \frac{d}{d t} \Omega{\left (t \right )} \right|_{\substack{ t=0 }} + \frac{1}{6} \left. \frac{d^{2}}{d t^{2}}  \Sigma_{n}{\left (t \right )} \right|_{\substack{ t=0 }} \right]\, \left.\vec{\Sigma}\right|_{t=0} \otimes_{\mathrm{s}} \left.\hat{\ell}\right|_{t=0} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{3} \Omega{\left (0 \right )} \Sigma_{\lambda}{\left (0 \right )} + \frac{1}{3} \left. \frac{d}{d t} \Sigma_{n}{\left (t \right )} \right|_{\substack{ t=0 }} \right]\, \left.\partial_t \vec{\Sigma}\right|_{t=0} \otimes_{\mathrm{s}} \left.\hat{\ell}\right|_{t=0} \right\} \right. \nonumber \\&\quad \left. + \left\{ \left[ \frac{1}{6} \Sigma_{n}{\left (0 \right )} \right]\, \left.\partial_t^{2} \vec{\Sigma}\right|_{t=0} \otimes_{\mathrm{s}} \left.\hat{\ell}\right|_{t=0} \right\} \right)\end{align*}$$

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