I would like sympy to confirm that, given the equation:
$$ r'(t) = A \times r(t) $$
(i.e. "the derivative of an unknown function r is the cross product of an unknown matrix A with r"), it follows that:
$$ r''(t) = A \times r'(t) $$
(i.e. "the second derivative of r is the cross product of A with the first derivative of r").
From the documentation it seems like I want to use MatrixSymbol for A, but MatrixSymbol doesn't define cross:
from sympy import *
from sympy.abc import *
r = Function('r')(t)
A = MatrixSymbol('A', 4, 4) # dummy dimensions
Derivative(A.cross(r))
gives me:
AttributeError Traceback (most recent call last)
<ipython-input-52-4c8dc7c142cf> in <module>
4 A = MatrixSymbol('A', 4, 4)
5
----> 6 Derivative(A.cross(r))
AttributeError: 'MatrixSymbol' object has no attribute 'cross'
What's the right way to do this?
SymPy's vector class is completely separate from the Matrix class which can be confusing if you're used to thinking of a vector as a particular kind of matrix: https://docs.sympy.org/latest/modules/vector/index.html
I'll demonstrate how to do this with the vector class. This can be done more compactly but I'm spelling it out in detail:
In [23]: from sympy.vector import CoordSys3D
...: N = CoordSys3D('N')
In [24]: a1, a2, a3 = symbols('a1:4')
In [25]: r1, r2, r3 = [ri(t) for ri in symbols('r1:4', cls=Function)]
In [26]: A = a1*i + a2*j + a3*k
In [27]: r = r1*i + r2*j + r3*k
In [28]: A
Out[28]: a1*N.i + a2*N.j + a3*N.k
In [29]: r
Out[29]: (r1(t))*N.i + (r2(t))*N.j + (r3(t))*N.k
In [30]: (A.cross(r.diff(t))).diff(t) == A.cross(r.diff(t, 2))
Out[30]: True