Vector Module¶

This document will describe some of the more advanced functionality that this module offers but which is not part of the “official” interface. Here, some of the features that will be implemented in the future will also be covered, along with unanswered questions about proper functionality. Also, common problems will be discussed, along with some solutions.

Inertia (Dyadics)¶

A dyadic tensor is a second order tensor formed by the juxtaposition of a pair of vectors. There are various operations defined with respect to dyadics, which have been implemented in vector in the form of class Dyadic. To know more, refer to the Dyadic and Vector class APIs. Dyadics are used to define the inertia of bodies within mechanics. Inertia dyadics can be defined explicitly but the inertia function is typically much more convenient for the user:

>>> from sympy.physics.mechanics import ReferenceFrame, inertia
>>> N = ReferenceFrame('N')

Supply a reference frame and the moments of inertia if the object
is symmetrical:

>>> inertia(N, 1, 2, 3)
(N.x|N.x) + 2*(N.y|N.y) + 3*(N.z|N.z)

Supply a reference frame along with the products and moments of inertia
for a general object:

>>> inertia(N, 1, 2, 3, 4, 5, 6)
(N.x|N.x) + 4*(N.x|N.y) + 6*(N.x|N.z) + 4*(N.y|N.x) + 2*(N.y|N.y) + 5*(N.y|N.z) + 6*(N.z|N.x) + 5*(N.z|N.y) + 3*(N.z|N.z)

Notice that the inertia function returns a dyadic with each component represented as two unit vectors separated by a |. Refer to the Dyadic section for more information about dyadics.

Inertia is often expressed in a matrix, or tensor, form, especially for numerical purposes. Since the matrix form does not contain any information about the reference frame(s) the inertia dyadic is defined in, you must provide one or two reference frames to extract the measure numbers from the dyadic. There is a convenience function to do this:

>>> inertia(N, 1, 2, 3, 4, 5, 6).to_matrix(N)
Matrix([
[1, 4, 6],
[4, 2, 5],
[6, 5, 3]])

Common Issues¶

Here issues with numerically integrating code, choice of \(dynamicsymbols\) for coordinate and speed representation, printing, differentiating, and substitution will occur.

Printing¶

The default printing options are to use sorting for Vector and Dyadic measure numbers, and have unsorted output from the vprint, vpprint, and vlatex functions. If you are printing something large, please use one of those functions, as the sorting can increase printing time from seconds to minutes.

Substitution¶

Substitution into large expressions can be slow, and take a few minutes.

Acceleration of Points¶

At a minimum, points need to have their velocities defined, as the acceleration can be calculated by taking the time derivative of the velocity in the same frame. If the 1 point or 2 point theorems were used to compute the velocity, the time derivative of the velocity expression will most likely be more complex than if you were to use the acceleration level 1 point and 2 point theorems. Using the acceleration level methods can result in shorted expressions at this point, which will result in shorter expressions later (such as when forming Kane’s equations).

Advanced Interfaces¶

Here we will cover advanced options in: ReferenceFrame, dynamicsymbols, and some associated functionality.

ReferenceFrame¶

ReferenceFrame is shown as having a .name attribute and .x, .y, and .z attributes for accessing the basis vectors, as well as a fairly rigidly defined print output. If you wish to have a different set of indices defined, there is an option for this. This will also require a different interface for accessing the basis vectors.

>>> from sympy.physics.vector import ReferenceFrame, vprint, vpprint, vlatex
>>> N = ReferenceFrame('N', indices=['i', 'j', 'k'])
>>> N['i']
N['i']
>>> N.x
N['i']
>>> vlatex(N.x)
'\\mathbf{\\hat{n}_{i}}'

Also, the latex output can have custom strings; rather than just indices though, the entirety of each basis vector can be specified. The custom latex strings can occur without custom indices, and also overwrites the latex string that would be used if there were custom indices.

>>> from sympy.physics.vector import ReferenceFrame, vlatex
>>> N = ReferenceFrame('N', latexs=['n1','\mathbf{n}_2','cat'])
>>> vlatex(N.x)
'n1'
>>> vlatex(N.y)
'\\mathbf{n}_2'
>>> vlatex(N.z)
'cat'

dynamicsymbols¶

The dynamicsymbols function also has ‘hidden’ functionality; the variable which is associated with time can be changed, as well as the notation for printing derivatives.

>>> from sympy import symbols
>>> from sympy.physics.vector import dynamicsymbols, vprint
>>> q1 = dynamicsymbols('q1')
>>> q1
q1(t)
>>> dynamicsymbols._t = symbols('T')
>>> q2 = dynamicsymbols('q2')
>>> q2
q2(T)
>>> q1
q1(t)
>>> q1d = dynamicsymbols('q1', 1)
>>> vprint(q1d)
q1'
>>> dynamicsymbols._str = 'd'
>>> vprint(q1d)
q1d
>>> dynamicsymbols._str = '\''
>>> dynamicsymbols._t = symbols('t')

Note that only dynamic symbols created after the change are different. The same is not true for the \(._str\) attribute; this affects the printing output only, so dynamic symbols created before or after will print the same way.

Also note that Vector‘s .dt method uses the ._t attribute of dynamicsymbols, along with a number of other important functions and methods. Don’t mix and match symbols representing time.