5  Basics of python

5.1 Symbolics using SymPy

To get started with symbolic computation in Python, we will use the SymPy library. SymPy is a Python library for symbolic mathematics, also known as computer algebra system (CAS). We can use SymPy to perform algebraic manipulations, calculus, solve equations, and much more. We load the SymPy library using the following command:

import sympy as sp

Let’s showcase some of the basic functionalities of SymPy using an example, we will solve the following equation: \(x^2 = 4\).

First we create a symbol \(x\) using the symbols function. We then define the equation \(x^2 = 4\) using the Eq function. Finally, we solve the equation using the solve function.

x = sp.symbols('x') # create a symbol x
eq = sp.Eq(x**2, 4) # define the equation x^2 = 4
sol = sp.solve(eq, x) # solve the equation
sol

[-2, 2]

We could also utillize assumptions to solve the equation. For example, if we know that \(x\) is a real number, we can use the solve function with the real=True argument. Or if we know that \(x\) is a positive number, we can use the positive=True argument.

x = sp.symbols('x', positive=True) # create a symbol x which is positive
eq = sp.Eq(x**2, 4) # define the equation x^2 = 4
sol = sp.solve(eq, x) # solve the equation
sol

[2]

5.1.1 Example 1 - Creating a vector and equation system

We will create a force vector with two unknown forces \(F_x\) and \(F_y\).

F_x, F_y = sp.symbols('F_x F_y') # create symbols F_x and F_y
FF = sp.Matrix([F_x, F_y, 0]) # create a matrix FF with F_x and F_y and 0
FF

\(\displaystyle \left[\begin{matrix}F_{x}\\F_{y}\\0\end{matrix}\right]\)

Next we create an equation system

# Import the pi function from sympy
from sympy import pi

# Define the functions sind and cosd
# sind and cosd are the sin and cos functions that take degrees as input
def sind(x):
    return sp.sin(x*pi/180)
def cosd(x):
    return sp.cos(x*pi/180)

# create a matrix FF1 = [250*cosd(38), 250*sind(38), 0]
FF1 = 250*sp.Matrix([cosd(38), sind(38), 0]) 

# define the equation FF + FF1 = [1000, 0, 0]
eq = sp.Eq(FF + FF1, sp.Matrix([1000, 0, 0])) 
eq

\(\displaystyle \left[\begin{matrix}F_{x} + 250 \cos{\left(\frac{19 \pi}{90} \right)}\\F_{y} + 250 \sin{\left(\frac{19 \pi}{90} \right)}\\0\end{matrix}\right] = \left[\begin{matrix}1000\\0\\0\end{matrix}\right]\)

Now, we can solve the equation system using the solve function. We specify which variables we want to solve for by passing them as a list to the solve function. In this case, we want to solve for \(F_x\) and \(F_y\).

sol = sp.solve(eq, [F_x, F_y]) # solve the equation
sol

{F_x: 1000 - 250*cos(19*pi/90), F_y: -250*sin(19*pi/90)}

The output could be printed in a more readable format using the following code (for Jupyter Notebooks):

from IPython.display import Math
display(Math(sp.latex(sol)))

\(\displaystyle \left\{ F_{x} : 1000 - 250 \cos{\left(\frac{19 \pi}{90} \right)}, \ F_{y} : - 250 \sin{\left(\frac{19 \pi}{90} \right)}\right\}\)

The output of solve is a dictionary where the keys are the variables and the values are the solutions to the equations. We can access the solutions using the variable names as keys. Also note that the variables are of type sympy expression, so we can use them in further calculations or if they do not contain any symbols, we can convert them to a float using the evalf method.

# extract the values of F_x and F_y into a matrix and evaluate the matrix
sp.Matrix([sol[F_x], sol[F_y], 0]).evalf() 

\(\displaystyle \left[\begin{matrix}802.997311598319\\-153.915368831415\\0\end{matrix}\right]\)