# Solving symbolic equations with SymPy

SymPy is a Python library for symbolic mathematics. It is one of the layers used in SageMath, the free open-source alternative to Maple/Mathematica/Matlab. When you have simple but big calculations that are tedious to be solved by hand, feed them to SymPy, and at least you can be sure it will make no calculation mistake ;-)

The basic functionalities of SymPy are expansion/factorization/simplification of symbolic expressions, limit calculations, differentiation, integration, algebraic equation solving, and some simple differential equation solving. In what follows, we will use it to solve a system of quadratic equations.

## Solving a system of quadratic equations

Consider the following system of quadratic equations:

(These notations come from physics, where these equations are used to calculate
the zero-tilting moment point.)
First, declare variables using the `var()` construct:

from sympy import var Ldy, Ldz = var('Ldy Ldz') g, x, y, z = var('g x y z') xZ, yZ, zZ = var('xZ yZ zZ') xdd, ydd, zdd = var('xdd ydd zdd')

You can then use them directly as Python variables, performing all common
operations such as addition or multiplication. Next, define the expressions to
be zeroed and pass them to the `solve()` function:

from sympy import solve E1 = z * xdd + (xZ - x) * (g + zdd) E2 = z * ydd + (yZ - y) * (g + zdd) - Ldy E3 = -y * xdd + x * ydd - zZ * (g + zdd) + Ldz sols = solve([E1, E2, E3], [xdd, ydd, Ldy]) print "xdd = ", (sols[xdd]).factor() print "ydd = ", (sols[ydd]).factor() print "Ldy = ", (sols[Ldy]).factor()

The second argument of `solve()` indicates the set of "output" variables.
Indeed, we have three equations for twelve variables. Each equation can be used
to express one variable as function of the others. Thus, we can pick three
variables and express them as functions of the remaining nine. This is what we
do here with \(\ddot{x}\), \(\ddot{y}\) and \(\dot{L}_y\).

## Reading the solution

The output from this code is:

xdd = (g + zdd)*(x - xZ)/z ydd = (-Ldz*z + g*x*y - g*xZ*y + g*z*zZ + x*y*zdd - xZ*y*zdd + z*zZ*zdd)/(x*z) Ldy = (-Ldz*z + g*x*yZ - g*xZ*y + g*z*zZ + x*yZ*zdd - xZ*y*zdd + z*zZ*zdd)/x

Which corresponds to the solution:

SymPy has some routines to make formulas more palatable. For instance, it can
print `sympy.Expr` objects (expressions) in \(\LaTeX\):

from sympy import latex print "\\begin{array}{rcl}" print "xdd & = & %s \\\\" % latex((sols[xdd]).factor()) print "ydd & = & %s \\\\" % latex((sols[ydd]).factor()) print "Ldy & = & %s" % latex((sols[Ldy]).factor()) print "\\end{array}"

If you use IPython's QTConsole, you can even render \(\LaTeX\) formulas directly in your console. See Printing from the SymPy documentation for details.