Examples¶

In the following sections we give few examples of what can be done with this module.

Dimensional analysis¶

We will start from Newton’s second law

\[m a = F\]

where \(m, a\) and \(F\) are the mass, the acceleration and the force respectively. Knowing the dimensions of \(m\) (\(M\)) and \(a\) (\(L T^{-2}\)), we will determine the dimension of \(F\); obviously we will find that it is a force: \(M L T^{-2}\).

From there we will use the expression of the gravitational force between the particle of mass \(m\) and the body of mass \(M\), at a distance \(r\)

\[F = \frac{G m M}{r^2}\]

to determine the dimension of the Newton’s constant \(G\). The result should be \(L^3 M^{-1} T^{-2}\).

>>> from sympy import symbols
>>> from sympy.physics.units.systems import SI
>>> from sympy.physics.units import length, mass, acceleration, force
>>> from sympy.physics.units import gravitational_constant as G
>>> from sympy.physics.units.systems.si import dimsys_SI
>>> F = mass*acceleration
>>> F
Dimension(acceleration*mass)
>>> dimsys_SI.get_dimensional_dependencies(F)
{'length': 1, 'mass': 1, 'time': -2}
>>> dimsys_SI.get_dimensional_dependencies(force)
{'length': 1, 'mass': 1, 'time': -2}

Dimensions cannot compared directly, even if in the SI convention they are the same:

>>> F == force
False

Dimension system objects provide a way to test the equivalence of dimensions:

>>> dimsys_SI.equivalent_dims(F, force)
True

>>> m1, m2, r = symbols("m1 m2 r")
>>> grav_eq = G * m1 * m2 / r**2
>>> F2 = grav_eq.subs({m1: mass, m2: mass, r: length, G: G.dimension})
>>> F2  
Dimension(mass*length*time**-2)
>>> F2.get_dimensional_dependencies()  
{'length': 1, 'mass': 1, 'time': -2}

Note that one should first solve the equation, and then substitute with the dimensions.

Equation with quantities¶

Using Kepler’s third law

\[\frac{T^2}{a^3} = \frac{4 \pi^2}{GM}\]

we can find the Venus orbital period using the known values for the other variables (taken from Wikipedia). The result should be 224.701 days.

>>> from sympy import solve, symbols, pi, Eq
>>> from sympy.physics.units import Quantity, length, mass
>>> from sympy.physics.units import day, gravitational_constant as G
>>> from sympy.physics.units import meter, kilogram
>>> T = symbols("T")
>>> a = Quantity("venus_a")

Specify the dimension and scale in SI units:

>>> SI.set_quantity_dimension(a, length)
>>> SI.set_quantity_scale_factor(a, 108208000e3*meter)

Add the solar mass as quantity:

>>> M = Quantity("solar_mass")
>>> SI.set_quantity_dimension(M, mass)
>>> SI.set_quantity_scale_factor(M, 1.9891e30*kilogram)

Now Kepler’s law:

>>> eq = Eq(T**2 / a**3, 4*pi**2 / G / M)
>>> eq
Eq(T**2/venus_a**3, 4*pi**2/(gravitational_constant*solar_mass))
>>> q = solve(eq, T)[1]
>>> q
2*pi*venus_a**(3/2)/(sqrt(gravitational_constant)*sqrt(solar_mass))

To convert to days, use the convert_to function (and possibly approximate the outcoming result):

>>> from sympy.physics.units import convert_to
>>> convert_to(q, day)
71.5112118495813*pi*day
>>> convert_to(q, day).n()
224.659097795948*day

We could also have the solar mass and the day as units coming from the astrophysical system, but we wanted to show how to create a unit that one needs.

We can see in this example that intermediate dimensions can be ill-defined, such as sqrt(G), but one should check that the final result - when all dimensions are combined - is well defined.

Philosophy behind unit systems

Dimensions and dimension systems