DifferentialBases.jl

DifferentialBases.jl is a Julia package for computing Groebner bases in differential settings.

How to use DifferentialBases.jl

The following example is derived from a simple pendulum. Calling simple_pendulum() loads the relevant equations.

using DifferentialBases: differential_basis, simple_pendulum
ideal, derivatives, R, Rv = simple_pendulum()

Then we have the following system:

julia> println(ideal)
QQMPolyRingElem[x^2 + y^2 - 1]

julia> println(derivatives)
Dict{QQMPolyRingElem, QQMPolyRingElem}(x => u, l => dl, u => x*l, v => y*l - 1, y => v)println(derivatives)
Dict{Nemo.FqMPolyRingElem, Nemo.FqMPolyRingElem}(y => v, x => u, u => x*l, v => y*l + 100, l => dl), Multivariate polynomial ring in 6 variables over GF(101)

Calling differential_basis(ideal, derivatives, R, Rv, false, 0) results in the following Gröbner basis:

julia> differential_basis(ideal, derivatives, R, Rv, false, 0)
7-element Vector{QQMPolyRingElem}:
 u^2 + v^2 - y + l
 x*u + y*v
 x^2 + y^2 - 1
 y*u*v - x*v^2 + x*y - x*l
 y^2*u - x*y*v - u
 y^3 - y^2*l + v^2 - y + l
 x*y^2 - x*y*l + u*v

We can uncover interesting additional constraints from those equations i.e $ux + vy = 0$ describes that the motion of the masspoint of the pendulum is tangent to the rod of the pendulum. Furthermore we uncover that $u^2 + v^2 - y + 1 = 0$ which describes that the sum of potential and kinetic energy is constant i.e. energy is conserved. We learn that $dl= 3v$.

Installation

To install DifferentialBases.jl enter the Pkg REPL by pressing ] and execute

add DifferentialBases