cinfsymmetries 1.2.3

Geometric study of differential equations (Lie symmetries, lambda-symmetries, Cinf-structures, solvable structures)

cinfsymmetries

A Python library for the geometric study of differential equations, bridging the gap from sympy to advanced concepts such as Lie symmetries, $\lambda$-symmetries, $C^{\infty}$-structures, and solvable structures.

Originally inspired by functionality from Maple packages for Differential Geometry.

Features

• VectorFields and Distributions: Define symbolic vector fields over an arbitrary set of variables.
• Lie Brackets: Compute Lie Brackets of vector fields analytically.
• Involutivity: Check if a distribution is involutive.
• Symmetries: Check if a given vector field is a symmetry of a distribution.
• C^inf-Symmetries: Specialized check for Generalized symmetries including $\lambda$-symmetries.
• Prolongations: Automatically prolong vector fields to higher-order jet spaces (any number of independent and dependent variables).
• Differential Forms: Support for $k$-forms, exterior derivative, wedge product, interior product, and unified Lie derivatives ($\mathcal{L}_X$).
• Symmetry Search: Automatically derive determining equations for Lie and lambda-symmetries of systems of ODEs or PDEs.
• Pfaffian Systems: Compute dual distributions from 1-forms and vice versa.
• Invariants: Calculate fundamental invariants of a vector field.

Documentation

For a detailed guide on how to use the library, including how to create vector fields and check for symmetries, please refer to the Manual.

Installation

pip install -e .

Quick Start

import sympy as sp
from cinfsymmetries import VectorField, LieBracket, Distribution

x, y = sp.symbols('x y')

# Define Vector Fields X = d/dx, Y = x*d/dy
X = VectorField({x: 1})
Y = VectorField({y: x})

# Lie Bracket [X, Y] = d/dy
bracket = LieBracket(X, Y)
print(bracket) # Outputs: (1)*d_y

# Create an ODE-like rank 1 distribution (spanned by X + Y)
D = Distribution([X + Y])

# Check if X is a symmetry of D
# [X, X+Y] = [X, Y] = d/dy, which is NOT in the span of (d/dx + x*d/dy), so False.
print(D.check_symmetry(X))

# Get invariants of X
print(X.get_invariants()) # No non-constant invariants in 1D