Source Of Complexity

Lesson

Introduction
Matter of scale
Variability
Non-linearity
Introduction to differential equations
Governing equations
Different approaches
- Navier-Stokes equations
- Use of supercomputer
- Symmetries of the equations of evolution
- Self similar organization in environment
- Self Similarity
Exercises
To go further
Conclusion

Symmetries of the equations of evolution

Scale invariance : the Navier-Stokes are invariant under isotropic "zoom"

What does it mean symmetry of differential equations ? This means a change of variables which may relate to the dependent or independent variables which leaves unchanged the equation. In other words, the equation is written in the same way in term of old and new variables.

$x \rightarrow \lambda^{-1} x' \quad \quad \quad t \rightarrow \lambda^{\gamma-1} t' \\ \quad\\ p \rightarrow p' \quad \quad \quad V \rightarrow \lambda^{- \gamma} V' \\ \quad\\ \nu \rightarrow \lambda^{-1- \gamma '} \nu' \quad \quad \quad \rho \rightarrow \lambda^{-\gamma} \rho' \\ \quad\\ f \rightarrow \lambda^{1 - \gamma -\gamma '} f'\\ \quad\\ \text{Where} \gamma, \gamma' \text{are an arbitrary exponents}$

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