Moser's trick

In differential geometry, a branch of mathematics, the Moser's trick (or Moser's argument) is a method to relate two differential forms ${\displaystyle \alpha _{0}}$ and ${\displaystyle \alpha _{1}}$ on a smooth manifold by a diffeomorphism ${\displaystyle \psi \in \mathrm {Diff} (M)}$ such that ${\displaystyle \psi ^{*}\alpha _{1}=\alpha _{0}}$, provided that one can find a family of vector fields satisfying a certain ODE.

More generally, the argument holds for a family ${\displaystyle \{\alpha _{t}\}_{t\in [0,1]}}$ and produce an entire isotopy ${\displaystyle \psi _{t}}$ such that ${\displaystyle \psi _{t}^{*}\alpha _{t}=\alpha _{0}}$.

It was originally given by Jürgen Moser in 1965 to check when two volume forms are equivalent, but its main applications are in symplectic geometry. It is the standard argument for the modern proof of Darboux's theorem, as well as for the proof of Darboux-Weinstein theorem and other normal form results.