Moser's trick
In differential geometry, a branch of mathematics, Moser's trick (or Moser's argument) is a method to relate two differential forms α 0 {\displaystyle \alpha _{0}} and α 1 {\displaystyle \alpha _{1}} on a smooth manifold by a diffeomorphism ψ ∈ D i f f ( M ) {\displaystyle \psi \in \mathrm {Diff} (M)} such that ψ ∗ α 1 = α 0 {\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 { α t } t ∈ [ 0 , 1 ] {\displaystyle \{\alpha _{t}\}_{t\in [0,1]}} and produce an entire isotopy ψ t {\displaystyle \psi _{t}} such that ψ t ∗ α t = α 0 {\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.