Consider a symplectic manifold $(M, \omega)$. We have the following isomorphism $i_{\omega} : TM \to T*M$ which induces an isomorphism between $\theta : \wedge^{2}(TM) \to \wedge^{2}(T*M)$, this induces a map on the global sections, $\tilde{\theta} : \Gamma(M, \wedge^2(TM) ) \to \Gamma(M, \wedge^2(T*M))$
I have to show that for $F,G \in C^{\infty}(M)$, $\{ F, G\} := (\Lambda, dF \wedge dG) = \omega(X_F,X_G)$ where $\Lambda$ is the pullback of the symplectic form by $\tilde{\theta}$ and $(\cdot ,\cdot)$ is just the contraction of an element of $\Gamma(M, \wedge^2TM)$ and $\Gamma(M, \wedge^2T…

@BalarkaSen As of now I only have a physics motivation and that too has to be made formal. So in physics, these functions $f \in C^{\infty}(M)$ are called classical observables. Noether's theorem states that such conservation of such observables have a "symmetry law" associated with them.
To formalise this, one can use Poisson brackets. So consider the vector field $\{K, \cdot \}$ on a symplectic manifold $M$ with a hamiltonian, then this vector field actually preserves the flow generated by $H$.