Definition. A Hamiltonian symmetry of aclassical Hamiltonian system $(M,\omega,H)$ is a symplectomorphism of $M$ into itself that also preserves the fixed function $H$ defined on $M$. $\blacksquare$
More precisely, a symmetry of a Hamiltonian system on a symplectic manifold $(M,\omega)$ with a Hamiltonian function $H$ is a diffeomorphism $\varphi: M \to M$ such that:
1. $\varphi^* \omega = \omega$: The transformation preserves the symplectic form $\omega$.
2. $\varphi^* H = H$: The transformation preserves the Hamiltonian function $H$.
They preserve the phase space structure, the Hamiltonian function and the equations of motion (i.e. $\varphi_*(X_H)=X_H$). It can be concluded from 1. and 2. and the definition of Hamiltonian vector field.
If we have a 1-parameter family of Hamiltonian symmetries, they constitute a symplectic group action. It could be the case (or not) that this group action have a momentum map, in whose case they would be a Hamiltonian group action.
Observe that in the latter case, the family is generated by a vector field $X_A$ and $X_A(H)=0$, so
$$ X_A(H) = dH(X_A) = \omega(X_A,X_H)=\{A,H\}=0 $$where $\{\cdot,\cdot\}$ denotes the Poisson bracket.
Hence, using the Jacobi identity for the Poisson bracket, we have:
$$ [X_A,X_H] = X_{\{H,A\}}=0 $$So $X_A$ is a specially good case of symmetry of a distribution, the one generated by $X_H$.
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Author of the notes: Antonio J. Pan-Collantes
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