Hamiltonian symmetry

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 $M$. $◼$

More precisely, a symmetry of a Hamiltonian system on a symplectic manifold $(M,\omega )$ with a Hamiltonian function $H$ is a diffeomorphism $\phi :M\to M$ such that:

1. ${\phi }^{\ast }\omega =\omega$: The transformation preserves the symplectic form $\omega$.

2. ${\phi }^{\ast }H=H$: The transformation preserves the Hamiltonian function $H$.

They preserve the phase space structure, the Hamiltonian function and the equations of motion (i.e. ${\phi }_{\ast }(X_H)=X_H$). It can be concluded from 1. and 2. and the definition of Hamiltonian vector field.

1-parameter families

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

where $\{\cdot ,\cdot \}$ denotes the Poisson bracket.

Hence, using the Jacobi identity for the Poisson bracket, we have:

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

antonio.pan@uca.es