0) Let us for simplicity assume that the Legendre transformation from Lagrangian to Hamiltonian formulation is regular.
1) The Lagrangian action SL[q]:=∫dt L is invariant under the infinite-dimensional group of diffeomorphisms of the n-dimensional (generalized) position space M.
2) The Hamiltonian action SH[q,p]:=∫dt(pi˙qi−H) is invariant (up to boundary terms) under the infinite-dimensional group of symplectomorphisms of the 2n-dimensional phase space T∗M.
3) The group of diffeomorphisms of position space can be prolonged onto a subgroup inside the group of symplectomorphisms. (But the group of symplectomorphism is much bigger.) The above is phrased in the active picture. We can also rephrase it in the passive picture of coordinate transformations. Then we can prolong a coordinate transformation
qi ⟶ q′j = q′j(q)
into the cotangent bundle T∗M in the standard fashion
pi = p′j∂q′j∂qi .
It is not hard to check that the symplectic two-form becomes invariant
dp′j∧dq′j = dpi∧dqi
(which corresponds to a symplectomorphism in the active picture).
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