When deriving the symplectic condition for a canonical transformation, this operation applied to a column matrix M, containing n generalized coordinates and n generalized momenta, is set equal to the product of a two n-by-two n block matrix times the partial of the Hamiltonian with respect to M.
When deriving the symplectic condition for a canonical transformation, this operation applied to a column matrix M, containing n generalized coordinates and n generalized momenta, is set equal to the product of a two n-by-two n block matrix times the partial of the Hamiltonian with respect to M. This operation is applied to the generating function in a canonical transformation. Adding this operation applied to an arbitrary function of the generalized coordinates to a Lagrangian gives another valid Lagrangian that yields the same equations of motion. This operation is applied to the generalized coordinates and momenta on the left side of Hamilton's equations. It is also applied to the derivative of the Lagrangian with respect to the generalized velocity in the Euler-Lagrange equation. Applying this operation to action gives the Lagrangian. For 10 points, name this operation often indicated by putting a single dot over a variable, which can be applied to position to give velocity.
When deriving the symplectic condition for a canonical transformation, this operation applied to a column matrix M, containing n generalized coordinates and n generalized momenta, is set equal to the product of a two n-by-two n block matrix times the partial of the Hamiltonian with respect to M.
time derivative [or partial derivative with respect to  time; or big/little d over/by big/little dt; prompt on derivative]