One of the chief virtues of the Lagrangian equations of motion is that they remain invariant under an arbitrary point transformation

Hamilton's equations of motion not only share this virtue but they take it to a higher level: they are invariant under certain more general transformations

which is to say,

Here is the transformed superhamiltonian obtained from with the help of Eq.(5):

A sufficient condition for a transformation, Eq.(5), to be canonical is that there exist a scalar function of and ,

Indeed, letting

one finds that

zero | ||

(8) |

*Problem 1*: Prove that the representation invariance of Eq.(8) implies the local existence of a scalar whose gradients yield Eq.(7).Thus the existence of a scalar is both a necessary and a sufficient condition for the invariance expressed by Eq.(8). It also is a sufficient condition for the invariance of Hamilton's equations of motion.

*Problem 2*: Show that a transformation such as the one given by Eq.(7) transforms the given euations of motion, Eqs.(3)-(4), into the same form, and given by Eq.(6).__Discussion:__Taking advantage of the chain rule, leta) Show that

and(9)

b) Point out whyd) Show that the introduction of the new coordinates into ,