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# Advantage of the Hamiltonian Formulation.

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

 (5)

which is to say,

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

Termed canonical, such transformations have the distinguishing property that they leave invariant the representation of the antisymmetric tensor

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

Indeed, letting

 and (7)

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, let

be a vector tangent to the phasespace trajectory relative to the given ( ) and the new ( ) coordinates respectively.

a) Show that

 and (9)

b) Point out why

c) Show that

implies Hamilton's equations of motion, Eq.(3)-(4).

d) Show that the introduction of the new coordinates into ,

yields Eq.(6), Hamilton's equations relative to the new coordinates .

Next: The Dynamical Phase Up: Laser-driven particle mechanics Previous: Lagrangian and Hamiltonian Formulation   Contents
Ulrich Gerlach 2005-11-07