I Lagrangian with generalized positions

[Dale]

Then I'd call $H$ the total energy of the particle, because then it's the conserved quantity related to the conserved quantity from time-translation invariance in the sense of Noether's theorem.

I would also. To me, the only reason that we are interested in energy is because it is conserved. So if there is a conserved quantity and a non-conserved quantity that both have claim to being "energy" then I will always pick the conserved one.

 

[Heidi]

And when there is no hamiltonian at all to describe the system is there still a notion of energy?

i read this:
The Hamiltonian method describes and keeps track of the state of the system of particles and fields at a given time. In the new theory, there are no field variables, and every radiative process depends on contributions from the future as well as from the past! One is forced to view the entire process from start to finish. The only existing classical approach of this kind for particles makes use of the principle of least action, and Feynman’s thesis project was to develop and generalize this approach so that it could be used to formulate the Wheeler–Feynman theory (a theory possessing an action, but without a Hamiltonian). If successful, he should then try to find a method to quantize the new theory.

 

[vanhees71]

Well, the Feynman-Wheeler absorber theory was not that an success. Particularly there has never been found a way to quantize it. A very nice treatment of various theories on radiation, including the absorber theory (modulo some typos) can be found in

A. O. Barut, Electrodynamics and classical theory of fields and particles, Dover Publications (1980)