Applications and limitations of the Hamilton-Jacobi formalism

Question

It was my understanding that the Hamiltonian formalism was inadequate to describe systems that are invariant under time reparametrization or that have gauge symmetries.

However, I see in Classical Dynamics by Jorge V. José and Eugene J. Saletan, that both a relativistic particle and a particle under an electromagnetic potential are described using the Hamilton-Jacobi formalism, dealing the right equations of motion.

I wonder why does this work: Are there systems that can be treated with by Hamilton-Jacobi formalism but yield false results when treated by Hamilton? Is there a way to adequately treat systems with the mentioned invariances through Hamiltonian mechanics? If so, are their Hamiltonians always of the form $H=T+V$?

Answer 1

1. Hamiltonian formalism also works for gauge systems, although one has to introduce constraints (and possibly the corresponding Lagrange multipliers), see e.g. Ref. 1. For the relativistic point particle, see e.g. this Phys.SE post.

2. In fact the Hamilton-Jacobi equation is derived from the Hamiltonian formalism, not the other way around.

3. The Hamiltonian $H$ is not always of the form $T+U$. See e.g. this Phys.SE post for the corresponding Lagrangian question.

References:

1. M. Henneaux & C. Teitelboim, Quantization of Gauge Systems, 1994.