# Applications and limitations of the Hamilton-Jacobi formalism

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 $$H=T+V$$?

• I am not sure what exactly is meant by "Hamilton formalism" vs "Hamilton-Jacobi formalism", but the Hamilton-Jacobi equation is a first order PDE whose characteristics are given by Hamilton's equations. Solving them is mathematically equivalent. It is true that Hamilton's formalism (unlike Lagrange's) does not have a good generalization to gauge fields, but motion of particles in external fields is far from that. Dec 22, 2019 at 4:36
• Which page in J&S? Dec 22, 2019 at 5:51

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