About the meaning "on-shell" vs "off-shell" in Hamiltonian mechanics

[cianfa72]

TL;DR

About the meaning "on-shell" vs "off-shell" in the context of Hamiltonian/Lagrangian mechanics

In the derivation of Hamiltonian mechanics, the concept of "on-shell" vs "off-shell" is involved in the calculation.

I searched it for like off-shelf, however it seems it makes sense in the context of four-momentum in special relativity.

What is the meaning of that concept in the context of Hamiltonian/Lagrangian mechanics ? Thanks.

 

[anuttarasammyak]

You might have already read https://en.wikipedia.org/wiki/On_shell_and_off_shell

 

[jbergman]

TL;DR Summary: About the meaning "on-shell" vs "off-shell" in the context of Hamiltonian/Lagrangian mechanics

In the derivation of Hamiltonian mechanics, the concept of "on-shell" vs "off-shell" is involved in the calculation.

I searched it for like off-shelf, however it seems it makes sense in the context of four-momentum in special relativity.

What is the meaning of that concept in the context of Hamiltonian/Lagrangian mechanics ? Thanks.

I'm not an expert on Hamiltonian Mechanics but as I understand it, we have a phase space with coordinates $(q_i,p_i)$. A trajectory is a path through phase space parametrized by time.

Such a path is considered on shell if $\frac{dq_i}{dt} = \frac{\partial H}{\partial p_i}$.

With a typical Hamiltonian of $H = m/2\sum p_i^2 + V(q_i)$ this just reduces to $\frac{dq_i}{dt} = p_i/m = \dot{q_i}$.

TLDR, On shell is restricted to paths where q and p are related via the Hamiltonian and off-shell refers to an arbitrary path through phase space where q and p aren't necessarily related and are treated as independent variables.

 

[cianfa72]

Ah ok, so basically the "time evolution" of an "on-shell" trajectory through phase space is "constrained" from Hamilton's equations.

 

[ohwilleke]

Ah ok, so basically the "time evolution" of an "on-shell" trajectory through phase space is "constrained" from Hamilton's equations.

Mostly on-shell and off-shell are used to distinguish between different kinds of path integral evaluations in quantum mechanics. In quantum mechanics, paths that would be forbidden by conservation laws are still possible so long as the end point where an observation if made complies with the relevant conservation laws.

In particular, off-shell analysis considers "virtual" paths from one state to another, sometimes involving "virtual particles." One of the most common cases where an off-shell analysis is needed is in quantum tunneling situations, and in W boson transitions where the system lacks the mass-energy to create a real, on-shell W boson in an intermediate step.