On shell and off shell

If we have a configuration space which is much larger than the space of valid solutions, then, at least for theoretical physicists, those points (i.e. configurations) which are valid are called on shell and those which are not are called off shell.

For instance, in classical mechanics in the action formulation, extremal solutions to the variational principle are on shell and the Euler-Lagrange equations are on shell equations (i.e. they do not hold off shell). Noether's theorem is also another on shell theorem.