The authors show that covariant field theory for sections of a fiber bundle

$\pi :E\to M$ lifts in a natural way to the bundle of vertically adapted linear frames

${L}_{\pi}E$. The advantage gained by this reformulation is that it allows to utilize the

$n$-symplectic geometry that is supported on

${L}_{\pi}E$. On

${L}_{\pi}E$ the canonical soldering 1-forms play the role of the contact structure of

${J}^{1}\pi $. A lifted Lagrangian is used to construct modified soldering 1-forms, which are referred to as Cartan-Hamilton-Poincaré 1-forms. These forms on

${L}_{\pi}E$ can be used to define the standard Cartan-Hamilton-Poincaré

$m$-form on

${J}^{1}\pi $. Then, the generalized Hamilton-Jacobi and Hamilton equations on

${L}_{\pi}E$ are derived. Finally, as special cases of the generalized Hamilton-Jacobi equations, the Carathéodory-Rund and the de Donder-Weyl Hamilton-Jacobi equations are obtained.