Correspondence of Manifestly and Implicit Lorentz-Covariant Treatments of the Relativistic Lagrangian Mechanics: Euler–Lagrange Covariant Equations Revisited

The equations of manifestly Lorentz-covariant Lagrangian mechanics are derived from implicitly Lorentz-covariant Euler–Lagrange equations. The resulting equations coincide with the Kalman equations, originally derived by a different way. However, the inverse transformation of the Kalman equations into Euler–Lagrange ones requires some restrictions of the manifestly covariant Lagrangian and the variational procedure. The Kalman equations are modified for one-to-one correspondence with implicitly covariant treatment of the mechanics. It is shown that the use of proper time as a non-variable parameter does not satisfy the correspondence principle of manifestly and implicit treatments.