Integrals of motion of a relativistic particle in 1 + 1 dimensions with coupled parameters

The eigenfunctions and eigenvalues of the integrals of motion γ and θ have been studied. An invariant form of motion was obtained for the derivatives of γ and θ, with respect to the proper time and velocity of a relativistic particle (RP). The integrals γ and θ were shown to be mutually expressible. Inverse values 1/E and 1/P were introduced for the energy and momentum of a free RP. A one-to-one correspondence of the RP energy and momentum was obtained. The properties of the γ integral expressed in terms of 1/E and 1/P were determined as a functional dependence γ = γ (1/E, 1/P). Forms of the motion equations depending on the γ and θ integrals were obtained using Lagrangian and Hamiltonian formalism. Based on the latter, a generalized integral of motion describing all types of motions in 1+1 dimensions was derived. Mutually expressive differential forms of RP motion were introduced.