Abstract: A relativistic Hamiltonian-Lagrangian formalism for a composite system submitted to conservative and non-conservative forces is developed. A block descending an incline with a frictional force, mechanical energy dissipation process, is described, obtaining an Euler-Lagrange equation including a Rayleigh's dissipation function. A cannonball rising on an incline, process evolving with mechanical energy production, is described by an Euler-Lagrange equation including a Gibbs' production function, with a chemical origin force. A matrix four-vector mechanical equation, considering processes' mechanical and phenomenological aspects, is postulated. This relativistic Hamiltonian-Lagrangian four-vector formalism complements the Einstein-Minkowski-Lorentz four-vector fundamental equation formalism. By considering a process' mechanical and thermodynamic description, temporal evolution equations, relating process' Hamiltonian (mechanical energy) evolution and the involved thermodynamic potentials (entropy of the universe, Helmholtz free energy, Gibbs free enthalpy) variations, are obtained.

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