The connection between the Hamilton and the standard Lagrange formalism is established for a generic quantum field theory with vanishing vacuum expectation values of the fundamental fields. The effective actions in both formalisms are the same if and only if the fundamental fields and the momentum fields are related by the stationarity condition. These momentum fields in general differ from the canonical fields as defined via the effective action. By means of functional methods a systematic procedure is presented to identify the full correlation functions, which depend on the momentum fields, as functionals of those usually appearing in the standard Lagrange formalism. Whereas Lagrange correlation functions can be decomposed into tree diagrams, the decomposition of Hamilton correlation functions involves loop corrections similar to those arising in n-particle effective actions. To demonstrate the method we derive for theories with linearized interactions the propagators of composite auxiliary fields and the ones of the fundamental degrees of freedom. The formalism is then utilized in the case of Coulomb gauge Yang-Mills theory for which the relations between the two-point correlation functions of the transversal and longitudinal components of the conjugate momentum to the ones of the gauge field are given.