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In the present work, we propose an Action Principle for Action-dependentLagrangians by generalizing the Herglotz variational problem for severalindependent variables. This Action Principle enables us to formulate Lagrangiandensities for non-conservative fields. In special, from a Lagrangian dependinglinearly on the Action, we obtain a generalized Einstein's field equations fora non-conservative gravity and analyze some consequences of their solutions tocosmology and gravitational waves. We show that the non-conservative part ofthe field equations depends on a constant cosmological four-vector. Dependingon this four-vector, the theory displays damped/amplified gravitational wavesand an accelerating Universe without dark energy.

From an Action Principle for Action-dependent Lagrangians toward non-conservative Gravity: accelerating Universe without dark energy