Based on a novel extended version of the Lagrange equations for systems containing non-material volumes, the nonlinear equations of motion for cantilever pipe systems conveying fluid are deduced. An alternative to existing methods utilizing Newtonian balance equations or Hamilton's principle is thus provided. The application of the extended Lagrange equations in combination with a Ritz method directly results in a set of nonlinear ordinary differential equations of motion, as opposed to the methods of derivation previously published, which result in partial differential equations. The pipe is modeled as a Euler elastica, where large deflections are considered without order-of-magnitude assumptions. For the equations of motion, a dimensional reduction with arbitrary order of approximation is introduced afterwards and compared with existing lower-order formulations from the literature. The effects of nonlinearities in the equations of motion are studied numerically. The numerical solutions of the extended Lagrange equations of the cantilever pipe system are compared with a second approach based on discrete masses and modeled in the framework of the multibody software HOTINT/MBS. Instability phenomena for an increasing number of discrete masses are presented and convergence towards the solution for pipes conveying fluid is shown.