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No Runge-Kutta method can be energy preserving for all Hamiltonian systems.But for problems in which the Hamiltonian is a polynomial, the Averaged VectorField (AVF) method can be interpreted as a Runge-Kutta method whose weights$b_i$ and abscissae $c_i$ represent a quadrature rule of degree at least thatof the Hamiltonian. We prove that when the number of stages is minimal, theRunge-Kutta scheme must in fact be identical to the AVF scheme.

The minimal stage, energy preserving Runge-Kutta method for polynomial Hamiltonian systems is the Averaged Vector Field method