Conjugate symplecticity of second-order linear multi-step methods

We review the two different approaches for symplecticity of linear multi-step methods (LMSM) by Eirola and Sanz-Serna, Ge and Feng, and by Feng and Tang, Hairer and Leone, respectively, and give a numerical example between these two approaches. We prove that in the conjugate relation G3λτ∘G1τ=G2τ∘G3λτ with G1τ and G3τ being LMSMs, if G2τ is symplectic, then the B-series error expansions of G1τ, G2τ and G3τ of the form Gτ(Z)=∑i=0+∞(τi/i!)Z[i]+τs+1A1+τs+2A2+τs+3A3+τs+4A4+O(τs+5) are equal to those of trapezoid, mid-point and Euler forward schemes up to a parameter θ (completely the same when θ=1), respectively, this also partially solves a problem due to Hairer. In particular we indicate that the second-order symmetric leap-frog scheme Z2=Z0+2τJ-1∇H(Z1) cannot be conjugate-symplectic via another LMSM.