Symplectic bipotentials

The paper defines a symplectic bipotential b̂ with axioms ensuring pointwise nonnegativity b̂(żI,ż) ≥ ω(żI,ż) and an equality–subdifferential equivalence, then states the SBEN functional Π(z) = ∫{b̂(ż−XH, ż) − ω(ż−XH, ż)} dt has minimum value 0 over admissible paths, with the ‘natural evolution’ realizing the minimum. These are precisely the ingredients used in the candidate solution: Step 1 invokes axiom (b) to deduce Π ≥ 0; Step 2 invokes axiom (c) to characterize Π = 0 via the symplectic subdifferential conditions; Step 3 notes minimizers are exactly those paths satisfying the pointwise extremality almost everywhere. This mirrors the paper’s statement and the standard SBEN argument adapted from the Fenchel/polar setting to the bipotential setting. The paper’s core claim (Eq. (24)) and the axioms (a)–(c) are aligned with the model’s reasoning, so both are correct and essentially the same proof in different words .