bipotential

Bipotential is a concept in convex analysis and non-smooth mechanics used to represent certain constitutive relations between dual variables, such as forces and motions, without relying on a single potential energy function. It centers on a function b that takes two arguments from dual spaces, typically denoted x and y*, and is designed so that the relation of interest can be characterized by an equality condition involving the duality pairing ⟨x, y*⟩.

Formally, let X and X* be a pair of dual vector spaces with pairing ⟨·,·⟩. A function b: X × X* → R is a bipotential for a relation G ⊆ X × X* if, for all x ∈ X and y* ∈ X*, b(x, y*) ≥ ⟨x, y*⟩, and the graph G can be recovered as G = { (x, y*) | b(x, y*) = ⟨x, y*⟩ }. In practice, b is convex and lower semicontinuous in each argument, and may be constructed as a sum or supremum of simpler convex functions. The graph of the associated relation consists of the pairs that achieve the equality.

Bipotentials generalize the classical potential framework and are especially useful for modeling dissipative or non-associated constitutive laws, such as certain friction, plasticity, and contact problems where a single energy potential does not suffice. They connect with monotone operator theory and convex analysis, providing a flexible tool for analysis and numerical treatment of non-smooth systems.

See also monotone operators, convex analysis, Fitzpatrick functions, and friction laws.