A remark on the Hamiltonian form of the magnetic-field-line equations

It is frequently assumed that a solenoidal vector field u in a simple toroid may be represented in the form of u=∇Ψx∇φ+∇χx∇θ with appropriate toroidal angle φ, poloidal angle θ, and two fluxes Ψ and χ. This expression allows one to write the field-line equations in a Hamiltonian form [Phys. Fluids 26, 1288 (1983)]. It is proved on a rigorous mathematical basis that every solenoidal field in an arbitrary smooth simple toroidal domain can be written in the above form