The topological charge or winding number of a solution φ φ is N = 1 2 π π ∫ ∫ R d φ φ = 1 2 π π [ φ φ ( x = ∞ ∞ , t ) − − φ φ ( x = − − ∞ ∞ , t ) ] . The energy of a solution φ φ is E = ∫ ∫ R d x ( 1 2 ( φ φ t 2 + φ φ x 2 ) + m 2 ( 1 − − cos ⁡ ⁡ φ φ ) ) where a constant energy density has been added so that the potential is non-negative. With it the first two terms in the Taylor expansion of the potential coincide with the potential of a massive scalar field, as mentioned in the naming section; the higher order terms can be thought of as interactions.