Standard Nearest Neighbor Discretizations of Klein-Gordon Models Cannot Preserve Both Energy and Linear Momentum

We consider nonlinear Klein-Gordon wave equations and illustrate that standard discretizations thereof (involving nearest neighbors) may preserve either standardly defined linear momentum or total energy but not both. This has a variety of intriguing implications for the ``non-potential'' discretizations that preserve only the linear momentum, such as the self-accelerating or self-decelerating motion of coherent structures such as discrete kinks in these nonlinear lattices.