Mechanization of a scalar field theory in 1+1 dimensions: Bogomol'nyi-Prasad-Sommerfeld mechanical kinks and their scattering

We present an updated version of a general-purpose collective coordinate model that aims to fully map out the dynamics of a single scalar field in $1+1$ dimensions. This is achieved by a procedure that we call a mechanization, in which we reduce the infinite number of degrees of freedom down to a finite and controllable number by chopping the field into flat segments connected via joints. In this paper we introduce two new ingredients to our procedure. The first is a manifestly Bogomol'nyi-Prasad-Sommerfeld (BPS) mechanization in which BPS mechanical kinks saturate the same bound on energy as their field-theoretic progenitors. The second is allowing the joints to switch, leading to an extended concept of the effective Lagrangian, through which we describe direct collisions of mechanical kinks and antikinks.