Induced dynamics

Induced dynamics is defined as dynamics of real zeros with respect to $x$ of equation $f(q_1-x,\ldots,q_N-x,p_1,\ldots,p_N)=0$, where $f$ is a function, and $q_i$ and $p_j$ are canonical variables obeying some (free) evolution. Identifying zero level lines with the world lines of particles, we show that the resulting dynamical system demonstrates highly nontrivial collisions of particles. In particular, induced dynamical systems can describe such ``quantum'' effects as bound states and creation/annihilation of particles, both in nonrelativistic and relativistic cases. On the other side, induced dynamical systems inherit properties of the $(p,q)$-systems being Hamiltonian and Liouville integrable.