In Hamiltonian Monte Carlo, what is the physical role of the 'momentum' variable during the simulation of trajectories across the parameter space?

In Hamiltonian Monte Carlo, the momentum variable acts as an auxiliary velocity that allows the simulation to traverse the target probability distribution by following the laws of classical mechanics. The algorithm treats the target parameter space as a physical landscape where the probability density is mapped to a potential energy surface; high-probability regions represent deep valleys, and low-probability regions represent high hills. The momentum variable is introduced to provide ....