Let us try to get more intuition from these equations. It is helpful to consider a simple physical analogy with a particle of mass \( m \) moving in a viscous medium with drag coefficient \( \mu \) and potential \( E(\mathbf{w}) \). If we denote the particle's position by \( \mathbf{w} \), then its motion is described by $$ m {d^2 \mathbf{w} \over dt^2} + \mu {d \mathbf{w} \over dt }= -\nabla_w E(\mathbf{w}). $$ We can discretize this equation in the usual way to get $$ m { \mathbf{w}_{t+\Delta t}-2 \mathbf{w}_{t} +\mathbf{w}_{t-\Delta t} \over (\Delta t)^2}+\mu {\mathbf{w}_{t+\Delta t}- \mathbf{w}_{t} \over \Delta t} = -\nabla_w E(\mathbf{w}). $$ Rearranging this equation, we can rewrite this as $$ \Delta \mathbf{w}_{t +\Delta t}= - { (\Delta t)^2 \over m +\mu \Delta t} \nabla_w E(\mathbf{w})+ {m \over m +\mu \Delta t} \Delta \mathbf{w}_t. $$