Velocity only

Let us focus on the velocity only. Discretizing and using the simplest possible approximation for the derivative, we have Euler's forward method for the updated velocity at a time step $i+1$ given by

$$v_{i+1}=v_i+\Delta t \frac{dv}{dt}_{\vert_{v=v_i}}=v_i+\Delta t\left(F_i-\eta v_i-x_i\right).$$

Defining a function

$$h_i(x_i,v_i,F_i)=v_i+\Delta t\left(F_i-\eta v_i-x_i\right),$$

we have

$$v_{i+1}=h_i(x_i,v_i,F_i).$$

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