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Achieving a target kinetic energy by varying the forcing strength...

Energy dissipates in a crossing time which is \tau=\frac{l}{v} which means that the specific kinetic energy dissipation rate is \approx \frac{1}{2} \frac{v^{3}}{l}. If we have a given accleration field a, it should inject specific kinetic energy at a rate proportional to a v

So assume we have a system

\dot{v^2} = \beta a v - \alpha \frac{1}{2} \frac{v^3}{l}

where \alpha and \beta are unknown constants \approx 1

and we want to adjust a so that {v^2} = v_0^2 or so that \beta a v_0 - \alpha \frac{1}{2} \frac{v_0^3}{l} \rightarrow a = \frac{\alpha}{\beta} \frac{v_0^2}{2l}

we could start by assuming that \alpha=1 and \beta=1 and set a = \frac{\alpha}{\beta} \frac{v_0^2}{2l} and then run until we reach a steady state energy. At that point we can evaluate \frac{\alpha}{\beta} = \frac{2la}{v_s^2} and then adjust the acceleration a \rightarrow a \frac{v_0^2}{v_s^2}

The kinetic energy could potentially be widely varying on short time scales - which makes it tricky to determine a steady state energy. Additionally any change in the forcing will take of order a dynamical time for the system to relax… So each adjustment to the forcing should be followed by 1 dynamical time of allowing the system to relax - followed by a half dynamical time of averaging. The initial state should probably be allowed to evolve for 2 dynamical times before averaging.

astrobear

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Achieving a target kinetic energy by varying the forcing strength...

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