this is for holding javascript data
Ghlen Livid added vec_x_t_Delta_t__1.tex
almost 8 years ago
Commit id: a7818da66dca14c1d14964e1bd91f78bd776e664
deletions | additions
diff --git a/vec_x_t_Delta_t__1.tex b/vec_x_t_Delta_t__1.tex
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+++ b/vec_x_t_Delta_t__1.tex
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$$\vec{x}(t + \Delta t) = \vec{x}(t) + \vec{v}(t) \Delta t + \frac{1}{2} \vec{a}(t) \Delta t^2 + \frac{1}{6} \vec{b}(t) \Delta t^3 + O (\Delta t^4) $$
$$\vec{v}(t + \Delta t) = \vec{v}(t) + \frac{1}{2}(\vec{a}(t) + \vec{a}(t + \Delta t)) \Delta t + \frac{1}{4}(\vec{b}(t) - \vec{b}(t+\Delta t)) \Delta t^2 + O(\Delta t^3)$$
Then
$$\vec{x}(t) = \vec{x}(t-\Delta t) + \vec{v}(t -\Delta t) \Delta t + \frac{1}{2} \vec{a}(t-\Delta t) \Delta t^2 + \frac{1}{6} \vec{b}(t-\Delta t) \Delta t^3 + O (\Delta t^4) $$
$$\vec{v}(t) = \vec{v}(t-\Delta t) + \frac{1}{2}(\vec{a}(t-\Delta t) + \vec{a}(t)) \Delta t + \frac{1}{4}(\vec{b}(t-\Delta t) - \vec{b}(t)) \Delta t^2 + O(\Delta t^3)$$