this is for holding javascript data
Ghlen Livid Deleted File
almost 8 years ago
Commit id: e5b8bebb9c98d1f1792c2505da994af67735b9d4
deletions | additions
diff --git a/vec_x_t_Delta_t__1.tex b/vec_x_t_Delta_t__1.tex
deleted file mode 100644
index eb667d0..0000000
--- a/vec_x_t_Delta_t__1.tex
+++ /dev/null
...
$$\vec{x}(t + \Delta t) = \vec{x}(t) + \vec{v}(t) \Delta t + \frac{1}{2} \vec{a}(t) \Delta t^2 + \frac{1}{6} \vec{b}(t) \Delta t^3 + O (\Delta t^4) $$
$$\vec{v}(t + \Delta t) = \vec{v}(t) + \frac{1}{2}(\vec{a}(t) + \vec{a}(t + \Delta t)) \Delta t + \frac{1}{4}(\vec{b}(t) - \vec{b}(t+\Delta t)) \Delta t^2 + O(\Delta t^3)$$
Then
$$\vec{x}(t) = \vec{x}(t-\Delta t) + \vec{v}(t -\Delta t) \Delta t + \frac{1}{2} \vec{a}(t-\Delta t) \Delta t^2 + \frac{1}{6} \vec{b}(t-\Delta t) \Delta t^3 + O (\Delta t^4) $$
$$\vec{v}(t) = \vec{v}(t-\Delta t) + \frac{1}{2}(\vec{a}(t-\Delta t) + \vec{a}(t)) \Delta t + \frac{1}{4}(\vec{b}(t-\Delta t) - \vec{b}(t)) \Delta t^2 + O(\Delta t^3)$$
Substituiting
$$\vec{x}(t + \Delta t) = \vec{x}(t) + (\vec{v}(t-\Delta t) + \frac{1}{2}(\vec{a}(t-\Delta t) + \vec{a}(t)) \Delta t + \frac{1}{4}(\vec{b}(t-\Delta t) - \vec{b}(t)) \Delta t^2 + O(\Delta t^3)) \Delta t + \frac{1}{2} \vec{a}(t) \Delta t^2 + \frac{1}{6} \vec{b}(t) \Delta t^3 + O (\Delta t^4) $$
