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    "## Derivative of Angular Momentum\n",
    "\n",
    "#### Background\n",
    "The derivative of momentum is torque. So we may always see equation like\n",
    "$$ \\frac{{\\rm d}}{{\\rm d}t}(Iw) = \\sum\\limits_{i = 1}^{n}r_i\\times f_i$$\n",
    "\n",
    "#### Question\n",
    "What is the derivative of angular momentum?\n",
    "$$\\frac{{\\rm d}}{{\\rm d}t}L  ?or \\frac{{\\rm d}}{{\\rm d}t} (Iw)?$$\n",
    "\n",
    "#### Result\n",
    "$$\\frac{{\\rm d}}{{\\rm d}t}(Iw) = I \\dot w + w\\times (Iw)$$\n",
    "\n",
    "#### Derive\n",
    "\n",
    "*No problem can be solved from the same level of consciousness that created it.*\n",
    "<div align=\"right\"> --- Albert Einstein\n",
    "\n",
    "To derive the equations:\n",
    "\n",
    "1. The $I_{body}$ which is a constant inertia tensor described in body frame. The $R(t)$ is rotation matrix relative to world frame.\n",
    "\n",
    "3. To find the inertia tensor in world frame you need the transformation $$I(t)=R(t)I_{body}R(t)^T$$\n",
    " \n",
    "4. The body at this moment has angular velocity $\\omega(t)$ described in world frame.\n",
    " \n",
    "5. The derivative of rotation matrix $R(t)$\n",
    "$$\n",
    "\\frac{{\\rm d}}{{\\rm d}t} R(t) = \\omega(t) \\times R(t)\n",
    "$$\n",
    "where $\\times$ is the vector cross product.\n",
    " \n",
    "6. The derivative of inertia tensor is derived with the chain rule \n",
    "$$\\begin{split}\n",
    "\\frac{\\rm d}{{\\rm d}t} I(t) & = \\frac{\\rm d}{{\\rm d}t} \\left( R(t) I_{body} R(t)^T\\right) = \\frac{{\\rm d}R(t)}{{\\rm d}t} I_{body} R(t)^T + R(t) I_{body} \\frac{{\\rm d}R(t)}{{\\rm d}t}^T \\\\\n",
    "& = (\\omega\\times R(t)) I_{body} R(t)^T + R(t) I_{body} (\\omega\\times R(t))^T) \\\\\n",
    "& = \\omega \\times I(t) - I(t) \\omega \\times \n",
    "\\end{split} $$ \n",
    "\n",
    "\n",
    "7. The derivative of angular momentum is derived with the chain rule \n",
    "$$\\begin{split}\n",
    "\\frac{{\\rm d}}{{\\rm d}t} L &= I(t) \\frac{{\\rm d} \\omega(t)}{{\\rm d}t} + \\frac{{\\rm d}I(t) }{{\\rm d}t} \\omega(t) \\\\ &= I(t) \\dot{\\omega} + \\left( \\omega \\times I(t) - I(t) \\omega \\times \\right) \\omega\n",
    "\\end{split}$$\n",
    "\n",
    "$$\\boxed{\\tau = \\dot{L}  = I(t) \\dot{\\omega} + \\omega \\times I(t) \\omega}$$\n",
    "\n",
    "The last is Euler's equations of rotational motion.\n",
    "\n",
    "\n",
    "Weita\n",
    "2021/12/03"
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