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   "source": [
    "# Point particle on the cone\n",
    "\n",
    "Consider the movement of the material point on the surface of the cone.\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
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   "outputs": [],
   "source": [
    "var('t')\n",
    "var('a g')\n",
    "xy_wsp = [('x','x'),('y','y'),('z','z')]\n",
    "uv_wsp = [('r','r'),('phi','\\phi'),('z','z')]\n",
    "\n",
    "for v,lv in uv_wsp+xy_wsp:\n",
    "    var(\"%s\"%v,latex_name=r'%s'%lv)\n",
    "    vars()[v.capitalize()] = function(v.capitalize())(t)\n",
    "    var(\"%sdd\"%v,latex_name=r'\\ddot %s'%lv)\n",
    "    var(\"%sd\"%v,latex_name=r'\\dot %s'%lv)\n",
    "    var(\"d%s\"%v,latex_name=r'\\delta %s'%lv)\n",
    "\n",
    "\n",
    "xy = [vars()[v] for v,lv in xy_wsp]\n",
    "uv = [vars()[v] for v,lv in uv_wsp]\n",
    "\n",
    "to_fun=dict()\n",
    "for v,lv in xy_wsp+uv_wsp:\n",
    "    to_fun[vars()[v]]=vars()[v.capitalize()]\n",
    "    to_fun[vars()[v+\"d\"]]=vars()[v.capitalize()].diff()\n",
    "    to_fun[vars()[v+\"dd\"]]=vars()[v.capitalize()].diff(2)\n",
    "\n",
    "to_var = dict((v,k) for k,v in to_fun.items())"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dAlemb = (X.diff(t,2))*dx +(Y.diff(t,2))*dy+ (Z.diff(t,2)+g)*dz\n",
    "f = x^2 + y^2 - tan(a)^2*z^2 \n",
    "\n",
    "dxy = [vars()['d'+repr(zm)] for zm in xy]\n",
    "constr =sum([dzm*f.diff(zm) for zm,dzm in zip(xy,dxy)])\n",
    "show(f)\n",
    "show(constr)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f.solve(z)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dAlemb_xy = dAlemb.subs(constr.solve(dz)[0]).subs(f.solve(z)[1])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "show(dAlemb_xy.subs(to_var).coeff(dx))\n",
    "show(dAlemb_xy.subs(to_var).coeff(dy))"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "lang": "en"
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   "source": [
    "Let's solve the problem in cylindrical coordinates. Because these coordinates are not consistent with constraints, but the cone has a particularly simple form in them. Therefore, we will have to perform the step of removing the dependent virtual offsets after moving to the cylindrical coordinate system."
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "x2u = {x:r*cos(phi),y:r*sin(phi),z:z}\n",
    "for w in xy:\n",
    "    vars()['d'+repr(w)+'_polar']=sum([w.subs(x2u).diff(w2)*vars()['d'+repr(w2)] for w2 in uv])\n",
    "show(dx_polar)\n",
    "show(dy_polar)\n",
    "show(dz_polar)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {
    "lang": "en"
   },
   "source": [
    "First, we write the d'Alembert's rule in cylindrical coordinates:\n"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dAlemb = (x.subs(x2u).subs(to_fun).diff(t,2))*dx_polar + \\\n",
    "         (y.subs(x2u).subs(to_fun).diff(t,2))*dy_polar+\\\n",
    "         (z.subs(x2u).subs(to_fun).diff(t,2)+g)*dz_polar \n",
    "dAlemb = dAlemb.subs(to_var)\n",
    "show(dAlemb)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "f_uv  = f.subs(x2u).trig_simplify()"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "duv = [vars()['d'+repr(zm)] for zm in uv]\n",
    "constr =sum([dzm*f_uv.diff(zm) for zm,dzm in zip(uv,duv)])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "show(constr.solve(dz)[0])"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dAlemb.subs( constr.solve(dz)[0] )"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "zdd_rdd = (f_uv.solve(z)[1]).subs(to_fun).diff(t,2).subs(to_var)\n",
    "zdd_rdd"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "dAlemb_uv = dAlemb.subs( constr.solve(dz)[0] ).subs(f_uv.solve(z)[1]).subs(zdd_rdd)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "r1 =  dAlemb_uv.coeff(dr)\n",
    "r2 =  dAlemb_uv.coeff(dphi)\n",
    "\n",
    "show(r1)\n",
    "show( r2)"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": [
    "table([rown.trig_simplify() for rown in solve([r1,r2],[rdd,phidd])[0]])"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "\\newpage"
   ]
  },
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   "cell_type": "code",
   "execution_count": null,
   "metadata": {},
   "outputs": [],
   "source": []
  },
  {
   "cell_type": "code",
   "execution_count": null,
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