diff --git a/docs/_config.yml b/docs/_config.yml
index e6598e6..fdf0871 100644
--- a/docs/_config.yml
+++ b/docs/_config.yml
@@ -1,4 +1,4 @@
-title: Solutions to STRON2020 HaSP School
+title: Solutions to STRONG2020 HaSP School
 author: Participants of the STRONG2020 HaSP School
 copyright: "2023"
 logo: https://indico.ific.uv.es/event/6803/logo-2761386249.png
diff --git a/docs/lecture02.ipynb b/docs/lecture02.ipynb
index 1c62bf8..90ad600 100644
--- a/docs/lecture02.ipynb
+++ b/docs/lecture02.ipynb
@@ -11,7 +11,808 @@
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "[Lecture 2](https://indico.ific.uv.es/event/6803/contributions/21220) by Vincent Mathieu contains a few data files containing four-momenta data samples. Our goal in this notebook is to identify which reaction was used to generate these data samples."
+    "[Lecture 2](https://indico.ific.uv.es/event/6803/contributions/21220) by Vincent Mathieu contains a few data files containing four-momenta data samples. \n",
+    "Our goal in this notebook is to identify which reaction was used to generate these data samples."
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": [
+     "remove-cell"
+    ]
+   },
+   "outputs": [],
+   "source": [
+    "%pip install -q gdown matplotlib numpy particle"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Two particles Final States"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The standard frames: CoM frame and Lab frame"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The scattering happens in the x-z plane for both frames."
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The two frames are related by a boost along the z axis"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Lorentz transformations"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "### Boost between Lab and CoM"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Only the energy and the z component change"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\\begin{pmatrix}\n",
+    " E^*\n",
+    " \\\\\n",
+    " p_z^*\n",
+    "\\end{pmatrix} =\n",
+    "\\begin{pmatrix}\n",
+    "  \\gamma & \\gamma \\beta \\\\\n",
+    "  \\gamma \\beta & \\gamma\n",
+    "\\end{pmatrix}\n",
+    "\\begin{pmatrix}\n",
+    " E^L\\\\\n",
+    " p^L_z\n",
+    "\\end{pmatrix}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Given the inverse relation"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\\begin{pmatrix}\n",
+    " E^L\n",
+    " \\\\\n",
+    " p_z^L\n",
+    "\\end{pmatrix} =\n",
+    "\\begin{pmatrix}\n",
+    "  \\gamma & -\\gamma \\beta \\\\\n",
+    "  -\\gamma \\beta & \\gamma\n",
+    "\\end{pmatrix}\n",
+    "\\begin{pmatrix}\n",
+    " E^*\\\\\n",
+    " p^*_z\n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "With "
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\beta = \\frac{p^*_b}{m_b}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "and"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\gamma \\beta = \\frac{E^*_b}{m_b}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "We check that it brings the target at rest in Exercise below"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    ":::{exercise}\n",
+    ":label: boost\n",
+    "Check that it brings the target at rest\n",
+    "\n",
+    ":::"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    ":::{solution} boost\n",
+    ":class: dropdown\n",
+    "\n",
+    "By the boost from CoM frame to Lab frame, and apply if to the 4-vector of particle b (target)\n",
+    "\n",
+    "\n",
+    "$$\\begin{pmatrix}\n",
+    " E_b^*\n",
+    " \\\\\n",
+    " p_{b,z}^*\n",
+    "\\end{pmatrix} =\n",
+    "\\begin{pmatrix}\n",
+    "  \\gamma & \\gamma \\beta \\\\\n",
+    "  \\gamma \\beta & \\gamma\n",
+    "\\end{pmatrix}\n",
+    "\\begin{pmatrix}\n",
+    " E_b^L\\\\\n",
+    " p^L_{b,z}\n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "using the relations (3) and (4) and subsititute into (2):\n",
+    "\n",
+    "\n",
+    "$$\\begin{pmatrix}\n",
+    " E_b^L\n",
+    " \\\\\n",
+    " p_{b,z}^L\n",
+    "\\end{pmatrix} =\n",
+    "\\begin{pmatrix}\n",
+    "  \\frac{E_b^*}{p_b^*} & -\\frac{E_b^*}{m_b} \\\\\n",
+    "  -\\frac{E_b^*}{m_b} & \\frac{E_b^*}{p_b^*}\n",
+    "\\end{pmatrix}\n",
+    "\\begin{pmatrix}\n",
+    " E_b^*\\\\\n",
+    " p^*_{b,z}\n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "Since it's only changing in z-componenet and energy, one yields:\n",
+    "\n",
+    "\n",
+    "$$\\begin{pmatrix}\n",
+    " E_b^L\n",
+    " \\\\\n",
+    " p_{b,z}^L\n",
+    "\\end{pmatrix} =\n",
+    "% \\begin{pmatrix}\n",
+    "%   \\frac{E_b^*}{p_b^*} & -\\frac{E_b^*}{m_b} \\\\\n",
+    "%   -\\frac{E_b^*}{m_b} & \\frac{E_b^*}{p_b^*}\n",
+    "% \\end{pmatrix}\n",
+    "\\begin{pmatrix}\n",
+    " \\frac{E_b^*}{p_{b,z}^*} \\cdot E_b^* - \\frac{E_b^*}{m_b}p_{b,z}^* \\\\\n",
+    " -\\frac{E_b^*}{m_b} \\cdot E_b^* + \\frac{E_b^*}{p_{b,z}^*} \\cdot p_{b,z}^*\n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "When assuming $p_{b,z}=0$, then \n",
+    "\n",
+    "$$p^L_{b,z} = 0$$\n",
+    "\n",
+    ":::"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- By the boost from CoM frame to Lab frame, and apply if to the 4-vector of particle b (target) -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "tags": []
+   },
+   "source": [
+    "<!-- % $$\\begin{pmatrix}\n",
+    " E_b^*\n",
+    " \\\\\n",
+    " p_{b,z}^*\n",
+    "\\end{pmatrix} =\n",
+    "\\begin{pmatrix}\n",
+    "  \\gamma & \\gamma \\beta \\\\\n",
+    "  \\gamma \\beta & \\gamma\n",
+    "\\end{pmatrix}\n",
+    "\\begin{pmatrix}\n",
+    " E_b^L\\\\\n",
+    " p^L_{b,z}\n",
+    "\\end{pmatrix}\n",
+    "$$ -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- using the relations (3) and (4) and subsititute into (2): -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- $$\\begin{pmatrix}\n",
+    " E_b^L\n",
+    " \\\\\n",
+    " p_{b,z}^L\n",
+    "\\end{pmatrix} =\n",
+    "\\begin{pmatrix}\n",
+    "  \\frac{E_b^*}{p_b^*} & -\\frac{E_b^*}{m_b} \\\\\n",
+    "  -\\frac{E_b^*}{m_b} & \\frac{E_b^*}{p_b^*}\n",
+    "\\end{pmatrix}\n",
+    "\\begin{pmatrix}\n",
+    " E_b^*\\\\\n",
+    " p^*_{b,z}\n",
+    "\\end{pmatrix}\n",
+    "$$ -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- Since it's only changing in z-componenet and energy, one yields: -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- $$\\begin{pmatrix}\n",
+    " E_b^L\n",
+    " \\\\\n",
+    " p_{b,z}^L\n",
+    "\\end{pmatrix} =\n",
+    "% \\begin{pmatrix}\n",
+    "%   \\frac{E_b^*}{p_b^*} & -\\frac{E_b^*}{m_b} \\\\\n",
+    "%   -\\frac{E_b^*}{m_b} & \\frac{E_b^*}{p_b^*}\n",
+    "% \\end{pmatrix}\n",
+    "\\begin{pmatrix}\n",
+    " \\frac{E_b^*}{p_{b,z}^*} \\cdot E_b^* - \\frac{E_b^*}{m_b}p_{b,z}^* \\\\\n",
+    " -\\frac{E_b^*}{m_b} \\cdot E_b^* + \\frac{E_b^*}{p_{b,z}^*} \\cdot p_{b,z}^*\n",
+    "\\end{pmatrix}\n",
+    "$$ -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- When assuming $p_{b,z}$=0, then  -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- $$p^L_{b,z} = 0$$ -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Active Rotation"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Under an active rotation, the momentum is changed and the axes are fixed"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Example: a momentum of unit length forming an angle θ with the z axis"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "\\begin{pmatrix}\n",
+    " p_x\n",
+    " \\\\\n",
+    " p_y\n",
+    " \\\\\n",
+    " p_z\n",
+    "\\end{pmatrix} =\n",
+    "\\begin{pmatrix}\n",
+    " sin(\\theta)\n",
+    " \\\\\n",
+    " 0\n",
+    " \\\\\n",
+    " cos(\\theta)\n",
+    "\\end{pmatrix}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "After a rotation of ω around y, it forms an angle θ + ω with the z axis"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "tags": []
+   },
+   "source": [
+    "$$\n",
+    "\\begin{pmatrix}\n",
+    " p_x'\n",
+    " \\\\\n",
+    " p_y'\n",
+    " \\\\\n",
+    " p_z'\n",
+    "\\end{pmatrix}\n",
+    "=\n",
+    "R_y(\\omega)\n",
+    "\\begin{pmatrix}\n",
+    " p_x\n",
+    " \\\\\n",
+    " p_y\n",
+    " \\\\\n",
+    " p_z\n",
+    "\\end{pmatrix}\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    " sin(\\theta+\\omega)\n",
+    " \\\\\n",
+    " 0\n",
+    " \\\\\n",
+    " cos(\\theta+\\omega)\n",
+    "\\end{pmatrix}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Now we check the result of the rotation above"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    ":::{exercise}\n",
+    ":label: rotation\n",
+    "Check the result of the rotation above\n",
+    "\n",
+    ":::"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    ":::{solution} rotation\n",
+    ":class: dropdown\n",
+    "\n",
+    "Recall the rotation matrix:\n",
+    "\n",
+    "$$\n",
+    "R_z(\\omega)\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "cos(\\omega) & -sin(\\omega) & 0\n",
+    "\\\\\n",
+    "sin(\\omega) & cos(\\omega) & 0\n",
+    "\\\\\n",
+    "0 & 0 & 1\n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "R_y(\\omega)\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "cos(\\omega) & 0 & sin(\\omega) \n",
+    "\\\\\n",
+    "0 & 1 & 0\n",
+    "\\\\\n",
+    "-sin(\\omega) &0 & cos(\\omega) \n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "Any rotation can be decomposed into rotations around z and y axes above\n",
+    "\n",
+    "\n",
+    "$$\n",
+    "R_y(\\omega)\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "1&0&0\n",
+    "\\\\\n",
+    "0 & cos(\\omega) & -sin(\\omega) \n",
+    "\\\\\n",
+    "0 & sin(\\omega) & cos(\\omega) \n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "The rotation matrix around x axes is also given for completeness.\n",
+    "\n",
+    "Thus equation (9) becomes:\n",
+    "\n",
+    "$$\n",
+    "\\begin{pmatrix}\n",
+    " p_x'\n",
+    " \\\\\n",
+    " p_y'\n",
+    " \\\\\n",
+    " p_z'\n",
+    "\\end{pmatrix}\n",
+    "=\n",
+    "R_y(\\omega)\n",
+    "\\begin{pmatrix}\n",
+    " p_x\n",
+    " \\\\\n",
+    " p_y\n",
+    " \\\\\n",
+    " p_z\n",
+    "\\end{pmatrix}\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "cos(\\omega) & 0 & sin(\\omega) \n",
+    "\\\\\n",
+    "0 & 1 & 0\n",
+    "\\\\\n",
+    "-sin(\\omega) &0 & cos(\\omega) \n",
+    "\\end{pmatrix}\n",
+    "\\begin{pmatrix}\n",
+    " p_x\n",
+    " \\\\\n",
+    " p_y\n",
+    " \\\\\n",
+    " p_z\n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "\\begin{pmatrix}\n",
+    " p_x'\n",
+    " \\\\\n",
+    " p_y'\n",
+    " \\\\\n",
+    " p_z'\n",
+    "\\end{pmatrix}\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "p_x \\cdot cos(\\omega)+ p_z \\cdot sin(\\omega) \n",
+    "\\\\\n",
+    "p_y\n",
+    "\\\\\n",
+    "-p_x \\cdot sin(\\omega) + p_z \\cdot cos(\\omega) \n",
+    "\\end{pmatrix}\n",
+    "$$\n",
+    "\n",
+    "since \n",
+    "$\\begin{pmatrix}\n",
+    " p_x\n",
+    " \\\\\n",
+    " p_y\n",
+    " \\\\\n",
+    " p_z\n",
+    "\\end{pmatrix}$ =\n",
+    "$\\begin{pmatrix}\n",
+    " sin(\\theta)\n",
+    " \\\\\n",
+    " 0\n",
+    " \\\\\n",
+    " cos(\\theta)\n",
+    "\\end{pmatrix}\n",
+    "$,\n",
+    "and using the trigonometric identities,\n",
+    "thus we have:\n",
+    "\n",
+    "$$\n",
+    "\\begin{pmatrix}\n",
+    " p_x'\n",
+    " \\\\\n",
+    " p_y'\n",
+    " \\\\\n",
+    " p_z'\n",
+    "\\end{pmatrix}\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "sin(\\theta) \\cdot cos(\\omega)+ cos(\\theta) \\cdot sin(\\omega) \n",
+    "\\\\\n",
+    "0\n",
+    "\\\\\n",
+    "-sin(\\theta) \\cdot sin(\\omega) + cos(\\theta) \\cdot cos(\\omega) \n",
+    "\\end{pmatrix}\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "sin(\\theta+\\omega)\n",
+    "\\\\\n",
+    "0\n",
+    "\\\\\n",
+    "cos(\\theta+\\omega) \n",
+    "\\end{pmatrix}\n",
+    "\n",
+    "$$\n",
+    "\n",
+    "\n",
+    "\n",
+    ":::"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- Recall the rotation matrix: -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- $$\n",
+    "R_z(\\omega)\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "cos(\\omega) & -sin(\\omega) & 0\n",
+    "\\\\\n",
+    "sin(\\omega) & cos(\\omega) & 0\n",
+    "\\\\\n",
+    "0 & 0 & 1\n",
+    "\\end{pmatrix}\n",
+    "$$ -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- $$\n",
+    "R_y(\\omega)\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "cos(\\omega) & 0 & sin(\\omega) \n",
+    "\\\\\n",
+    "0 & 1 & 0\n",
+    "\\\\\n",
+    "-sin(\\omega) &0 & cos(\\omega) \n",
+    "\\end{pmatrix}\n",
+    "$$ -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- Any rotation can be decomposed into rotations around z and y axes above  -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- $$\n",
+    "R_y(\\omega)\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "1&0&0\n",
+    "\\\\\n",
+    "0 & cos(\\omega) & -sin(\\omega) \n",
+    "\\\\\n",
+    "0 & sin(\\omega) & cos(\\omega) \n",
+    "\\end{pmatrix}\n",
+    "$$ -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- The rotation matrix around x axes is also given for completeness. -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- Thus equation (9) becomes: -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- $$\n",
+    "\\begin{pmatrix}\n",
+    " p_x'\n",
+    " \\\\\n",
+    " p_y'\n",
+    " \\\\\n",
+    " p_z'\n",
+    "\\end{pmatrix}\n",
+    "=\n",
+    "R_y(\\omega)\n",
+    "\\begin{pmatrix}\n",
+    " p_x\n",
+    " \\\\\n",
+    " p_y\n",
+    " \\\\\n",
+    " p_z\n",
+    "\\end{pmatrix}\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "cos(\\omega) & 0 & sin(\\omega) \n",
+    "\\\\\n",
+    "0 & 1 & 0\n",
+    "\\\\\n",
+    "-sin(\\omega) &0 & cos(\\omega) \n",
+    "\\end{pmatrix}\n",
+    "\\begin{pmatrix}\n",
+    " p_x\n",
+    " \\\\\n",
+    " p_y\n",
+    " \\\\\n",
+    " p_z\n",
+    "\\end{pmatrix}\n",
+    "$$ -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- $$\n",
+    "\\begin{pmatrix}\n",
+    " p_x'\n",
+    " \\\\\n",
+    " p_y'\n",
+    " \\\\\n",
+    " p_z'\n",
+    "\\end{pmatrix}\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "p_x \\cdot cos(\\omega)+ p_z \\cdot sin(\\omega) \n",
+    "\\\\\n",
+    "p_y\n",
+    "\\\\\n",
+    "-p_x \\cdot sin(\\omega) + p_z \\cdot cos(\\omega) \n",
+    "\\end{pmatrix}\n",
+    "$$ -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- since \n",
+    "$\\begin{pmatrix}\n",
+    " p_x\n",
+    " \\\\\n",
+    " p_y\n",
+    " \\\\\n",
+    " p_z\n",
+    "\\end{pmatrix}$ =\n",
+    "$\\begin{pmatrix}\n",
+    " sin(\\theta)\n",
+    " \\\\\n",
+    " 0\n",
+    " \\\\\n",
+    " cos(\\theta)\n",
+    "\\end{pmatrix}\n",
+    "$,\n",
+    "and using the trigonometric identities,\n",
+    "thus we have: -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- $$\n",
+    "\\begin{pmatrix}\n",
+    " p_x'\n",
+    " \\\\\n",
+    " p_y'\n",
+    " \\\\\n",
+    " p_z'\n",
+    "\\end{pmatrix}\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "sin(\\theta) \\cdot cos(\\omega)+ cos(\\theta) \\cdot sin(\\omega) \n",
+    "\\\\\n",
+    "0\n",
+    "\\\\\n",
+    "-sin(\\theta) \\cdot sin(\\omega) + cos(\\theta) \\cdot cos(\\omega) \n",
+    "\\end{pmatrix}\n",
+    "=\n",
+    "\\begin{pmatrix}\n",
+    "sin(\\theta+\\omega)\n",
+    "\\\\\n",
+    "0\n",
+    "\\\\\n",
+    "cos(\\theta+\\omega) \n",
+    "\\end{pmatrix}\n",
+    "\n",
+    "$$ -->\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {
+    "tags": []
+   },
+   "source": [
+    "## `Two-particles-1.dat`\n",
+    "\n",
+    "\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "The data are in the lab frame, Compute and plot the cos θL distribution\n",
+    "Boost to the CoM and Compute and plot\n",
+    "the cos θ* distribution"
    ]
   },
   {
@@ -21,45 +822,417 @@
     "jupyter": {
      "source_hidden": true
     },
+    "mystnb": {
+     "code_prompt_show": "Import Python libraries"
+    },
     "tags": [
-     "remove-cell"
+     "hide-cell"
     ]
    },
    "outputs": [],
    "source": [
-    "%pip install -q gdown matplotlib numpy particle"
+    "import warnings\n",
+    "\n",
+    "import gdown\n",
+    "import numpy as np\n",
+    "from IPython.display import display\n",
+    "\n",
+    "warnings.filterwarnings(\"ignore\")"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "jupyter": {
+     "source_hidden": true
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# https://indico.ific.uv.es/event/6803/contributions/21220/attachments/11209/15506/Two-particles-1.dat\n",
+    "filename = gdown.cached_download(\n",
+    "    url=\"https://indico.ific.uv.es/event/6803/contributions/21220/attachments/11209/15506/Two-particles-1.dat\",\n",
+    "    # path=\"data/Two-particles-1.dat\",\n",
+    "    # md5=\"a49ebfd97ae6a02023291df665ab924c\",\n",
+    "    quiet=True,\n",
+    "    verify=False,\n",
+    ")\n",
+    "data = np.loadtxt(filename)\n",
+    "data.shape"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "jupyter": {
+     "outputs_hidden": true
+    },
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "print(data)"
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "# p ="
+   ]
+  },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {
+    "tags": []
+   },
+   "outputs": [],
+   "source": [
+    "# n_final_state = 2\n",
+    "# pa, p1, p2 = (data[i::4].T for i in range(n_final_state + 1))\n",
+    "# p0 = p1 + p2\n",
+    "# pb = p0 - pa"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Mandelstam variables\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$ p_i^2 = m^2_i $$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$ s =(p_a + p_b)^2 = (p_1+p_2)^2 $$\n",
+    "$$ t =(p_a - p_1)^2 = (p_b-p_2)^2 $$\n",
+    "$$ u =(p_a - p_2)^2 = (p_b-p_1)^2 $$\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "s+t+u = m_a^2+m_b^2+m_1^2+m^2_2\n",
+    "$$ (label:mv)\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- Check the relation in equation {eq}`label:` by using  -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!--  $$p_a+p_b=p_1+p_2$$ -->"
    ]
   },
   {
    "cell_type": "markdown",
    "metadata": {},
    "source": [
-    "## `Three-particles-1.dat`"
+    ":::{exercise}\n",
+    ":label: mv\n",
+    "Check the relation in equation {eq}`label:mv` by using $p_a+p_b=p_1+p_2$\n",
+    ":::"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    ":::{solution} mv\n",
+    ":class: dropdown\n",
+    "\n",
+    "$$ s+t+u = (p_a + p_b)^2 + (p_a - p_1)^2 + (p_a - p_2)^2$$\n",
+    "\n",
+    "$$ s+t+u = p_a^2 +p_b^2 + 2 p_a p_b + p_a^2 -2 p_a p_1 + p_1^2 + p_a^2 - 2p_a p_2 +p_2^2$$\n",
+    "\n",
+    "$$\n",
+    "s+t+u = 3p_a^2 + p_b^2 +p_1^2 + p_2^2 + 2p_a p_b - 2p_a p_1 -2 p_a p_2\n",
+    "$$ (label:proving)\n",
+    "\n",
+    "from equation {eq}`label:mv` we have:\n",
+    "\n",
+    "$$ p_b = p_1 + p_2 - p_a $$\n",
+    "\n",
+    "$$ \n",
+    "p_a \\cdot p_b = p_a \\cdot p_1 + p_a \\cdot p_2 - p_a^2 \n",
+    "$$ (label:use)\n",
+    "\n",
+    "By equation {eq}`label:proving` and substitute it into equation {eq}`label:use`:\n",
+    "\n",
+    "$$\n",
+    "s+t+u = 3p_a^2 + p_b^2 +p_1^2 + p_2^2 + 2p_a \\cdot p_1 + 2p_a \\cdot p_2 - 2p_a^2 -2p_a p_1 -2 p_a p_2\n",
+    "$$\n",
+    "\n",
+    "$$\n",
+    "s+t+u = p_a^2 + p_b^2 +p_1^2 + p_2^2 \n",
+    "$$\n",
+    "\n",
+    "by the mass shell relation, and thus:\n",
+    "\n",
+    "$$s+t+u = m_a^2 + m_b^2 + m_1^2 + m_2^2$$\n",
+    ":::"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- As following: -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "% $$ s+t+u = (p_a + p_b)^2 + (p_a - p_1)^2 + (p_a - p_2)^2$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "% $$ s+t+u = p_a^2 +p_b^2 + 2 p_a p_b + p_a^2 -2 p_a p_1 + p_1^2 + p_a^2 - 2p_a p_2 +p_2^2$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- $$\n",
+    "s+t+u = 3p_a^2 + p_b^2 +p_1^2 + p_2^2 + 2p_a p_b - 2p_a p_1 -2 p_a p_2\n",
+    "$$ (label:proving) -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- from equation {eq}`label:mv` we have: -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "% $$ p_b = p_1 + p_2 - p_a $$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- $$ \n",
+    "p_a \\cdot p_b = p_a \\cdot p_1 + p_a \\cdot p_2 - p_a^2 \n",
+    "$$ (label:use) -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- By equation {eq}`label:proving` and substitute it into equation {eq}`label:use`: -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- $$\n",
+    "s+t+u = 3p_a^2 + p_b^2 +p_1^2 + p_2^2 + 2p_a \\cdot p_1 + 2p_a \\cdot p_2 - 2p_a^2 -2p_a p_1 -2 p_a p_2\n",
+    "$$ -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- $$\n",
+    "s+t+u = p_a^2 + p_b^2 +p_1^2 + p_2^2 \n",
+    "$$ -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- by the mass shell relation, and thus: -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- $$s+t+u = m_a^2 + m_b^2 + m_1^2 + m_2^2$$ -->"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## CoM Kinematics"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Every frame dependent quantities is expressed with Mandelstam variables\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "<!-- Check all relations! -->\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "% $$t = m_a^2 +m_1^2 -2 E^*_a E^*_1 + 2|\\vec{p_a}^*| |\\vec{p_1}^*| cos \\theta^*$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Equivalent expression for the scattering angle in the CoM"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\n",
+    "cos \\theta^* = 1 + \\frac{t-t_{min}}{2|\\vec{v}_a^*||\\vec{p}_1^*|}\n",
+    "=\n",
+    "\\frac{s(t-u)+(m_a^2-m-b^2)(m_1^2-m_2^2)}{\\lambda^{1/2}(s,m_a^2,m_b^2)\\lambda^{1/2}(s,m_1^2,m_2^2)}\n",
+    "$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Compute $cos \\theta^L$ as a function of s,t,u"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$ cos \\theta^L $$"
+   ]
+  },
+  {
+   "cell_type": "raw",
+   "metadata": {},
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "## Decay into three particles\n"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "$$\\theta^*_{12}+\\theta^*_{23}+\\theta^*_{31}=2 \\pi$$"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    "Check relation (33) using $s_{12}+s_{23}+s_{31}$"
    ]
   },
   {
    "cell_type": "code",
    "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": []
+  },
+  {
+   "cell_type": "markdown",
    "metadata": {
-    "jupyter": {
-     "source_hidden": true
-    },
-    "mystnb": {
-     "code_prompt_show": "Import Python libraries"
-    },
-    "tags": [
-     "hide-cell"
-    ]
+    "tags": []
    },
-   "outputs": [],
    "source": [
-    "import warnings\n",
+    "## `Three-particles-1.dat`"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    ":::{exercise}\n",
+    ":label: three-particles-1\n",
+    "File: Three-Particles-1.dat is \n",
+    "The data are in the lab frame, \n",
+    "with Format:\n",
     "\n",
-    "import gdown\n",
-    "import numpy as np\n",
-    "from IPython.display import display\n",
+    "Ea, pa,x, pa,y, pa,z\n",
     "\n",
-    "warnings.filterwarnings(\"ignore\")"
+    "E1, p1,x, p1,y, p1,z\n",
+    "\n",
+    "E2, p2,x, p2,y, p2,z \n",
+    "\n",
+    "E3, p3,x, p3,y, p3,z\n",
+    "\n",
+    ".    .    .    .\n",
+    "\n",
+    ".    .    .    .\n",
+    "\n",
+    ".    .    .    .\n",
+    "\n",
+    "\n",
+    "Three questions to be found out.\n",
+    "\n",
+    "Which reaction is it?
",
+    "\n",
+    "\n",
+    "What resonances are included?
",
+    " \n",
+    "\n",
+    "What is the spin of the resonances?\n",
+    ":::"
+   ]
+  },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    ":::{solution-start} three-particles-1\n",
+    ":class: dropdown\n",
+    ":::"
    ]
   },
   {
@@ -81,6 +1254,15 @@
     "data.shape"
    ]
   },
+  {
+   "cell_type": "code",
+   "execution_count": null,
+   "metadata": {},
+   "outputs": [],
+   "source": [
+    "print(data)"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": null,
@@ -97,9 +1279,6 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
-    "jupyter": {
-     "source_hidden": true
-    },
     "tags": []
    },
    "outputs": [],
@@ -120,9 +1299,6 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
-    "jupyter": {
-     "source_hidden": true
-    },
     "tags": [
      "hide-input"
     ]
@@ -141,9 +1317,6 @@
    "cell_type": "code",
    "execution_count": null,
    "metadata": {
-    "jupyter": {
-     "source_hidden": true
-    },
     "mystnb": {
      "code_prompt_show": "Identify final state particles"
     },
@@ -194,6 +1367,14 @@
     "So this is a photon&nbsp;$\\gamma$ hitting a proton&nbsp;$p$ and producing a meson&nbsp;$\\eta$, pion&nbsp;$\\pi^0$, and proton&nbsp;$p$."
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    ":::{solution-end} \n",
+    ":::"
+   ]
+  },
   {
    "cell_type": "markdown",
    "metadata": {},
@@ -370,11 +1551,50 @@
   },
   {
    "cell_type": "markdown",
-   "metadata": {},
+   "metadata": {
+    "tags": []
+   },
    "source": [
     "## `Three-particles-2.dat`"
    ]
   },
+  {
+   "cell_type": "markdown",
+   "metadata": {},
+   "source": [
+    ":::{exercise}\n",
+    ":label: three-particles-2\n",
+    "File: Three-Particles-2.dat is also\n",
+    "The data are in the lab frame, \n",
+    "with Format:\n",
+    "\n",
+    "Ea, pa,x, pa,y, pa,z\n",
+    "\n",
+    "E1, p1,x, p1,y, p1,z\n",
+    "\n",
+    "E2, p2,x, p2,y, p2,z \n",
+    "\n",
+    "E3, p3,x, p3,y, p3,z\n",
+    "\n",
+    ".    .    .    .\n",
+    "\n",
+    ".    .    .    .\n",
+    "\n",
+    ".    .    .    .\n",
+    "\n",
+    "\n",
+    "Three questions to be found out.\n",
+    "\n",
+    "Which reaction is it?
",
+    "\n",
+    "\n",
+    "What resonances are included?
",
+    " \n",
+    "\n",
+    "What is the spin of the resonances?\n",
+    ":::"
+   ]
+  },
   {
    "cell_type": "code",
    "execution_count": null,
@@ -561,7 +1781,7 @@
    "name": "python",
    "nbconvert_exporter": "python",
    "pygments_lexer": "ipython3",
-   "version": "3.10.12"
+   "version": "3.10.13"
   }
  },
  "nbformat": 4,