Example showing that subs doesn't work as you might expect with derivatives.

subs_derivatives.ipynb

{
 "cells": [
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "# Be careful about substituting for derivatives!"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 1,
   "metadata": {},
   "outputs": [],
   "source": [
    "import sympy as sm\n",
    "import sympy.physics.mechanics as me"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 2,
   "metadata": {},
   "outputs": [],
   "source": [
    "q1, q2 = me.dynamicsymbols('q1, q2')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 3,
   "metadata": {},
   "outputs": [],
   "source": [
    "a, b, c = sm.symbols('a, b, c')"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 4,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle \\operatorname{q_{1}}{\\left(t \\right)} \\left(\\frac{d^{2}}{d t^{2}} \\operatorname{q_{1}}{\\left(t \\right)}\\right)^{2} + \\frac{d}{d t} \\operatorname{q_{1}}{\\left(t \\right)} \\frac{d}{d t} \\operatorname{q_{2}}{\\left(t \\right)}$"
      ],
      "text/plain": [
       "q1(t)*Derivative(q1(t), (t, 2))**2 + Derivative(q1(t), t)*Derivative(q2(t), t)"
      ]
     },
     "execution_count": 4,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr = q1.diff().diff()**2*q1 + q1.diff()*q2.diff()\n",
    "expr"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "It is worth looking at the raw form (the non-rendered form) to see what is going on:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 5,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'q1(t)*Derivative(q1(t), (t, 2))**2 + Derivative(q1(t), t)*Derivative(q2(t), t)'"
      ]
     },
     "execution_count": 5,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "repr(expr)"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "We are going to be substiting expressions for $q$'s, $\\dot{q}$'s, and $\\dot{u}$'s. There are a couple of key things to be aware of:\n",
    "\n",
    "- `subs()` and `xreplace()` do not substitute the expression in a fixed order (i.e. Python dictionaries are not ordered), so if you pass in more than one item in a single `subs()` call, things might get substituted in an unexpected order.\n",
    "- If you have a `dynamicsymbol` and its deritivate in an expression and substitute something for the symbol, it will get replaced inside the derivative too."
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice how $c$ is not replaced first below and thus derivatives are left in the expression:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 6,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a b c \\frac{d}{d t} \\operatorname{q_{2}}{\\left(t \\right)} + a \\left(\\frac{\\partial}{\\partial t} a b c\\right)^{2}$"
      ],
      "text/plain": [
       "a*b*c*Derivative(q2(t), t) + a*Derivative(a*b*c, t)**2"
      ]
     },
     "execution_count": 6,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.subs({q1: a, q1.diff(): a*b*c, q1.diff().diff(): c})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 7,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'a*b*c*Derivative(q2(t), t) + a*Derivative(a*b*c, t)**2'"
      ]
     },
     "execution_count": 7,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "repr(expr.subs({q1: a, q1.diff(): a*b*c, q1.diff().diff(): c}))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "Notice how all the $q_1$ are replaced with $a$, even \"inside\" the derivatives:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 8,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a \\left(\\frac{d^{2}}{d t^{2}} a\\right)^{2} + \\frac{d}{d t} a \\frac{d}{d t} \\operatorname{q_{2}}{\\left(t \\right)}$"
      ],
      "text/plain": [
       "a*Derivative(a, (t, 2))**2 + Derivative(a, t)*Derivative(q2(t), t)"
      ]
     },
     "execution_count": 8,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.subs({q1: a})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 9,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'a*Derivative(a, (t, 2))**2 + Derivative(a, t)*Derivative(q2(t), t)'"
      ]
     },
     "execution_count": 9,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "repr(expr.subs({q1: a}))"
   ]
  },
  {
   "cell_type": "markdown",
   "metadata": {},
   "source": [
    "To ensure proper substitution make sure to substitute the highest order deriavtives first in a chain of `subs()` calls:"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 10,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/latex": [
       "$\\displaystyle a c^{2} + b \\frac{d}{d t} \\operatorname{q_{2}}{\\left(t \\right)}$"
      ],
      "text/plain": [
       "a*c**2 + b*Derivative(q2(t), t)"
      ]
     },
     "execution_count": 10,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "expr.subs({q1.diff().diff(): c}).subs({q1.diff(): b}).subs({q1: a})"
   ]
  },
  {
   "cell_type": "code",
   "execution_count": 11,
   "metadata": {},
   "outputs": [
    {
     "data": {
      "text/plain": [
       "'a*c**2 + b*Derivative(q2(t), t)'"
      ]
     },
     "execution_count": 11,
     "metadata": {},
     "output_type": "execute_result"
    }
   ],
   "source": [
    "repr(expr.subs({q1.diff().diff(): c}).subs({q1.diff(): b}).subs({q1: a}))"
   ]
  }
 ],
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