Roadmap for Formal Mathematical Physics Content

 

Explanation: Content MathML using the command 

latexmlmath --cmml=- "v=0"
Not all of the nuances from each expression have been correctly translated in Content MathML. Some manual cleanup is required.

Because this page uses MathJax, the Presentation MathML is available for each expression by right-clicking on any rendered expression and selecting "Show Math As" and then "MathML code".

MathJax has an extension for generation of Content MathML but it is not available for v3 of MathJax as of August 2020.

Who does this work: human using LatexML

Motive for doing this work: preparation for use with Computer Algebra System. Caveat: no CAS currently consumes Content MathML as input.

Navigation:
  1. lecture video
  2. hand-written notes
  3. Latex document
  4. Content tags
  5. tags for sections and words and expressions
  6. concepts to variables
  7. all steps
  8. derivation graph
  9. replace symbols with numeric ID
  10. validation of steps
  11. validation of dimension
  12. proof of inference rule
Back to layers overview

source: "Derivation of Gravitational Potential Energy" by Rhett Allain

Suppose an object starts an infinite distance from a moon and is dropped, falling towards the moon due to gravitational acceleration. What is the speed of the object when it is distance \(r\) from the moon?

Figure 1: small mass falling towards a moon from initial position at infinity.

The initial condition is

expression as Content MathML \begin{equation} v(x=\infty) =0 \label{eq:initial_velocity} \end{equation} 
    <?xml version="1.0" encoding="UTF-8"?>
          <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v=0" display="block">
            <apply>
              <eq/>
              <ci>v</ci>
              <cn type="integer">0</cn>
            </apply>
          </math>
The force acting on the object is
expression as Content MathML \begin{equation} \vec{F} = \frac{-G m_1 m_2}{x^2} \hat{x} \label{eq:gravitational force} \end{equation} 
    <?xml version="1.0" encoding="UTF-8"?>
    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="\vec{F}=\frac{-Gm_{1}m_{2}}{x^{2}}\hat{x}" display="block">
      <apply>
        <eq/>
        <apply>
          <ci>→</ci>
          <ci>𝐹</ci>
        </apply>
        <apply>
          <times/>
          <apply>
            <divide/>
            <apply>
              <minus/>
              <apply>
                <times/>
                <ci>𝐺</ci>
                <apply>
                  <csymbol cd="ambiguous">subscript</csymbol>
                  <ci>𝑚</ci>
                  <cn type="integer">1</cn>
                </apply>
                <apply>
                  <csymbol cd="ambiguous">subscript</csymbol>
                  <ci>𝑚</ci>
                  <cn type="integer">2</cn>
                </apply>
              </apply>
            </apply>
            <apply>
              <csymbol cd="ambiguous">superscript</csymbol>
              <ci>𝑥</ci>
              <cn type="integer">2</cn>
            </apply>
          </apply>
          <apply>
            <ci>^</ci>
            <ci>𝑥</ci>
          </apply>
        </apply>
      </apply>
    </math>
The work is calculated using W = \(\Delta E\) since the force changes. To find the cumulative work done on the object, integrate over all positions between \(\infty\) and \(r\)
expression as Content MathML \begin{equation} W = \int_{\infty}^r \vec{F}\cdot d\vec{r} \label{eq:work as function of force} \end{equation} 
<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="W=\int_{\infty}^{r}\vec{F}\cdot d\vec{r}" display="block">
  <apply>
    <eq/>
    <ci>𝑊</ci>
    <apply>
      <apply>
        <csymbol cd="ambiguous">superscript</csymbol>
        <apply>
          <csymbol cd="ambiguous">subscript</csymbol>
          <int/>
          <infinity/>
        </apply>
        <ci>𝑟</ci>
      </apply>
      <apply>
        <ci>⋅</ci>
        <apply>
          <ci>→</ci>
          <ci>𝐹</ci>
        </apply>
        <apply>
          <csymbol cd="latexml">differential-d</csymbol>
          <apply>
            <ci>→</ci>
            <ci>𝑟</ci>
          </apply>
        </apply>
      </apply>
    </apply>
  </apply>
</math>
Substituting the gravitational force into Eq. \ref{eq:work as function of force},
expression as Content MathML \begin{equation} W = \int_{\infty}^r \frac{-G m_1 m_2}{x^2} dx \end{equation} 
<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="W=\int_{\infty}^{r}\frac{-Gm_{1}m_{2}}{x^{2}}dx" display="block">
  <apply>
    <eq/>
    <ci>𝑊</ci>
    <apply>
      <apply>
        <csymbol cd="ambiguous">superscript</csymbol>
        <apply>
          <csymbol cd="ambiguous">subscript</csymbol>
          <int/>
          <infinity/>
        </apply>
        <ci>𝑟</ci>
      </apply>
      <apply>
        <times/>
        <apply>
          <divide/>
          <apply>
            <minus/>
            <apply>
              <times/>
              <ci>𝐺</ci>
              <apply>
                <csymbol cd="ambiguous">subscript</csymbol>
                <ci>𝑚</ci>
                <cn type="integer">1</cn>
              </apply>
              <apply>
                <csymbol cd="ambiguous">subscript</csymbol>
                <ci>𝑚</ci>
                <cn type="integer">2</cn>
              </apply>
            </apply>
          </apply>
          <apply>
            <csymbol cd="ambiguous">superscript</csymbol>
            <ci>𝑥</ci>
            <cn type="integer">2</cn>
          </apply>
        </apply>
        <apply>
          <csymbol cd="latexml">differential-d</csymbol>
          <ci>𝑥</ci>
        </apply>
      </apply>
    </apply>
  </apply>
</math>
Factor out the constants,
expression as Content MathML \begin{equation} W = -G m_1 m_2\int_{\infty}^r \frac{1}{x^2} dx \end{equation} 
<?xml version="1.0" encoding="UTF-8"?>
    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="W=-Gm_{1}m_{2}\int_{\infty}^{r}\frac{1}{x^{2}}dx" display="block">
      <apply>
        <eq/>
        <ci>𝑊</ci>
        <apply>
          <minus/>
          <apply>
            <times/>
            <ci>𝐺</ci>
            <apply>
              <csymbol cd="ambiguous">subscript</csymbol>
              <ci>𝑚</ci>
              <cn type="integer">1</cn>
            </apply>
            <apply>
              <csymbol cd="ambiguous">subscript</csymbol>
              <ci>𝑚</ci>
              <cn type="integer">2</cn>
            </apply>
            <apply>
              <apply>
                <csymbol cd="ambiguous">superscript</csymbol>
                <apply>
                  <csymbol cd="ambiguous">subscript</csymbol>
                  <int/>
                  <infinity/>
                </apply>
                <ci>𝑟</ci>
              </apply>
              <apply>
                <times/>
                <apply>
                  <divide/>
                  <cn type="integer">1</cn>
                  <apply>
                    <csymbol cd="ambiguous">superscript</csymbol>
                    <ci>𝑥</ci>
                    <cn type="integer">2</cn>
                  </apply>
                </apply>
                <apply>
                  <csymbol cd="latexml">differential-d</csymbol>
                  <ci>𝑥</ci>
                </apply>
              </apply>
            </apply>
          </apply>
        </apply>
      </apply>
    </math>
which leads to
expression as Content MathML \begin{equation} W = \frac{G m_1 m_2}{r} \end{equation} 
<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="W=\frac{Gm_{1}m_{2}}{r}" display="block">
  <apply>
    <eq/>
    <ci>𝑊</ci>
    <apply>
      <divide/>
      <apply>
        <times/>
        <ci>𝐺</ci>
        <apply>
          <csymbol cd="ambiguous">subscript</csymbol>
          <ci>𝑚</ci>
          <cn type="integer">1</cn>
        </apply>
        <apply>
          <csymbol cd="ambiguous">subscript</csymbol>
          <ci>𝑚</ci>
          <cn type="integer">2</cn>
        </apply>
      </apply>
      <ci>𝑟</ci>
    </apply>
  </apply>
</math>
Another definition of work is that it is the change in energy for a system: \(W = \Delta E \) Because the initial velocity was zero, the work here is
expression as Content MathML \begin{equation} W = \Delta KE \end{equation} 
<?xml version="1.0" encoding="UTF-8"?>
    <math xmlns="http://www.w3.org/1998/Math/MathML" alttext="W=\Delta KE" display="block">
      <apply>
        <eq/>
        <ci>𝑊</ci>
        <ci>ΔKE</ci>
      </apply>
    </math>
Thus we can combine the two definitions of work to get
expression as Content MathML \begin{equation} W = \frac{1}{2} m_1 v^2 = \frac{G m_1 m_2}{r} \end{equation} 
<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="W=\frac{1}{2}m_{1}v^{2}=\frac{Gm_{1}m_{2}}{r}" display="block">
  <apply>
    <and/>
    <apply>
      <eq/>
      <ci>𝑊</ci>
      <apply>
        <times/>
        <apply>
          <divide/>
          <cn type="integer">1</cn>
          <cn type="integer">2</cn>
        </apply>
        <apply>
          <csymbol cd="ambiguous">subscript</csymbol>
          <ci>𝑚</ci>
          <cn type="integer">1</cn>
        </apply>
        <apply>
          <csymbol cd="ambiguous">superscript</csymbol>
          <ci>𝑣</ci>
          <cn type="integer">2</cn>
        </apply>
      </apply>
    </apply>
    <apply>
      <eq/>
      <share href="#Ex1.m1.sh1"/>
      <apply>
        <divide/>
        <apply>
          <times/>
          <ci>𝐺</ci>
          <apply>
            <csymbol cd="ambiguous">subscript</csymbol>
            <ci>𝑚</ci>
            <cn type="integer">1</cn>
          </apply>
          <apply>
            <csymbol cd="ambiguous">subscript</csymbol>
            <ci>𝑚</ci>
            <cn type="integer">2</cn>
          </apply>
        </apply>
        <ci>𝑟</ci>
      </apply>
    </apply>
  </apply>
</math>
The \(m_1\) cancels, leaving
expression as Content MathML \begin{equation} v(r) = \sqrt{\frac{2Gm_2}{r}} \end{equation} 
<?xml version="1.0" encoding="UTF-8"?>
<math xmlns="http://www.w3.org/1998/Math/MathML" alttext="v=\sqrt{\frac{2Gm_{2}}{r}}" display="block">
  <apply>
    <eq/>
    <ci>𝑣</ci>
    <apply>
      <root/>
      <apply>
        <divide/>
        <apply>
          <times/>
          <cn type="integer">2</cn>
          <ci>𝐺</ci>
          <apply>
            <csymbol cd="ambiguous">subscript</csymbol>
            <ci>𝑚</ci>
            <cn type="integer">2</cn>
          </apply>
        </apply>
        <ci>𝑟</ci>
      </apply>
    </apply>
  </apply>
</math>