Euler's Equation for Vanishing Variation in Canonical Variables

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Theorem

Consider the following system of differential equations:

$\begin{cases}F_{y_i}-\dfrac \d {\d x} F_{y_i'}=0\\ \dfrac{\d {y_i} }{\d x}= y_i'\end{cases}$

where $i\in\set{1,\ldots,n}$.


Let the coordinates $\paren{x,\langle y_i\rangle_{1\mathop\le i\mathop\le n},\langle y_i' \rangle_{1\mathop\le i\mathop\le n},F}$ be transformed to canonical variables:

$\paren{x,\langle y_i\rangle_{1\mathop\le i\mathop\le n},\langle p_i\rangle_{1\mathop\le i\mathop\le n},H}$


Then the aforementioned system of differential equations is transformed into:

$ \begin{cases} \dfrac {\d y_i} {\d x}=\dfrac {\partial H} {\partial p_i} \\ \dfrac {\d p_i} {\d x}=-\dfrac {\partial H} {\partial y_i} \end{cases}$


Proof

Find the full differential of Hamiltonian:

\(\displaystyle \rd H\) \(=\) \(\displaystyle -\rd F+\rd {\sum_{i\mathop=1}^n y_i' p_i}\) $\quad$ Definition of Hamiltonian $\quad$
\(\displaystyle \) \(=\) \(\displaystyle -\rd F+\sum_{i\mathop=1}^n \paren{\rd {y_i'} p_i+y_i'\rd p_i}\) $\quad$ Full differential of a product $\quad$
\(\displaystyle \) \(=\) \(\displaystyle -\frac {\partial F} {\partial x} \rd x-\sum_{i\mathop=1}^n \frac {\partial F} {\partial y_i} \rd y_i-\sum_{i\mathop=1}^n \frac {\partial F} {\partial y_i'} \rd y_i'+{\sum_{i\mathop=1}^n\rd y_i' p_i}+\sum_{i\mathop=1}^n y_i'\rd p_i\) $\quad$ Full differential of real multivariate function F w.r.t. its own variables $\quad$
\(\displaystyle \) \(=\) \(\displaystyle -\frac {\partial F} {\partial x} \rd x-\sum_{i\mathop=1}^n\frac {\partial F} {\partial y_i} \rd y_i-\sum_{i\mathop=1}^n p_i \rd y_i'+{\sum_{i\mathop=1}^n\rd y_i' p_i}+\sum_{i\mathop=1}^n y_i'\rd p_i\) $\quad$ Definition of $p_i$ $\quad$
\(\displaystyle \) \(=\) \(\displaystyle -\frac {\partial F} {\partial x} \rd x-\sum_{i\mathop=1}^n\frac {\partial F} {\partial y_i} \rd y_i+\sum_{i\mathop=1}^n y_i'\rd p_i\) $\quad$ Terms with $p_i\rd y_i'$ cancel $\quad$
\(\displaystyle \) \(=\) \(\displaystyle \frac {\partial H} {\partial x} \rd x+\sum_{i\mathop=1}^n\frac {\partial H} {\partial y_i} \rd y_i+\sum_{i\mathop=1}^n \frac {\partial H} {\partial p_i} \rd p_i\) $\quad$ Full differential of real multivariate function H w.r.t. its own variables $\quad$

By equating coefficients of differentials in last two equations we find that:

$\dfrac {\partial H} {\partial x}=-\dfrac {\partial F} {\partial x},\quad \dfrac {\partial H} {\partial y_i}=-\dfrac {\partial F} {\partial y_i},\quad\dfrac {\partial H} {\partial p_i}=y_i'$

From the third identity it follows that:

$\paren{\dfrac {\d y_i} {\d x} = y_i}\implies\paren{\dfrac {\d y_i} {\d x}=\dfrac {\partial H} {\partial p_i} }$

while the second identity together with the definition of $p_i$ assures that:

$\paren{\dfrac {\partial F} {\partial y_i}-\dfrac \d {\d x} \dfrac {\partial F} {\partial y_i}=0}\implies\paren{\dfrac {\d p_i} {\d x}=-\dfrac {\partial H} {\partial y_i} }$

$\blacksquare$


Source of Name

This entry was named for Leonhard Paul Euler.


Sources