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 $\paren{x, \family {y_i}_{1 \mathop \le i \mathop \le n},\family {y_i'}_{1 \mathop \le i \mathop \le n}, F}$ be transformed to canonical variables:

$\paren{x,\family {y_i}_{1 \mathop \le i \mathop \le n},\family {p_i}_{1 \mathop \le i \mathop \le n} ,H}$


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:

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

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