Toronto Math Forum
APM3462018S => APM346Lectures => Topic started by: Jingxuan Zhang on March 30, 2018, 09:37:25 AM

From (what I consider to be EulerLagrange)
\begin{equation}\label{1}L_u=(L_{u'})_t\end{equation}
how can I derive
\begin{equation}\label{2}L=u'L_{u'}+C?\end{equation}
Or is \eqref{2} even right? are they derived independently? if I integrate both sides of \eqref{1}, what exactly will be on the left?

Start from E.L.:
\begin{equation}
\frac{d\ }{dt} L_{q'_j}(q,q',t) L_{q_j}=0,
\tag{A}
\end{equation}
consider
\begin{align*}
\frac{d\ }{dt} \Bigl(\sum_j q'_j L_{q'_j} L\Bigr)=&
\sum_j q''_j L_{q'_j} + \sum_j{\color{blue}{ q'_j\frac{d\ }{dt} L_{q'_j}}}  \frac{d\ }{dt}L \\
=&\sum_j q''_j L_{q'_j} + \sum_j {\color{blue}{q'_j L_{q_j}}}{\color{magenta}{\frac{d\ }{dt}L}}\\
=&\sum_j q''_j L_{q'_j} + \sum_j q'_j L_{q_j} 
{\color{magenta}{\Bigl(\sum_j q''_j L_{q'_j} + \sum_j q'_j L_{q_j}+L_t\Bigr)}},
\end{align*}
where transition from the first line to the second is due to (A) and from the second to the third due to chain rule.
Therefore
\begin{equation}
\frac{d\ }{dt} \Bigl(\sum_j q'_j L_{q'_j} L\Bigr)=L_t
\tag{B}
\end{equation}
and if $L$ does not depend explicitly on $t$ we have $L_t=0$ and
\begin{equation}
\frac{d\ }{dt} \Bigl(\sum_j q'_j L_{q'_j} L\Bigr)=0.
\tag{C}
\end{equation}
Note: $q=(q_1,\ldots, q_n)$ etc

Do you mean
\begin{equation}
\frac{d\ }{dt} L_{q'_j}(q,q',t) L_{q_j}=0
\tag{A1}
\end{equation}
and
\begin{equation}
\frac{d\ }{dt} \Bigl(\sum_j q'_j L_{q'_j} L\Bigr)=0?
\label{C1}
\end{equation}
and do you mean $q=(q_j)?$

Corrected.
In the Hamiltonian dynamics (beyond scope of this class)
\begin{equation}
\frac{dq_j}{dt}=H_{p_j}, \quad \frac{dp_j}{dt}=H_{q_j}
\tag{D}
\end{equation}
we have
\begin{equation}
\frac{d\ }{dt}H = \sum_j H_{q_j} q'_j +\sum_j H_{p_j} p'_j +H_t= H_t
\tag{E}
\end{equation}
and
\begin{equation}
\frac{d\ }{dt}H = 0
\tag{F}
\end{equation}
as long as $H_t=0$.
Thus, $\frac{\partial L}{\partial t}=\frac{\partial H}{\partial t}$, but the leftside is calculated in independent variables $(q,q',t)$ and the rightside is calculated in independent variables $(q,p,t)$; $p_j=L_{q'_j}$ calculated in independent variables $(q,q',t)$ and $q'_j=H_{p_j}$ calculated in independent variables $(q,p,t)$.