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Let's assume a conservative holonomic system with $n$ independent generalized coordinates and a Lagrangian $L(q,\dot{q},t)$, resulting in a system of $n$ Lagrange differential equations of second order. Building the respective Hamiltonian $H(q,p,t)$ with the Legendre transform (given the Lagrangian is not singular), we can derive a set of $2n+1$ Hamilton equations

\begin{align*} \dot{q}_i &= \frac{\partial H}{\partial p_i}, \quad i=1,\dots,n, \\ -\dot{p}_i &= \frac{\partial H}{\partial q_i}, \quad i=1,\dots,n, \\ -\frac{\partial L}{\partial t} &= \frac{\partial H}{\partial t}. \end{align*}

The first $2n$ equations are equivalent to the $n$ Lagrange equations, so these are the ones we are primarily interested in, and their interpretation is well-known. But what about the last equation? All my text books mention it, some count it to the Hamiltonian equations, others don't, but none of them explains its physical significance. Is there any at all? From the examples I have been computing so far, I always get rather complicated expressions, which I have not been able to interpret in any way.

Obviously, if $L$ does not depend explicitly on $t$, then the equation says that $H$ does not either, and vice versa. But is there a significance of this equation beyond that?

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    $\begingroup$ The last should be $-\partial L/\partial t= dH/dt$, with full derivative of $H$. If the Lagrangian does not depend explicitly on $t$ then $H$ is conserved. $\endgroup$ – ZeroTheHero Feb 19 at 17:41
  • $\begingroup$ @ZeroTheHero, I think we have $ -\frac{\partial L}{\partial t} = \frac{\partial H}{\partial t} = \frac{dH}{dt} $, so there is no difference. $\endgroup$ – Roland Salz Feb 19 at 21:00
  • $\begingroup$ $\partial H/\partial t \ne dH/dt$ in general. $\endgroup$ – ZeroTheHero Feb 19 at 21:02
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    $\begingroup$ @ZeroTheHero, with the help of the first 2n equations we find $ \frac{dH}{dt} = \frac{\partial H}{\partial q_i} \dot{q}_i + \frac{\partial H}{\partial p_j} \dot{p}_j + \frac{\partial H}{\partial t} = \frac{\partial H}{\partial q_i} \frac{\partial H}{\partial p_i} - \frac{\partial H}{\partial p_j} \frac{\partial H}{\partial q_j} + \frac{\partial H}{\partial t} = \frac{\partial H}{\partial t} $. $\endgroup$ – Roland Salz Feb 20 at 10:34
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The relations $$\frac{\partial (L+H)}{\partial t}~=~0\qquad\text{and}\qquad\frac{\partial (L+H)}{\partial q^i}~=~0\tag{1}$$ signify/illustrate the fact that time $t$ and positions $q^i$ are external parameters/passive spectators in the $v\leftrightarrow p$ Legendre transformation $$ L(q,v,t)+H(q,p,t)~=~p_iv^i.\tag{2}$$ Eqs. (1) are direct consequences of eq. (2). Eqs. (1) are not EOMs.

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