In a 1931 paper Hotelling gives the discounted profit of a mining operation as:
$$P=\int_{0}^{\infty} \dot{x} p(x,\dot{x},t) e^{-rt} \:\:dt$$
Note that this is, for the most part, a typical calculus of variations set up. The integrand could be defined in the more familiar notation $L(x,\dot{x},t;r)=\dot{x} p(x,\dot{x},t) e^{-rt}$. In physics, $P$ would be the "action."
(I won't go into the economic significance Hotelling attaches to the variables because I am interested in this only as a calculus of variations problem.)
In Hotelling's mine problem, the point is to solve for the $\dot{x}$ that maximizes $P$. To achieve this one of course solves the Euler-Lagrange equation for $\dot{x}$ (and then checks for sufficiency via the Legendre and Jacobi conditions).
Before solving the Euler-Lagrange equation, however, Hotelling reduces it from its usual form ($\frac{d}{dt}\frac{\partial L}{\partial{\dot{x}}}-\frac{\partial L}{\partial x}=0$) to this:
$$L-\dot{x}\frac{\partial L}{\partial{\dot{x}}}=0$$
This is my question. On what basis can you justify this reduction? Hotelling's justification is that he invokes the transversality condition that $P$ is a maximum. Another way he states this is that $x$ is a limited quantity (the amount of gold or whatever in the mine) and will approach an asymptote.
I don't get how this means you can reduce the E-L equation as he does. If someone understands, please help.
Hotelling's paper is: "The Economics of Exhaustible Resources"
Imagine a small perturbation in which one increases $\dot{x}_t$ (the rate of mining at time $t$) by 1 at time $t$; at all other times, the plan stays the same. This causes an extra $dt$ units to be mined at time $t$, and thus $x_t$ (the amount of resources mined so far) rises by $dt$ at time $t$ and for all future times.
We know that the cost of increasing $x_t$ by 1 unit is exactly the (present value of) the price of the good at time $t$, $L/\dot{x}$. For example, if it were less costly to increase $x_t$ than the price of the good, one should just increase $x_t$ by 1 to raise profits.
Thus, the cost of a perturbation of size $dt$ is $\frac{L}{\dot{x}} dt$.
The benefit is $\frac{\partial L}{\partial \dot{x}} dt$ as one directly increases flow profits ($Ldt$) at time $t$.
Because $P$ is already maximized, this perturbation must lead to no change in profits. Hence we have the result
$\frac{L}{\dot{x}} = \frac{\partial L}{\partial \dot{x}}$
Which is Hotelling's equation.