Page:PoincareDynamiqueJuillet.djvu/9

assuming that f, α, F, u, etc.. are subject to the following conditions and the ones deduced by symmetry:

(2)	$\sum {\frac {df}{dx}}=\rho ,\quad \alpha ={\frac {dH}{dy}}-{\frac {dG}{dz}},\quad u={\frac {df}{dt}}+\rho \xi .$

Regarding the integral J, it must be extended:

I° in relation to the volume element dτ = dx dy dz over the whole space;

2° in relation to time t, over the interval between the limits t = t₀, t = t₁.

According to the principle of least action, the integral J must be a minimum, if one sets the various quantities which appear in:

1° the conditions (2);

2° the condition that the state of the system is determined by both limiting times t = t₀, t = t₁.

This last condition allows us to transform our integral by partial integration with respect to time. If we have indeed an integral of the form

$\int dt\ d\tau A{\frac {dB\delta C}{dt}},$

where C is a quantity that defines the system state and its variation δC, it will be equal to (by partial integration with respect to time):

$\int d\tau \mid AB\delta C\mid {t=t_{1} \atop t=t_{0}}-\int dt\ d\tau {\frac {dA}{dt}}dB\delta C.$

Since the system state is determined by both limiting times, it is δC = 0 for t = t₀, t = t₁, so the first integral which is related to these two periods is zero, and the 2^nd one remains.

We can also integrate by parts with respect to x, y or z, we have indeed

$\int A{\frac {dB}{dx}}dx\ dy\ dz\ dt=\int AB\ dy\ dz\ dt-\int B{\frac {dA}{dx}}dx\ dy\ dz\ dt.$

Our integrations are extended to infinity, it must be $x=\pm \infty$ in the first integral on the right-hand side; so, since we always assume that all our functions vanish at infinity, this integral will be zero and it follows

$\int A{\frac {dB}{dx}}d\tau \ dt=-\int B{\frac {dA}{dx}}d\tau \ dt.$

If the system is supposed to be subject to bindings, the binding conditions should be connected to the conditions imposed on the various quantities appearing in the integral J.

Let us first give to F, G, H the increasements δF, δG, δH; where:

$\delta \alpha ={\frac {d\delta H}{dy}}-{\frac {d\delta G}{dz}}.$