particle in moving from A to B. If ds be the element of the curve, the finite value of vds between A and B will depend on the nature of the path or curve in which the body moves. The principle of Least Action consists in this, that if the particle be free to move in every direction between these two points, except in so far as it obeys the action of the forces X, Y, Z, it will in virtue of this action, choose the path in which the integral ∫vds is a minimum; and if it be constrained to move on a given surface, it will still move in the curve in which ∫vds is a minimum among all those that can be traced on the surface between the given points.
To demonstrate this principle, it is required to prove the variation of ∫vds to the zero, when A and B, the extreme points of the curve are fixed.
By the method of variations δ∫vds = ∫δ.vds: for ∫ the mark of integration being relative to the differentials, is independent of the variations.
Now
δ.vds = δv.ds + vδ.ds, but v = dsdt or ds = vdt;
hence
δv.ds = vδvdt = dt12δ.v²,
and therefore
δ.vds = dt . 12δ.v² + v.δ.ds.
The values of the two last terms of this equation must be found separately. To find dt.12δ.v². It has been shown that
v² = c + 2∫(Xdx + Ydy + Zdz),
its differential is vdv = (Xdx + Ydy + Zdz),
and changing the differentials into variations,
12δ.v² = Xdx + Ydy + Zdz.
If 12δ.v² be substituted in the general equation of the motion of a particle on its surface, it becomes
12δ.v² = d²xdt²δx + d²ydt²δy + d²zdt²δz + λδu = 0.
But λδu does not enter into this equation when the particle is free; and when it must move on the surface whose equation is u = 0, δu is also zero; hence in every case the term λδu vanishes; therefore
dt12δ.v² = d²xdt²δx + d²ydt²δy + d²zdt²δz
is the value of the first term required.
A value of the second term v.δ.ds must be found. Since
ds² = dx² + dy² + dz².
