provided the varied motion takes place with the same constant value of the energy, and with the same constant cyclic momenta, between the same two configurations, these being regarded as defined by the palpable co-ordinates alone.
§ 8. Hamilton’s Principal and Characteristic Functions.
In the investigations next to be described a more extended meaning is given to the symbol δ. We will, in the first instance, denote by it an infinitesimal variation of the most Principal function. general kind, affecting not merely the values of the co-ordinates at any instant, but also the initial and final configurations and the times of passing through them. If we put
S = ∫t′t (Τ − V) dt,
(1)
we have, then,
δS = (T′ − V′) δt′ − (T − V) δt + ∫t′t (δΤ − δV) dt
= (T′ − V′) δt′ − (T − V) δt + [Σm (ẋδx + ẏδy + z˙δz)]t′t.
(2)
Let us now denote by x′ + δx′, y′ + δy′, z′ + δz′, the final co-ordinates (i.e. at time t′ + δt′) of a particle m. In the terms in (2) which relate to the upper limit we must therefore write δx′ − ẋ′δt′, δy′ − ẏ′δt′, δz′ − z˙′δt′ for δx, δy, δz. With a similar modification at the lower limit, we obtain