Case of varying relations.
It has so far been assumed that the geometrical relations, if any,
which exist between the various parts of the system are of the type Sec.
1 (1), and so do not contain t explicitly. The extension of Lagrange's
equations to the case of "varying relations" of the type
x = f(t, q1, q2,...q_n), y = &c., z = &c., (1)
was made by J.M.L. Vieille. We now have
dPx dPx dPx
x` = --- + ---- q`1 + ---- q`2 + ..., &c., &c., (2)
dPt dPq1 dPq2
dPx dPx
dPx = ---- [delta]q1 + ---- [delta]q2 + ..., &c., &c., (3)
dPq1 dPq2
so that the expression Sec. 1 (8) for the kinetic energy is to be
replaced by
2[Tau] = [alpha]0 + 2[alpha]1q`1 + 2[alpha]2q`2 + ... + A11q`1 squared + A22q`2 squared + ... + A12q`1q`2 + ..., (4)
where
_ _ \
| /dPx\ squared /dPy\ squared /dPz\ squared | |
a0 = [Sigma]m |( --- ) + ( --- ) + ( --- ) |, |
|_\dPt/ \dPt/ \dPt/ _| |
_ _ > (5)
| dPx dPx dPy dPy dPz dPz | |
a_r = [Sigma]m | --- ----- + --- ----- + --- ----- |, |
|_dPt dPq_r dPt dPq_r dPt dPq_r_| |
/
and the forms of A_rr, A_rs are as given by Sec. 1 (7). It is to be
remembered that the coefficients [alpha]0, [alpha]1, [alpha]2, ...
A11, A22, ... A12 ... will in general involve t explicitly as well as
implicitly through the co-ordinates q1, q2,... Again, we find
[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) =
([alpha]1 + A11q`1 + A12q`2 + ...)[delta]q1
+ ([alpha]2 + A21q`1 + A22q`2 + ...)dPq2 + ...
dP[Tau] dP[Tau]
= ------- [delta]q1 + ------- [delta]q2 + ...
dPq`1 dPq`2
= p1[delta]q1 + p2[delta]q2 + ..., (6)
where p_r is defined as in Sec. 1 (13). The derivation of Lagrange's
equations then follows exactly as before. It is to be noted that the
equation Sec. 2 (15) does not as a rule now hold. The proof involved the
assumption that [Tau] is a homogeneous quadratic function of the
velocities q`1, q`2....
It has been pointed out by R.B. Hayward that Vieille's case can be
brought under Lagrange's by introducing a new co-ordinate ([chi]) in
place of t, so far as it appears explicitly in the relations (1). We
have then
2[Tau] = [alpha]0[chi]` squared + 2([alpha]1q`1 + [alpha]2q`2 + ...)[chi]`
+ A11q`1 squared + A22q`2 squared + ... + 2A12q`1q`2 + .... (7)
The equations of motion will be as in Sec. 2 (10), with the additional
equation
d dP[Tau] dP[Tau]
-- -------- - ------- = X, (8)
dt dP[chi]` dP[chi]
where X is the force corresponding to the co-ordinate [chi]. We may
suppose X to be adjusted so as to make [chi]" = 0, and in the
remaining equations nothing is altered if we write t for [chi] before,
instead of after, the differentiations. The reason why the equation Sec.
2 (15) no longer holds is that we should require to add a term X[chi]`
on the right-hand side; this represents the rate at which work is
being done by the constraining forces required to keep [chi]`
constant.
As an example, let x, y, z be the co-ordinates of a particle relative
to axes fixed in a solid which is free to rotate about the axis of z.
If [phi] be the angular co-ordinate of the solid, we find without
difficulty
2[Tau] = m(x` squared + y` squared +z` squared) + 2[phi]`m(xy` - yx`) + {I + m(x squared + y squared)}[phi]` squared, (9)
where I is the moment of inertia of the solid. The equations of
motion, viz.
d dP[Tau] dP[Tau] d dP[Tau] dP[Tau] d dP[Tau] dP[Tau]
-- ------ - ------ = X, -- ------- - ------- = Y, -- ------- - ------- = Z, (10)
dt dPx` dPx dt dPy` dPy dt dPz` dPz
d dP[Tau] dP[Tau]
and -- -------- - ------- = [Phi], (11)
dt dP[phi]` dP[phi]
become
m(x" - 2[phi]`y` - x[phi]` squared - y[phi]") = X, m(y" + 2[phi]`x` - y[phi]` squared + x[phi]`) = Y, mz" = Z, (12)
_ _
d | / \ |
and -- |(I + m(x squared + y squared)) [phi]` + m(xy` - yx`)| = [Phi]. (13)
dt |_\ / _|
If we suppose [Phi] adjusted so as to maintain [phi]" = 0, or (again)
if we suppose the moment of inertia I to be infinitely great, we
obtain the familiar equations of motion relative to moving axes, viz.
m(x" - 2[omega]y` - [omega] squaredx) = X, m(y" + 2[omega]x` - [omega] squaredy) = Y, mz" = Z, (14)
where [omega] has been written for [phi]. These are the equations
which we should have obtained by applying Lagrange's rule at once to
the formula
2[Tau] = m(x` squared + y` squared + z` squared) + 2m[omega](xy` - yx`) + m[omega] squared(x squared + y squared), (15)
which gives the kinetic energy of the particle referred to axes
rotating with the constant angular velocity [omega]. (See MECHANICS, Sec.
13.)
More generally, let us suppose that we have a certain group of
co-ordinates [chi], [chi]', [chi]", ... whose absolute values do not
affect the expression for the kinetic energy, and that by suitable
forces of the corresponding types the velocity-components [chi]`,
[chi]`', [chi]`", ... are maintained constant. The remaining
co-ordinates being denoted by q1, q2, ... q_n, we may write
2[Tau] = [@] + [Tau]0 + 2([alpha]1q`1 + [alpha]2q`2 + ...)[chi]`
+ 2([alpha]'1q`1 + [alpha]'2q`2 + ...)[chi]`' + ..., (16)
where [@] is a homogeneous quadratic function of the velocities q`1,
q`2, ... q`_n of the type Sec.1 (8), whilst [Tau]0 is a homogeneous
quadratic function of the velocities [chi]`,[chi]`', [chi]`", ...
alone. The remaining terms, which are bilinear in respect of the two
sets of velocities, are indicated more fully. The formulae (10) of Sec. 2
give n equations of the type
d /dP[@]\ /dP[@]\ dP[Tau]0
--( ----- ) - ( ----- ) + (r, 1)q`1 + (r, 2)q`2 + ... - -------- = Q_r (17)
dt \dPq_r/ \dPq_r/ dPq_r
where
/dPa_r dPa_s\ /dPa'_r dPa'_s\
(r, s) = ( ----- - ----- )[chi]` + ( ------ - ------ )[chi]`' + .... (18)
\dPq_s dPq_r/ \dPq_s dPq_r/