The systems contemplated by Lagrange are composed of discrete
particles, or of rigid bodies, in finite number, connected (it may be)
in various ways by invariable geometrical relations, the fundamental
postulate being that the position of every particle of the system at
any time can be completely specified by means of the instantaneous
values of a finite number of independent variables q1, q2, ... q_n,
each of which admits of continuous variation over a certain range, so
that if x, y, z be the Cartesian co-ordinates of any one particle, we
have for example
x = [f](q1, q2, ... q_n), y = &c., z = &c., (1)
where the functions [f] differ (of course) from particle to particle.
In modern language, the variables q1, q2, ... q_n are _generalized
co-ordinates_ serving to specify the _configuration_ of the system;
their derivatives with respect to the time are denoted by q`1, q`2,
... q`_n, and are called the _generalized components of velocity_. The
continuous sequence of configurations assumed by the system in any
actual or imagined motion (subject to the given connexions) is called
the _path_.
Impulsive motion.
For the purposes of a connected outline of the whole subject it is
convenient to deviate somewhat from the historical order of
development, and to begin with the consideration of _impulsive_
motion. Whatever the actual motion of the system at any instant, we
may conceive it to be generated instantaneously from rest by the
application of proper impulses. On this view we have, if x, y, z be
the rectangular co-ordinates of any particle m,
mx` = X', my` = Y', mz` = Z', (2)
where X', Y', Z' are the components of the impulse on m. Now let
[delta]x, [delta]y, [delta]z be any infinitesimal variations of x, y,
z which are consistent with the connexions of the system, and let us
form the equation
[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) =
[Sigma](X'[delta]x + Y'[delta]y + Z'[delta]z), (3)
where the sign [Sigma] indicates (as throughout this article) a
summation extending over all the particles of the system. To transform
(3) into an equation involving the variations [delta]q1, [delta]q2,
... of the generalized co-ordinates, we have
dPx dPx
x` = ---- q`1 + ---- q`2 + ..., &c., &c. (4)
dPq1 dPq2
dPx dPx
[delta]x = ---- [delta]q1 + ---- [delta]q2 + ..., &c., &c. (5)
dPq1 dPq2
and therefore
[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) =
(A11q`1 + A12q`2 + ...)[delta]q1 +
(A21q`1 + A22q`2 + ...)[delta]q2 + ..., (6)
where
_ _
| / dPx \ squared / dPy \ squared / dPz \ squared | \
A_rr = [Sigma]m | ( ----- ) + ( ----- ) + ( ----- ) |, |
|_ \dPq_r/ \dPq_r/ \dPq_r/ _| |
_ _ > (7)
| dPx dPx dPy dPy dPz dPz | |
A_rs = [Sigma]m | ----- ----- + ----- ----- + ----- ----- | = A_sr. |
|_dPq_r dPq_s dPq_r dPq_s dPq_r dPq_s _| /
If we form the expression for the kinetic energy [Tau] of the system,
we find
2[Tau] = [Sigma]m(x` squared + y` squared + z` squared) = A11q`1 squared + A22q`2 squared + ...
+ 2A12q`1q`2 + ... (8)
The coefficients A11, A22, ... A12, ... are by an obvious analogy
called the _coefficients of inertia_ of the system; they are in
general functions of the co-ordinates q1, q2, ... . The equation (6)
may now be written
dP[Tau] dP[Tau]
[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) = ------- [delta]q1 + ------- [delta]q2 + ... (9)
dPq`1 dPq`2
This maybe regarded as the cardinal formula in Lagrange's method. For
the right-hand side of (3) we may write
[Sigma](X'[delta]x + Y'[delta]y + Z'[delta]z) = Q'1[delta]q1 + Q'2[delta]q2 + ... , (10)
where
/ dPx dPy dPz \
Q'_r = [Sigma]( X'----- + Y'----- + Z'----- ). (11)
\ dPq_r dPq_r dPq_r/
The quantities Q1, Q2, ... are called the _generalized components of
impulse_. Comparing (9) and (10), we have, since the variations
[delta]q1, [delta]q2,... are independent,
dP[Tau] dP[Tau]
------- = Q'1, ------- = Q'2, ... (12)
dPq`1 dPq`2
These are the general equations of impulsive motion. It is now usual
to write
dP[Tau]
p_r = ------- (13)
dPq`_r
The quantities p1, p2, ... represent the effects of the several
component impulses on the system, and are therefore called the
_generalized components of momentum_. In terms of them we have
[Sigma]m(x`[delta]x + y`[delta]y + z`[delta]z) = p1[delta]q1 + p2[delta]q2 + ... (14)
Also, since [Tau] is a homogeneous quadratic function of the
velocities q`1, q`2 ...,