Lagrange's equations of motion in

terms of

function of zero degree and first and second degrees, for the kinetic energy, as follows:

generalized TK+(4) ↓ + ($) $ + ... + 1 { (4, 4) 42 + (Þ, $) $2 + ... 2(4, 6) ÿ¿...}.. (27),

co-ordinates

deduced

direct by transforma

tion from the equations of motion in terms of Cartesian co-ordinates.

K, (4), (4), (4, 4), (4, 4), etc. being thus in general each a known function of t, 4, 4, etc.

Equations (24) above are Lagrange's celebrated equations of motion in terms of generalized co-ordinates. It was first pointed out by Vieille that they are applicable not only when

,, etc. are related to x, y, z,, 2, etc. by invariable relations as supposed in Lagrange's original demonstration, but also when the relations involve t in the manner shown in equations (25). Lagrange's original demonstration, to be found in the Fourth Section of the Second Part of his Mécanique Analytique, consisted of a transformation from Cartesian to generalized co-ordinates of the indeterminate equation of motion; and it is the same demonstration with unessential variations that has been hitherto given, so far as we know, by all subsequent writers including ourselves in our first edition (§ 329). It seems however an unnecessary complication to introduce the indeterminate variations dx, dy, etc.; and we find it much simpler to deduce Lagrange's generalized equations by direct transformation from the equations of motion (19) of a free particle.

* Sur les équations différentielles de la dynamique, Liouville's Journal, 1849, p. 201.

generalized

When the kinematic relations are invariable, that is to say Lagrange's when t does not appear in the equations of condition (25), we form of the find from (27) and (28),

T = { {(4, 4) ¿↓3 + 2 (4, 6) $$ + (Þ; 6) $3 + ...} .............. (29),

= (4, 4) Ÿ + (4, 6) $ + ...

equations of motion expanded.

d dT

dt dy

Hence the

equation of motion expanded in this, the most

important class of cases, is as follows:

(4, 4) Ÿ + (4, 6) Ï + ... + Qu (T) = ¥,

Remark that Q (T) is a quadratic function of the velocity-com-
ponents derived from that which expresses the kinetic energy
(T) by the process indicated in the second of these equations,
in which appears singularly, and the other co-ordinates sym-
metrically with one another.

Multiply the

equation by , the

equation by 4, and so Equation of on; and add. In what comes from Q (T) we find terms

energy.

With this, and the rest simply as shown in (29"), we find

[(4, 4) Ÿ + (4, 6) $ + ... ] &

+ [(4, 6) 4 + (4, 4) ☀ + ... ] $

+

VOL. I.

20

Equation of energy.

Hamilton's form.

When the kinematical relations are invariable, that is to say, when t does not appear in the equations of condition (25), the equations of motion may be put under a slightly different form first given by Hamilton, which is often convenient; thus :-Let T,,,..., be expressed in terms of έ, n,..., the impulses required to produce the motion from rest at any instant [§ 313 (d)]; so that I will now be a homogeneous quadratic function, and ,,... each a linear function, of these elements, with coefficients-functions of 4, 4, etc., depending on the kinematical conditions of the system, but not on the particular motion. Thus, denoting, as in § 322 (29), by a, partial differentiation with reference to έ, n, ..., 4, 4,..., considered as independent variables, we have [§ 313 (10)]

and, allowing d to denote, as in what precedes, the partial differentiations with reference to the system &, 4,

we

(31).

T = { {(4, 4) ¿2 + ...+2(4, 6) $$+.....} = {{[4, ¥]§3+...+2[4, ¢]§n+.....}(32), the second of these is to be obtained from the first by substituting for , ...., their expressions in terms of έ, 7, ... Hence

form.

In § 327 below a purely analytical proof will be given of Hamilton's Lagrange's generalized equations of motion, establishing them directly as a deduction from the principle of "Least Action," independently of any expression either of this principle or of the equations of motion in terms of Cartesian co-ordinates. In their Hamiltonian form they are also deduced in § 330 (33) from the principle of Least Action ultimately, but through the beautiful "Characteristic Equation" of Hamilton.

319. Hamilton's form of Lagrange's equations of motion in terms of generalized co-ordinates expresses that what is required to prevent any one of the components of momentum. from varying is a corresponding component force equal in amount to the rate of change of the kinetic energy per unit increase of the corresponding co-ordinate, with all components of momentum constant: and that whatever is the amount of the component force, its excess above this value measures the rate of increase of the component momentum.

In the case of a conservative system, the same statement takes the following form:-The rate at which any component momentum increases per unit of time is equal to the rate, per unit increase of the corresponding co-ordinate, at which the sum of the potential energy, and the kinetic energy for constant momentums, diminishes. This is the celebrated "canonical form" of the equations of motion of a system, though why it has been so called it would be hard to say.

Let V denote the potential energy, so that [§ 293 (3)]

Let now U denote the algebraic expression for the sum of the potential energy, V, in terms of the co-ordinates, , ..., and the kinetic energy, T', in terms of the co-ordinates and the

components

"Canonical form" of Hamilton's general equations of motion of a

conserva

tive system.

Examples of the use of Lagrange's generalized equations of motion;polar coordinates.

the latter being equivalent to (30), since the potential energy does not contain έ, n, etc.

In the following examples we shall adhere to Lagrange's form (24), as the most convenient for such applications.

Example (A).-Motion of a single point (m) referred to polar co-ordinates (r, 0, 4). From the well-known geometry of this case we see that dr, r80, and r sin 084 are the amounts of linear displacement corresponding to infinitely small increments, dr, de, do, of the co-ordinates: also that these displacements are respectively in the direction of r, of the arc rde (of a great circle) in the plane of r and the pole, and of the arc r sin 0ồp (of a small circle in a plane perpendicular to the axis); and that they are therefore at right angles to one another. Hence if F, G, H denote the components of the force experienced by the point, in these three rectangular directions, we have

FR, Gr=, and Hr sin 0 Þ ;

R, O, being what the generalized components of force (§ 313) become for this particular system of co-ordinates. We also see that ở, rẻ, and r sin 0 are three components of the velocity, along the same rectangular directions. Hence

or, according to the ordinary notation of the differential calculus,