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Lagrange's function of zero degree and first and second degrees, for the equations of motion in kinetic energy, as follows :terms of generalized co-ordinates

T = K+(\)j +(0) + ... +}{(4,4)}+($,$) $+...2(4,4)od...}..(27), deduced direct by

where transformation from the equations of

K = } Em motion in

dt terms of Cartesian

dx dr co-ordi

dy dy dz) dz) (4) = {m

etc. nates.

dt) dy

dt) dys

dy dz (4,4)= Em

..(28); +


(dx dx dy dy dz dz
(4,0)= {m

dy dodų do dy do



at) dų

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K, (4), (6), (4,4), (4, 6), etc. being thus in general each a known function of t, 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 4, $, etc. are related to X, Y, Z, X,, 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 8x, 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, 1819, p. 201.

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

equations of find from (27) and (28),

expauded. T' = 1 {(484) 42 + 2 (4,0) 40 + (0,0) $+ ...} ...(29), d d

(4,4) ° +(\, *) 8+ ...


at dj

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d (4, 5) + ...} (20”).

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° +2 d (4,4)

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dT (d (4, 4)
{ “ + 2


29 dy dy


Hence the y-equation of motion expanded in this, the most
important class of cases, is as follows:

(4,4) Ü + (4, ) Ö+ ... +Q+(T") = 4,
fd (4,4)

d (T)=

a(4,0) d (0,0) )

+ 2


Remark that Qu (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 y appears singularly, and the other co-ordinates sym-
metrically with one another.

Multiply the y-equation by 4, the p-equation by 0, and so Equation of on; and add. In what comes from Q¢ (T) we find terms

d(, v)

4$.4, and



d (4,4).j'.$;



which together yield

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

[(y, v) + (v, ) 6+ ...];
+ [(y, 4)$ + (6, 6)+ ...];




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dT or

- ¥° +00+ dt

.(29). Hamilton's When the kinematical relations are invariable, that is to say, form,

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, 4, 6, ..., be expressed in terms of , n,..., the impulses re quired to produce the motion from rest at any instant ($ 313 (d)]; so that T will now be a homogeneous quadratic function, and j, s, ... each a linear function, of these elements, with coefficients-functions of y, 0, etc., depending on the kinematical conditions of the system, but not on the particular motion. Thus, denoting, as in § 322 (29), by 0, partial differentiation with reference to £, n, ..., y, $..., considered as independent Fariables, we have [S 313 (10)]


ar į

= ds'


dn' and, allowing d to denote, as in what precedes, the partial differentiations with reference to the system ý, , have [ 313 (8)]



= dj'


to The two expressions for T being, as above, $ 313, T= }{(4,4)ť?+...+2(4, 0){$+...} = }{{y, y]&+...+2[4, +] $n+... }(32),

the second of these is to be obtained from the first by substituting for 4, ...., their expressions in terms of $, », ... Hence ar dr dr aj dT

dT адт a aT

* dy dy dj dy do dy dy

dy dę dy dn
da ar ат

dT ar

dy dų αξ

From this we conclude

aT d

; and, similarly,

..... (33).

Hence Lagrange's equations become

de OT
¥, etc.......

... (34).

..., 4, 6, ..., we







) =

+ 2




at du


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)]

Ψδψ + Φδφ +

. =-8V, dV

IV and therefore


tive system. Let now U denote the algebraic expression for the sum of the potential energy, V, in terms of the co-ordinates, 4, ..., and the kinetic energy, T', in terms of the co-ordinates and the components of momentum, &, n,.... Then


dt dy

.....(35), U

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

etc. dt वह'


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

In the following examples we shall adhere to Lagrange's form

(24), as the most convenient for such applications. Examples of E.cample (A).-Motion of a single point (m) referred to polar the use of Lagrange's co-ordinates (r, 0, ). From the well-known geometry of this generalized equations of case we see that dr, r80, and r sin 084 are the amounts of linear motion ;polar co- displacement corresponding to infinitely small increments, ôr, 86, ordinates.

84, of the co-ordinates : also that these displacements are respectively in the direction of r, of the arc rdo (of a great circle) in the plane of r and the pole, and of the arc r sin 686 (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

F=R, Gr=, and IIr sin 6 = 0;
R, 0, being what the generalized components of force (313)
become for this particular system of co-ordinates. We also see
that i, ro, and r sin 0 are three components of the velocity,
along the same rectangular directions. Hence

T' = } m(** + r ® + på sino 0*).
From this we have


Emre sin'00;


= mr(Q® + sino 0%"), mre sin 0 cos 00, = 0.

Hence the equations of motion become


r(0€ + sino 008)} = F, lat



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B :')}
80"} = Gr,


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-p? sin cos 04
d (rsin'00)

Hr sin 8;

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



+ sinto des




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