Varying action. Same propositions for generalized coordinates. of rigid bodies connected with one another or not, we might, it is true, be contented to regard it still as a system of free particles, by taking into account among the impressed forces, the forces necessary to compel the satisfaction of the conditions of connexion. But although this method of dealing with a system of connected particles is very simple, so far as the law of energy merely is concerned, Lagrange's methods, whether that of "equations of condition," or, what for our present purposes is much more convenient, his "generalized co-ordinates," relieve us from very troublesome interpretations when we have to consider the displacements of particles due to arbitrary variations in the configuration of a system. ( Let us suppose then, for any particular configuration (x1, y1, ≈1) m ̧ (x ̧§x ̧+ÿ ̧§î ̧+8≈1) + etc., to become έd¥ + ŋdp +¿80+etc. (15), when transformed into terms of y, p, 0..., generalized co-ordinates, as many in number as there are of degrees of freedom for the system to move [§ 313, (c)]. The same transformation applied to the kinetic energy of the system would obviously give 1 2 } m ̧ (¿‚3 + ÿ ̧2 + ż ̧3) + etc. = § (§4 + n$ + ¿Õ + etc.) .. .(16), and hence έ,,, etc., are those linear functions of the generalized velocities which, in § 313 (e), we have designated as "generalized components of momentum ;" and which, when T, the kinetic energy, is expressed as a quadratic function of the velocities (of course with, in general, functions of the co-ordinates 4, 4, 0, etc., for the coefficients) are derivable from it thus: Hence, taking as before non-accented letters for the second, and accented letters for the initial, configurations of the system respectively, we have action. These equations (18), including of course (14) as a particular case, Varying express in mathematical terms the proposition stated in words above, as the Principle of Varying Action. The values of the momentums, thus, (14) and (18), expressed in terms of differential coefficients of A, must of course satisfy the equation of energy. Hence, for the case of free particles, Or, in general, for a system of particles or rigid bodies connected in any way, we have, (16) and (18), Hamilton's "characteristic equation" of motion in Cartesian co-ordi nates. where 4, 4, etc., are expressible as linear functions of etc., by the solution of the equations (4, 4) 4 + (4, 6) $ + (4, 0) 8 + etc. dA dA dy' do' (4, 4) ↓ + (4, 6) $ + (4, 0) 8 + etc. = n = Hamilton's characteristic equation of motion in generalized co-ordi nates. where it must be remembered that (4, 4), (4, 4), etc., are functions of the specifying elements, V, 4, 0, etc., depending on the kinematical nature of the co-ordinate system alone, and quite independent of the dynamical problem with which we are now concerned; being the coefficients of the half squares and the products of the generalized velocities in the expression for the Varying action. Proof that the charac teristic equation defines the motion, for free particles. kinetic energy of any motion of the system; and that (V', v′), It is remarkable that the single linear partial differential equa- + + dV =- 2 dx Using these properly in the preceding and taking half; and Now if we multiply the first member by dt, we have clearly the change of the value of me, due to varying, still on the Hamil action. tonian supposition, the co-ordinates of all the points, that is to say, Varying the configuration of the system, from what it is at any moment to Proof that what it becomes at a time dt later; and it is therefore the actual the charac change in the value of me,, in the natural motion, from the time, equation t, when the configuration is (x,, Y, Z, X, ..., E), to the time motion, for t + dt. It is therefore equal to midt, and hence (25) becomes particles. But these are [§ 293, (4)] the elementary differential equations of the motions of a conservative system composed of free mutually influencing particles. If next we regard x,, y1, z1, x,, etc., as constant, and go through precisely the same process with reference to x', y', z', x', etc., we have exactly the same equations among the accented letters, with only the difference that A appears in place of A; and end with mï' dV' from which we infer that, if (20) is satisfied, the motion represented by (14) is a natural motion through the configuration (x,', y'', z'', x', etc.). Hence if both (19) and (20) are satisfied, and if when x1 = x,', Y1 =Y1', z1 =≈,', x = x,, etc., we have dA dA dx" etc., the motion represented by (14) is a natural motion through the two configurations (x, y, z, x,', etc.), and (x1, Y1, %1, X, etc.). Although the signs in the preceding expressions have been fixed on the supposition that the motion is from the former, to the latter configuration, it may clearly be from either towards the other, since whichever way it is, the reverse is also a natural motion (§ 271), according to the general property of a conservative system. teristic defines the free position To prove the same thing for a conservative system of particles Same proor rigid bodies connected in any way, we have, in the first place, for a from (18) dn de de de where, on the Hamiltonian principle, we suppose 4, 4, etc., and έ, 7, etc., to be expressed as functions of , p, etc., y', p', etc., connected system, and generalized co-ordi nates. Varying action. and the sum of the potential and kinetic energies. On the same supposition, differentiating (21), we have dV + $ dy +0 + etc. + έ +n +8 + etc. di аф dė 2 ....(27). dy dy But, by (26), and by the considerations above, we have αξ dn dr i + $ +0 + etc. = & + $ +6 dy dy where & denotes the rate of variation of έ per unit of time in the actual motion. de de αξ + etc. = § .... (28), dy аф de if, as in Hamilton's system of canonical equations of motion, we suppose, 4, etc., to be expressed as linear functions of έ, 7, etc., with coefficients involving 4, 4, 6, etc., and if we take a to denote the partial differentiation of these functions with reference to the system &,,..., 4,..., regarded as independent variables. Let the coefficients be denoted by [, ], etc., according to the plan followed above; so that, if the formula for the kinetic energy be T = { {[4, 4] §2 + [6, 6] n2 + ... + 2 [4, 4] §n + etc.}.....(30), we have dT --- dT dn = [4, 4] § + [4, $] n + [4, 0] + etc. = [4, 4] § + [6, $] n + [6, 0] ¿ + etc. etc. etc. where of course [4, 4], and [4, 4], mean the same. Hence did [v, v ] & + _ == dy dy = = [4, 4], ..... ; = .(31), 08 _ d[b, 4] & + etc. etc., = [4, 4], αξ αξ dy = dy + etc. = {[4,4] § +[4,4]n+etc.} ~~+~+{[4,]+[6,6]n+etc.} d[4,] + etc.+] {= d[8, 9] n2 + etc. + 2~ dy dy ат dy ; dy dy dy |