By John David Logan

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We now compute a typical transformed velocity-squared term in the kinetic energy in order to obtain the exact form of these divergence terms. = C:t (x, + Eh)y + 1;8)2 = x/ + 21;8 Xk + (1;8)2 = (,x k = x• k 2 + I; 8 dtd (2xd + 0 (I; 8) . 46). 47) 10, 8,9. Whence, it follows from Eqs. 22) that ot. 48) k constant. 49) We remark that (xC' Ye, ze) are the coordinates of the center of mass of the system, and M is of course just the total mass of the n particles. 50) for constants A I, A z, and A 3' Denoting the constants on the right-hand sides of Eqs.

In our analysis, however, this degree of generality will not be required. 16 1 Necessary Conditions for an Extremum (r', ... • dt". The boundary of D will be denoted by Bd D. The Divergence Theorem, which states that the volume integral of a divergence expression equals a certain integral over the Bd D, is needed in our calculation of necessary conditions, so we shall now state it but without proof. Any standard advanced calculus text can be consulted for the proof. 3 Theorem Let D <;; R" and let f(t) = [it", ...

1 Necessary Conditions for an Extremum 24 EXERCISES Consider the variational problem J = J6" x2 dt. (t) = t + e sin t. Sketch this family geometrically, compute J(e) explicitly, and prove directly that M(x, 11) = 0, where x = t,11 = sin t, Deduce that x(t) = t is an extremal. 1-1 1-2 For the simplest variational problem J = J:~ L(t, x, x) dt, prove that the Euler-Lagrange equations may be written in the form 1-3 Analyze the following variational problems: (a) J(x) = (b) J(x) = (c) J(x) = fx dt -+ Ext, fxx f dt -+ Ext, txx dt -+ x(O) = 0, x(1) = 1.