296 Index isoperimetric, 107, 117, 132, 136, 168 nonholonomic, 110, 111, 132, 136 one-sided, 112 positional, 109 convexity, 244, 266 coordinate(s) cyclic, 51 generalized, 80, 82, 89, 99, 202 ignorable, 51, 86, 87 corner, 44, 175, 176, 220, 226, 230–233, 235 curvature, 73, 102, 184, 185, 187 mean, 17, 99 principal, 17, 99 cusp, 250 cycloid, 3, 20, 61 D’Alembert’s wave equation, 99 direct method(s), 45 Euler’s, 28, 46 Galerkin method, 45 Kantorovich method, 45 of finite differences, 45 Rayleigh-Ritz method, 45 diving board, 106 du Bois-Reymond’s simplification, 41 dynamic programming, 273 effective potential, 87 elastic body, 72 elastic strain energy, 72 elastica, 123, 128, 136 elastostatics, 212 energy function, 92 envelope, 157, 161, 206–208, 258, 272 equivalent variational problems, 255, 267, 268, 273 Euler’s (isoperimetric) rule, 119 Euler’s identity, 181–183, 227 Euler–Lagrange equation(s), 28, 31, 32, 38, 41, 44, 46–52, 59, 62–66, 68, 71, 78, 79, 81, 83–85, 88, 90, 91, 94, 95, 98, 100, 104, 105, 121, 131, 132, 135, 151, 152, 172, 173, 180–185, 187, 190, 194, 200, 216, 218, 235, 260, 263, 264, 266, 270, 273 for a degenerate functional, 48 integrated form, 44 ultradifferentiated form, 40, 47, 84, 183, 260 Euler–Poisson equation(s), 71, 76, 102, 103, 123 evolute, 208, 209 extrema, 33, 47, 81, 204, 263, 274 extremal(s), 47, 64, 81, 156, 270 broken, 215, 222, 230, 231, 238, 240 extremaloids, 231 extremum, 28, 30, 41, 49, 115, 117–119, 130, 176, 204, 212, 244, 245, 260, 263, 264 Fermat-type integral(s), 203, 207, 266 fibers graded-index, 94, 101 longitudinal, 72 optical, 94 step-index, 94 field central, 257, 264 of extremals, 255, 258, 259, 263, 264 proper, 256, 257 first fundamental form, 11, 23 first-order fundamental quantities, 11, 62 focal point(s), 206, 209 Frosh Pond, 113 functional, 27 degenerate, 48, 50, 101 fundamental lemma, 38–40, 45, 78 Gauss’s law, 22 geodesics, 7, 19, 62 in the plane, 31, 146, 184 numerical techniques for, 64 on a cylinder, 138 on a pseudosphere, 13 on a right circular cone, 67 on a right circular cylinder, 66
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