Index accessory equations, 174 accessory variational problem, 151 Atwood’s machine, 109, 137 autonomous, 65, 86 Bernoulli equation, 147 Bernoulli–Euler beam theory, 72 brachistochrone, 1–4, 18, 59–61, 64, 196, 203, 267, 273, 274 on a cylinder, 66 terrestrial, 5–7, 19, 66 to a vertical line, 212 with variable endpoints, 211 c-discriminant, 157, 207, 258 canonical equations, 90–92, 105, 190, 270 canonical momenta, 88–92, 269, 270 cantilever beam, 71–76, 103, 123 Carath´ eodory equations, 269 catenary, 53–56, 58, 65, 108, 160–167 catenoid, 15–17, 23, 63, 160–167 characteristic, 219–222, 224, 226, 238, 240, 244 chute-the-chute, 212 circle of latitude, 62, 63 complementary curve of descent, 20 condition(s) Euler–Lagrange, 41, 245 Jacobi, 147, 154, 155, 160, 165, 167–170, 172, 190, 232, 245, 252, 258 Legendre, 142–144, 158, 160, 169, 172, 184, 245, 251, 252, 254 natural boundary, 191, 194–197, 200, 211 necessary, 31, 172, 245 second-variation, 141 strengthened Jacobi, 173, 235, 237, 240, 259, 260, 263, 264 strengthened Legendre, 143, 158, 173, 235, 236, 240, 266 sufficient, 173–175, 255, 263, 264, 266, 273 transversality, 197, 200–207, 212, 213, 215, 217, 227, 242 variable-endpoint, 176 weak sufficiency, 149 Weierstrass, 242–247, 254, 273 Weierstrass–Erdmann corner, 176, 215–219, 221, 222 conjugate point, 155, 159, 165, 168, 237, 240, 258, 266 conjugate value, 155, 159 constraint(s) holonomic, 109, 110, 129–132 inequality, 112, 136 295
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