Index 297 on a sphere, 8–10, 68, 146 on a toroidal anchor ring, 19 on a triaxial ellipsoid, 14 on convex surfaces, 14 on general tubular surfaces, 19 on surfaces of revolution, 14, 62 Goldschmidt solution, 15, 58 Green’s theorem, 97 Hamilton’s action, 81 Hamilton’s first principle function, 81 Hamilton–Jacobi equation, 269–274 Hamiltonian, 52, 89–92, 105, 190, 202, 203, 271 Hamiltonian equations of motion, 270 harmonic oscillator, 83, 90, 91 helicoid, 17, 23, 24 Hilbert’s invariant integral, 261, 262, 273 Hooke’s law, 73 hyperbolic paraboloid, 23, 24 hypocycloid, 7 indicatrix, 219, 222, 226, 238, 244 Carath´ eodory’s, 224–228, 232 inverse problem of Lagrangian mechanics, 84, 85 inverse problem of the calculus of variations, 84, 85 involutive, 89 Jacobi conservation law, 92 Jacobi equation, 148, 149, 151, 152, 155, 158, 163, 172, 236 Jacobi integral, 92 Jacobi’s criterion, 152 Jacobi’s equation of the last multiplier, 84 Jacobi’s theorem, 152 Lagrange multiplier rule, 117 Lagrange multiplier(s), 114, 118, 120, 131, 132, 137, 138, 186 Lagrange’s simplification, 37, 41 Lagrangian, 80, 81, 83, 85, 86, 91, 92, 99, 100, 104, 105, 109, 131, 134, 135, 138, 169 Lagrangian density, 99–101, 106 Laplace’s equation, 98 Legendre transformation, 89, 91, 269 Legendre’s fallacy, 143 Legendre’s transformation of the second variation, 143, 149 lemma of du Bois-Reymond, 42, 45, 78 lemniscate, 20, 116 lima¸ con of Pascal, 229 mechanics Hamiltonian, 88, 101, 202 Lagrangian, 79, 80, 101, 202 meridian, 62, 63 meridional rays, 95, 105 minimal surface equation, 16, 98 minimal surface(s), 14, 16, 18, 19 of revolution, 52–55, 58, 146, 190, 196, 203, 267 natural boundary conditions, 71 Neile’s parabola, 206, 208 neutral axis, 72, 73 norm strong, 33, 35, 140 weak, 33, 237 optical path length, 93 optics, 190 optimal control theory, 273 osculating planes, 14 parabola of safety, 169 parabolic wire, 137 paraboloid, 105, 138 pencil, 57, 161, 257, 264 pendulum double, 80 simple, 80, 82, 87, 104 spherical, 85, 86, 105 positive homogeneity, 179, 181, 185, 186, 197, 201, 225, 228 principle Fermat’s, 3, 93, 95, 101 Hamilton’s, 81, 132, 169 of minimum total potential energy, 71
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