|
|
 |
| Àâòîðèçàöèÿ |
|
|
|
| Ïîèñê ïî óêàçàòåëÿì |
|
|
|
|
|
|
|
 |
|
|
|
| Logan J.D. — Invariant Variational Principles |
|
|
|
| Ïðåäìåòíûé óêàçàòåëü |
Absolute invariance of higher-order integrals 117—118 122
Absolute invariance of multiple integrals 64
Absolute invariance of single integrals 30—34
action 11
Action integral for dynamical systems 12
Action integral for electrodynamics 104
Action integral for multiple integral problems 17 113
Action integral for physical fields 92
Action integral for single integral problems 5 113
Adjoint operator 159
Admissible function 1
Angular momentum tensor 96
Arclength 33 151
Brachistochrone problem 26
Canonical momentum 37 38 73 121
Center of mass 46 47
Configuration space 10
Conformal factor 133
Conformal invariance 158—161
Conformal point transformation 132
Conformal transformation of metric tensor 133
Conservation law 22 37 70—74
Conservation of angular momentum 44 97
Conservation of energy 37—38 42 96—97 100 107 109 120—121
Conservation of linear momentum 42
contravariance 79
Covariance 79
Covariance, conformal 145—147
Degree of contravariance 79
Degree of contravariance of covariance 79
Dilation 131 135 139—140 146
Dilation, infinitesimal generators for 140
Dilation, invariance under 141—143
Divergence theorem 16
Divergence-invariance 34
Divergence-invariance for n-body problem 44 45
Dual vector space 79 80
Electric field intensity 99
electromagnetic field 93 98—104 107—109 146—147 160
Electromagnetic field, tensor 103
Electromagnetic potential see “Four-potential”
Emden’s equation 52
Energy see “Conservation of energy”
Energy, electric 99
Energy, magnetic 99
Energy-momentum tensor 108
Euler equations see “Euler — Lagrange equations”
Euler expressions for multiple integrals 21
Euler expressions for second-order problems 115 119 124
Euler expressions for single integrals 10 36 159 161
Euler — Lagrange equations for dynamical systems 20
Euler — Lagrange equations for higher-order problems 128
Euler — Lagrange equations for multiple integrals 12
Euler — Lagrange equations for physical fields 92—93
Euler — Lagrange equations for second-order problems 112—117
Euler — Lagrange equations for single integrals 7
Euler — Lagrange expressions see “Euler expressions”
Euler’s theorem 143
Extremal 8
Extremal surface 21
Fermat’s principle 11 50
Field equations 21 92
Field functions 92
Field, scalar 77 82
Field, tensor 81—82
Field, vector 77
First integrals 8 27 37 120—121
First integrals for n-body problem 39—37
Four-potential 93 101
Functional 1
Fundamental integral see “Action integral”
Fundamental invariance identities for higher-order problems 128
Fundamental invariance identities for multiple integrals 66
Fundamental invariance identities for second-order problems 118 122
Fundamental invariance identities for single integrals 34 52
Fundamental lemma of calculus of variations for multiple integrals 19
Fundamental lemma of calculus of variations for single integrals 6
Fundamental variational formula 23 162
Galilean invariance 41 77
Galilean transformations 41 76
Gauge transformation 102 160
Generators 29 63
Group, Galilean 41
Group, general Lorentz 85
Group, inhomogeneous Lorentz 86
Group, Poincare 86 88 93
Group, proper Lorentz 86 89
Group, r-parameter local Lie 28
Group, special conformal 138—139
Hamiltonian 37 107
Hamiltonian complex 73
Hamilton’s principle 11 14
Homogeneous function 143
Inertial frame 76
Infinitesimal generator see “Generators”
Infinitesimal transformation 89
Invariance identities see “Fundamental invariance identities”
Invariance of action integral see “Absolute invariance”
Invariance problems 2 22
Invariant see “Absolute invariance”
Inverse problem 59
Involution 136
Jacobi matrix 78
| Jacobian 78
Killing’s equations 55
Killing’s equations, generalized 56
Kinctic energy II 13
Klein — Gordon equation 92 97 149
Korteweg — de Vries equation 74—75 124
Korteweg — de Vries equation, conservation laws for 74—75 126—127
Lagrange’s equations see “Euler — Lagrange equations”
Lagrangian 4 11 16
Lagrangian density 92
Lagrangian for area of surface of revolution 9
Lagrangian for brachistochrone problem 26
Lagrangian for central force field 38
Lagrangian for damped harmonic oscillator 56—57
Lagrangian for electromagnetic field 101 104
Lagrangian for harmonic oscillator 12
Lagrangian for Klein — Gordon equation 97
Lagrangian for Korteweg — de Vries equation 125
Lagrangian for n-body problem 13 40
Lagrangian for Plateau’s problem 21
Lagrangian for vibrating rod 116
Lagrangian for wave equation 64
Laplacian operator 92
Liouville’s theorem 136 138
Lorentz condition 102
Lorentz transformation 86
Lorentz transformation, general 84
Lorentz transformation, infinitesimal 90
Lorentz transformation, proper 86
Magnetic deflection 99
Manifold 78
Maupertuis’ principle 11
Maxwell stress tensor 109
Maxwell’s equations 93
Maxwell’s equations in tensor form 104
Maxwell’s equations in vector form 100
Metric tensor 92
Minimum, absolute 1
Minimum, relative 1 3
Minkowski metric 91
n-body problem 13 14 39
Necessary conditions for an extremum 1
Necessary conditions for an extremum for functionals 3
Necessary conditions for an extremum for second-order problems 112—115
Necessary conditions for an extremum, multiple integral problems 20
Necessary conditions for an extremum, single integral problems 7
Noether’s identities see “Noether’s theorem”
Noether’s theorem 22
Noether’s theorem for higher-order multiple integrals 123
Noether’s theorem for higher-order single integrals 119
Noether’s theorem for infinite continuous groups 158—161
Noether’s theorem for multiple integrals 67
Noether’s theorem for single integrals 36
Norm 2
Norm, weak 4
Normal region 15
Normed linear space 2
Parameter-invariant 150
Pendulum 49—50
Plateau’s problem 21
Poincare group see “Group”
Poynting vector 100 109
Rotation of plane 29
Rotation, infinitesimal 29 139
Rotation, spatial 41 135
Scalar field 77 82 92 95—98 139—145
Scalar field, conformal invariance identities 140—143
Scalar field, conservation laws for 95—96 144—145
Scalar field, general invariance identities 97—98
Scalar potential see “Four-potential”
Second Noether theorem 158—161
Smith — Hemholtz invariant 50
Snell’s law 50
Special conformal transformation 138—139
Stationary 8
Sufficient conditions for parameter-invariance 153—155
Summation convention 6
Surface of revolution 9
Tensor operations, contraction 81
Tensor operations, product 80
Tensor operations, raising and lowering indices 82 83
Tensor operations, sum 110
Tensors 79—83
Tensors, rank of 79
Tensors, transformation law for 79
Transformation, r-parameter family 28
Translation, space 41
Translation, space-time 93
Translation, time 41
Variation, first 3
Variation, Gateaux 3
Variation, weak 3
Variational integral see “Action integral”
Vector field 77 82 92 105—107 145—147
Vector field, conservation laws for 106 145—147
Vector potential see “Four-potential”
Vector, contravariant 81
Vector, covariant 81
Vector, timelike 109
Velocity transformations 41 76
Wave equation 65 93 111
Weierstrass representation 156—157 161
Zermelo’s condition 155—157
|
|
|
| Ðåêëàìà |
|
|
|
|
|
Î ïðîåêòå