1. notice
  3. English
  4. 1. feature
  6. logic-topic
  7. 2. logic
  8. 3. set-theory
  9. 4. map
  10. 5. order
  11. 6. combinatorics
  13. calculus
  14. 7. real-numbers
  15. 8. limit-sequence
  16. 9. โ„^n
  17. 10. Euclidean-space
  18. 11. Minkowski-space
  19. 12. polynomial
  20. 13. analytic-Euclidean
  21. 14. analytic-Minkowski
  22. 15. analytic-struct-operation
  23. 16. ordinary-differential-equation
  24. 17. volume
  25. 18. integral
  26. 19. divergence
  27. 20. limit-net
  28. 21. compact
  29. 22. connected
  30. 23. topology-struct-operation
  31. 24. exponential
  32. 25. angle
  34. geometry
  35. 26. manifold
  36. 27. metric
  37. 28. metric-connection
  38. 29. geodesic-derivative
  39. 30. curvature-of-metric
  40. 31. Einstein-metric
  41. 32. constant-sectional-curvature
  42. 33. simple-symmetric-space
  43. 34. principal-bundle
  44. 35. group-action
  45. 36. stereographic-projection
  46. 37. Hopf-bundle
  48. field-theory
  49. 38. point-particle-non-relativity
  50. 39. point-particle-relativity
  51. 40. scalar-field
  52. 41. scalar-field-current
  53. 42. scalar-field-non-relativity
  54. 43. projective-lightcone
  55. 44. spacetime-momentum-spinor-representation
  56. 45. Lorentz-group
  57. 46. spinor-field
  58. 47. spinor-field-current
  59. 48. electromagnetic-field
  60. 49. Laplacian-of-tensor-field
  61. 50. Einstein-metric
  62. 51. interaction
  63. 52. harmonic-oscillator-quantization
  64. 53. spinor-field-misc
  65. 54. reference
  67. ไธญๆ–‡
  68. 55. notice
  69. 56. feature
  71. ้€ป่พ‘
  72. 57. ้€ป่พ‘
  73. 58. ้›†ๅˆ่ฎบ
  74. 59. ๆ˜ ๅฐ„
  75. 60. ๅบ
  76. 61. ็ป„ๅˆ
  78. ๅพฎ็งฏๅˆ†
  79. 62. ๅฎžๆ•ฐ
  80. 63. ๆ•ฐๅˆ—ๆž้™
  81. 64. โ„^n
  82. 65. Euclidean ็ฉบ้—ด
  83. 66. Minkowski ็ฉบ้—ด
  84. 67. ๅคš้กนๅผ
  85. 68. ่งฃๆž (Euclidean)
  86. 69. ่งฃๆž (Minkowski)
  87. 70. ่งฃๆž struct ็š„ๆ“ไฝœ
  88. 71. ๅธธๅพฎๅˆ†ๆ–น็จ‹
  89. 72. ไฝ“็งฏ
  90. 73. ็งฏๅˆ†
  91. 74. ๆ•ฃๅบฆ
  92. 75. ็ฝ‘ๆž้™
  93. 76. ็ดง่‡ด
  94. 77. ่ฟž้€š
  95. 78. ๆ‹“ๆ‰‘ struct ็š„ๆ“ไฝœ
  96. 79. ๆŒ‡ๆ•ฐๅ‡ฝๆ•ฐ
  97. 80. ่ง’ๅบฆ
  99. ๅ‡ ไฝ•
  100. 81. ๆตๅฝข
  101. 82. ๅบฆ่ง„
  102. 83. ๅบฆ่ง„็š„่”็ปœ
  103. 84. Levi-Civita ๅฏผๆ•ฐ
  104. 85. ๅบฆ่ง„็š„ๆ›ฒ็Ž‡
  105. 86. Einstein ๅบฆ่ง„
  106. 87. ๅธธๆˆช้ขๆ›ฒ็Ž‡
  107. 88. simple-symmetric-space
  108. 89. ไธปไธ›
  109. 90. ็พคไฝœ็”จ
  110. 91. ็ƒๆžๆŠ•ๅฝฑ
  111. 92. Hopf ไธ›
  113. ๅœบ่ฎบ
  114. 93. ้ž็›ธๅฏน่ฎบ็‚น็ฒ’ๅญ
  115. 94. ็›ธๅฏน่ฎบ็‚น็ฒ’ๅญ
  116. 95. ็บฏ้‡ๅœบ
  117. 96. ็บฏ้‡ๅœบ็š„ๅฎˆๆ’ๆต
  118. 97. ้ž็›ธๅฏน่ฎบ็บฏ้‡ๅœบ
  119. 98. ๅ…‰้”ฅๅฐ„ๅฝฑ
  120. 99. ๆ—ถ็ฉบๅŠจ้‡็š„่‡ชๆ—‹่กจ็คบ
  121. 100. Lorentz ็พค
  122. 101. ๆ—‹้‡ๅœบ
  123. 102. ๆ—‹้‡ๅœบ็š„ๅฎˆๆ’ๆต
  124. 103. ็”ต็ฃๅœบ
  125. 104. ๅผ ้‡ๅœบ็š„ Laplacian
  126. 105. Einstein ๅบฆ่ง„
  127. 106. ็›ธไบ’ไฝœ็”จ
  128. 107. ่ฐๆŒฏๅญ้‡ๅญๅŒ–
  129. 108. ๆ—‹้‡ๅœบๆ‚้กน
  130. 109. ๅ‚่€ƒ

note-math

Non-relativistic spacetime

[action-point-particle-non-relativity] Action of path

let be a time-varying vector field or time-varying ฮด diffeomorphism, or is a special type of vector field of non-relativistic spacetime

let is zero at the boundary โ€” fix the endpoints of the path

let the differential of the action be zero

where

use product rule

and is zero at the boundary, such that

So the differential of the action is

holds for all ฮด diffeomorphism , thus giving the Lagrange-equation (alias Eulerโ€“Lagrange-equation), for non-relativistic point particles, [Newton-equation]

The momentum part of the action does not use the volume-form of , but instead uses the quadratic form of and the volume-form of time

The Lagrangian can be written as a function (a function on the tangent bundle)

[point-particle-Lagrange-equation]

For a general , repeat the above process. Action

Differential of the action

use

is zero at the boundary + integral is zero for all ฮด diffeomorphism ==> [point-particle-Lagrange-equation]

Euclidean metric-manifold

Generalize to Euclidean metric-manifold

Needs metric-connection

Although the metric volume form is not used, due to the form of the kinetic energy part of the action, it is still related to the metric geodesic

Symmetry and conserved quantity (Noether theorem)

Handling the symmetry of non-relativistic spacetime alias the Galileo group, generated from the translation of , rotation, non-relativistic boost

let be the solution to the action equation

Note that the variation of along the symmetry may make it no longer a solution to the equation, i.e., the symmetric ฮด diffeomorphism may not be zero at the boundary, i.e., it will change the endpoints of the path, so the relevant derivative of the action at the solution may not be zero

  • Time translation

In non-relativity, the mapping that preserves the measure and direction of time is the time translation

ฮด variation of the integral area [calculation-1-action-point-particle-non-relativity]

The first equation can come from the fundamental theorem of calculus + derivative of composite functions

In general, changing the region by by ฮด diffeomorphism

On the other side, use change of variable formula

Apply it to

Then use the exchange of differential and integral

Derivative of composite function

is the variation of the action on the (changing endpoints) ฮด differentiation at the solution [calculation-2-action-point-particle-non-relativity]

recall

use , merge the previous item with the next item

Get

Quantity

Called the energy of the action , is invariant along time , forall , i.e. conserved. This is true for also imply

For the energy is [energy-point-particle-non-relativity]

Homogeneity of Time ==> Conservation of Energy

  • Spatial translation

Kinetic energy part of the action

Although the spatial translation ฮด diffeomorphism is not zero at the boundary or changes the path endpoints, the time endpoints remain unchanged, and spatial translation does not change kinetic energy. with ,

So similar to the case of energy, with ฮด diffeomorphism

[momentum-point-particle-non-relativity] The momentum of the action

is invariant along time , forall , i.e. conserved

More generally, let the action with such that the endpoints in this direction do not affect the action, then the momentum

is conserved

Homogeneity of Space ==> Translation Invariance of ==> Conservation of Total Linear Momentum

Lagrangian

forall (by ) so the momentum

is conserved

  • Spatial rotation

Choose an origin. Lagrangian

is represented as a rotation around axis , cross product is ฮด rotation

Rotation around axis ==>

in , with , and magnitude

Length is invariant to direction

Similar to the case of momentum, if the Lagrangian is invariant to rotation, the ฮด diffeomorphism (tangent vector field) is , thus

[rotation-momentum-point-particle-non-relativity] Rotation momentum rotation-momentum alias angular momentum angular-momentum

is invariant along time , forall

quantity

is also called rotation momentum

More generally, with such that the Lagrangian is invariant under rotation about , then the rotational momentum about is

Isotropy of Space ==> Rotational Invariance of ==> Conservation of Total Angular Momentum

= parallelepiped directed volume span by in Euclidean

The rotational momentum is constant with respect to time, so is a constant 2d plane. Since , , is in the constant two-dimensional plane

For the Lagrangian of a system of point particles

Total rotational momentum

is invariant along time

  • Non-relativistic boost

Non-relativistic boost

The conserved quantity of the action is

forall

  • The action has all ฮด symmetries of non-relativistic spacetime , a 10 dimensional conserved quantity

  • The action has conserved energy

  • The action has conserved energy and momentum

  • The action has conserved energy and rotational momentum

  • The action has conserved energy, momentum, rotational momentum, 7 dimension

Non-relativistic potential

  • Rigid body

Parameterized by (or ), so it can be regarded as a non-relativistic particle on the Euclidean manifold . But the use of metric or the use of kinetic energy is not the Killing-form of , because for objects that are not uniformly mass-distributed spheres, rotations in different directions have different inertias. Moment of inertia i.e. metric may need to be calculated additionally

The moment of inertia can also be used as a symmetric operator under the Killing-form, with the characteristic basis being the principal axes of inertia and the eigenvalues being the moments of inertia.