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