Page 1
1Prof. Sergio B. MendesSpring 2018
Relativistic Mechanics and Electromagnetic Field Theory
• Chapter 2 of “Modern Problems in Classical Electrodynamics” by Brau
• Chapter 12 “Classical Electrodynamics” by Jackson, 3rd ed.
• Chapter 2-4 “The Classical Theory of Fields” by Landau and Lifshitz, 4th ed.
• Chapter 6 “Electrodynamics” by Melia
• Special Relativity and Electrodynamics by Prof. Leonard Susskind, part of the Theoretical Minimum Lectures from Stanford University
Page 2
2Prof. Sergio B. MendesSpring 2018
𝑆 = න𝑡1
𝑡2
𝐿 𝑞𝑖 , ሶ𝑞𝑖 , 𝑡 𝑑𝑡
𝐿 = 𝐿 𝑞𝑖 , ሶ𝑞𝑖 , 𝑡
𝑑
𝑑𝑡
𝜕𝐿
𝜕 ሶ𝑞𝑖=𝜕𝐿
𝜕𝑞𝑖
Lagrangian of Particles
Page 3
3Prof. Sergio B. MendesSpring 2018
𝐻 = 𝑃𝑛 ሶ𝑞𝑛 − 𝐿
𝑃𝑖 ≡𝜕𝐿
𝜕 ሶ𝑞𝑖
Hamiltonian of Particles
Page 4
Relativistic Lagrangian of a Free Particle
Spring 2020 Prof. Sergio B. Mendes 4
𝑡𝑡𝑖 𝑡𝑓
𝑥𝑖
𝑥𝑖, 𝑖𝑛
𝑥𝑖, 𝑓𝑖
𝜕𝐿
𝜕𝑥𝑖=
𝑑
𝑑𝑡
𝜕𝐿
𝜕 ሶ𝑥𝑖
𝐿 𝑑𝑡 ∝ 𝑑𝜏
𝑆 = න𝑡𝑖
𝑡𝑓
𝐿 𝑑𝑡
𝐿 = −𝑚 𝑐2
𝛾
𝜕𝐿
𝜕 ሶ𝑥𝑖= 𝑃𝑖
0 =𝑑
𝑑𝑡𝛾 𝑚 ሶ𝑥𝑖
= −𝑚 𝑐2 1 −ሶ𝑥𝑖2
𝑐2
= 𝑚ሶ𝑥𝑖
1 −ሶ𝑥𝑖2
𝑐2
= 𝛾 𝑚 ሶ𝑥𝑖
: must be a scalar (invariant)
=𝑑𝑡
𝛾
Page 5
Relativistic Hamiltonian of a Free Particle
Spring 2020 Prof. Sergio B. Mendes 5
𝐿 = −𝑚 𝑐2
𝛾𝐻 = 𝑃𝑖 ሶ𝑥𝑖 − 𝐿
= 𝛾 𝑚 ሶ𝑥𝑖2 +
𝑚 𝑐2
𝛾
= 𝛾 𝑚 𝑐2ሶ𝑥𝑖2
𝑐2+
1
𝛾2
= 𝛾 𝑚 𝑐2ሶ𝑥𝑖2
𝑐2+ 1 − 𝛽2
𝐻 = 𝛾 𝑚 𝑐2
𝑃𝑖 = 𝛾 𝑚 ሶ𝑥𝑖
Page 6
Relativistic Lagrangian of a Charged Particle on Given EM Fields
Spring 2020 Prof. Sergio B. Mendes 6
𝐿 = −𝑚 𝑐2
𝛾− 𝑞 Φ + 𝑞
𝑑𝑥
𝑑𝑡𝐴𝑥 +
𝑑𝑦
𝑑𝑡𝐴𝑦 +
𝑑𝑧
𝑑𝑡𝐴𝑧
𝑆 = න𝑡𝑖
𝑡𝑓
𝐿 𝑑𝑡 = න𝑡𝑖
𝑡𝑓
−𝑚 𝑐2
𝛾𝑑𝑡 − 𝑞න
𝑡𝑖
𝑡𝑓
𝐴𝜇 𝑑𝑥𝜇
𝐴𝜇 =
𝛷/𝑐− 𝐴𝑥− 𝐴𝑦− 𝐴𝑧
𝑑𝑥𝜇 =
𝑐 𝑑𝑡𝑑𝑥𝑑𝑦𝑑𝑧
= න𝑡𝑖
𝑡𝑓
−𝑚 𝑐2
𝛾− 𝑞 Φ + 𝑞
𝑑𝑥
𝑑𝑡𝐴𝑥 +
𝑑𝑦
𝑑𝑡𝐴𝑦 +
𝑑𝑧
𝑑𝑡𝐴𝑧 𝑑𝑡
Page 7
Relativistic Equations of Motion
Spring 2020 Prof. Sergio B. Mendes 7
𝐿 = −𝑚 𝑐2
𝛾− 𝑞 Φ + 𝑞 ሶ𝑥𝑚 𝐴𝑚
𝜕𝐿
𝜕𝑥𝑖=
𝑑
𝑑𝑡
𝜕𝐿
𝜕 ሶ𝑥𝑖
− 𝑞𝜕Φ
𝜕𝑥𝑖+ 𝑞 ሶ𝑥𝑚
𝜕𝐴𝑚𝜕𝑥𝑖
=
=𝑑
𝑑𝑡𝛾 𝑚 ሶ𝑥𝑖 + 𝑞
𝜕𝐴𝑖𝜕𝑡
+ 𝑞 ሶ𝑥𝑚𝜕𝐴𝑖𝜕𝑥𝑚
𝑑
𝑑𝑡𝛾 𝑚 ሶ𝑥𝑖 + 𝑞 𝐴𝑖
Page 8
Relativistic Lorentz Force
Spring 2020 Prof. Sergio B. Mendes 8
− 𝑞𝜕Φ
𝜕𝑥𝑖− 𝑞
𝜕𝐴𝑖𝜕𝑡
+ 𝑞 ሶ𝑥𝑚𝜕𝐴𝑚𝜕𝑥𝑖
−𝜕𝐴𝑖𝜕𝑥𝑚
=𝑑
𝑑𝑡𝛾 𝑚 ሶ𝑥𝑖
𝑞 𝐸𝑖 + 𝑞 𝜖𝑖𝑚𝑘 ሶ𝑥𝑚 𝐵𝑘 =𝑑
𝑑𝑡𝛾 𝑚 ሶ𝑥𝑖
𝐵𝑘 = 𝜖𝑘𝑛𝑙𝜕𝐴𝑙
𝜕𝑥𝑛
𝜖𝑚𝑘𝑖 𝐵𝑘= 𝜖𝑚𝑘𝑖 𝜖𝑘𝑛𝑙
𝜕𝐴𝑙
𝜕𝑥𝑛= − 𝜖𝑘𝑚𝑖 𝜖𝑘𝑛𝑙
𝜕𝐴𝑙
𝜕𝑥𝑛=𝜕𝐴𝑚
𝜕𝑥𝑖−𝜕𝐴𝑖
𝜕𝑥𝑚
Page 9
Relativistic Lorentz Force
Spring 2020 Prof. Sergio B. Mendes 9
− 𝑞𝜕Φ
𝜕𝑥𝑖− 𝑞
𝜕𝐴𝑖𝜕𝑡
+ 𝑞 ሶ𝑥𝑚𝜕𝐴𝑚𝜕𝑥𝑖
−𝜕𝐴𝑖𝜕𝑥𝑚
=𝑑
𝑑𝑡𝛾 𝑚 ሶ𝑥𝑖
− 𝑞 𝑐𝜕𝛷𝑐
𝜕𝑥𝑖− 𝑞 𝑐
𝜕𝐴𝑖𝑐 𝜕𝑡
+ 𝑞 ሶ𝑥𝑚𝜕𝐴𝑚𝜕𝑥𝑖
−𝜕𝐴𝑖𝜕𝑥𝑚
=
− 𝑞 𝑐 𝜕𝑖𝐴0 + 𝜕0𝐴
𝑖 + 𝑞 ሶ𝑥𝑚 𝜕𝑖𝐴𝑚 − 𝜕𝑚𝐴
𝑖 =
𝑞 𝑐 𝜕𝑖𝐴0 − 𝜕0𝐴𝑖 + 𝑞 ሶ𝑥𝑚 𝜕𝑖𝐴𝑚 − 𝜕𝑚𝐴
𝑖 =
Page 10
Relativistic Lorentz Force, cont.
Spring 2020 Prof. Sergio B. Mendes 10
𝑞 𝑐 𝜕𝑖𝐴0 − 𝜕0𝐴𝑖 + 𝑞 ሶ𝑥𝑚 𝜕𝑖𝐴𝑚 − 𝜕𝑚𝐴
𝑖 =𝑑
𝑑𝑡𝛾 𝑚 ሶ𝑥𝑖
𝑑𝑡
𝑑𝜏𝑞 𝑐 𝜕𝑖𝐴0 − 𝜕0𝐴𝑖 + 𝑞
𝑑𝑡
𝑑𝜏
𝑑𝑥𝑚
𝑑𝑡𝜕𝑖𝐴
𝑚 − 𝜕𝑚𝐴𝑖 =
𝑑𝑡
𝑑𝜏
𝑑
𝑑𝑡𝛾 𝑚 ሶ𝑥𝑖
𝑑𝑡
𝑑𝜏×
𝑞𝑑𝑥0
𝑑𝜏𝜕𝑖𝐴0 − 𝜕0𝐴𝑖 + 𝑞
𝑑𝑥𝑚
𝑑𝜏𝜕𝑖𝐴
𝑚 − 𝜕𝑚𝐴𝑖 =
𝑑
𝑑𝜏𝛾 𝑚 ሶ𝑥𝑖
− 𝑞 𝑢𝜇 𝐹𝜇𝑖 =
𝑑𝑝𝑖
𝑑𝜏
− 𝑞𝑑𝑥0𝑑𝜏
𝜕0𝐴𝑖 − 𝜕𝑖𝐴0 − 𝑞𝑑𝑥𝑚𝑑𝜏
𝜕𝑚𝐴𝑖 − 𝜕𝑖𝐴𝑚 =𝑑
𝑑𝜏𝛾 𝑚 ሶ𝑥𝑖
Page 11
Relativistic Canonical Momenta ofa Charged Particle on Given EM Fields
Spring 2020 Prof. Sergio B. Mendes 11
𝑃𝑖 ≡𝜕𝐿
𝜕 ሶ𝑥𝑖= 𝛾 𝑚 ሶ𝑥𝑖 + 𝑞 𝐴𝑖
𝐿 = −𝑚 𝑐2
𝛾− 𝑞 Φ + 𝑞 ሶ𝑥𝑚 𝐴𝑚
Page 12
Spring 2020 Prof. Sergio B. Mendes 12
𝐻 = 𝑃𝑛 ሶ𝑥𝑛 +𝑚 𝑐2
𝛾+ 𝑞 Φ − 𝑞 ሶ𝑥𝑚 𝐴𝑚
= 𝛾 𝑚 ሶ𝑥𝑛 + 𝑞 𝐴𝑛 ሶ𝑥𝑛 +𝑚 𝑐2
𝛾+ 𝑞 Φ − 𝑞 ሶ𝑥𝑚 𝐴𝑚
= 𝛾 𝑚 𝑐2ሶ𝑥𝑛2
𝑐2+
1
𝛾2+ 𝑞 Φ
= 𝛾 𝑚 𝑐2 + 𝑞 Φ
𝐻 = 𝑃𝑛 ሶ𝑥𝑛 − 𝐿
𝐿 = −𝑚 𝑐2
𝛾− 𝑞 Φ + 𝑞 ሶ𝑥𝑚 𝐴𝑚
𝑃𝑖 ≡𝜕𝐿
𝜕 ሶ𝑥𝑖= 𝛾 𝑚 ሶ𝑥𝑖 + 𝑞 𝐴𝑖
Relativistic Hamiltonian of a Charged Particle on Given EM Fields
Page 13
13Prof. Sergio B. MendesSpring 2019
= න −𝜇 𝑐2
𝛾− 𝜌 Φ + 𝜌 ሶ𝑥𝑚 𝐴𝑚 𝑑𝑉
= න −𝜇 𝑐2
𝛾− 𝜌 Φ + 𝐽𝑚 𝐴𝑚 𝑑𝑉
= න −𝜇 𝑐2
𝛾− 𝐽𝜇 𝐴
𝜇 𝑑𝑉
𝜇 𝑑𝑉 ≡ 𝑑𝑚
𝐿 = −𝑚 𝑐2
𝛾− 𝑞 Φ + 𝑞 ሶ𝑥𝑚 𝐴𝑚
ℒ𝑚𝑒𝑐ℎ + ℒ𝑖𝑛𝑡 = −𝜇 𝑐2
𝛾− 𝐽𝜇 𝐴
𝜇
Relativistic Lagrangian Density of a Charged Particle on Specified EM Fields
𝜌 𝑑𝑉 ≡ 𝑑𝑞
𝜌 ሶ𝑥𝑚 𝑑𝑉 ≡ 𝐽𝑚 𝑑𝑉
Page 14
14Prof. Sergio B. MendesSpring 2018
ℒ = ℒ 𝐴𝜇 , 𝜕𝛽𝐴𝛼
𝜕𝛽𝜕ℒ
𝜕 𝜕𝛽𝐴𝛼=
𝜕ℒ
𝜕𝐴𝛼
Euler-Lagrange Equations for Continuous Fields
Page 15
15Prof. Sergio B. MendesSpring 2018
𝜕ℒ
𝜕𝐴𝛼= − 𝐽𝛼
𝜕𝛽𝜕ℒ
𝜕𝛽𝐴𝛼
𝜕𝛽𝜕ℒ
𝜕 𝜕𝛽𝐴𝛼=
𝜕ℒ
𝜕𝐴𝛼
1
𝜇0𝜕𝛽𝐹𝛽𝛼 = 𝐽𝛼
Relativistic Lagrangian Density of EM Fields on Specified Charge Motion
proof on next slide
ℒ𝑖𝑛𝑡 + ℒ𝑓𝑖𝑒𝑙𝑑 = − 𝐽𝜇 𝐴𝜇 −
1
4 𝜇0𝐹𝜇𝜈 𝐹
𝜇𝜈
= 𝜕𝛽 −1
𝜇0𝐹𝛽𝛼
Page 16
16Prof. Sergio B. MendesSpring 2018
ℒ = −1
4 𝜇0𝐹𝜇𝜈 𝐹
𝜇𝜈 − 𝐽𝜇 𝐴𝜇
𝜕ℒ
𝜕 𝜕𝛽𝐴𝛼= −
1
4 𝜇0
𝜕 𝐹𝜇𝜈 𝐹𝜇𝜈
𝜕 𝜕𝛽𝐴𝛼
= −1
4 𝜇0
𝜕 𝑔𝜇𝛾 𝑔𝜈𝛿 𝜕𝛾𝐴𝛿 − 𝜕𝛿𝐴𝛾 𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇
𝜕 𝜕𝛽𝐴𝛼
= −𝑔𝜇𝛾 𝑔𝜈𝛿
4 𝜇0𝛿𝛽𝛾𝛿𝛼𝛿 − 𝛿𝛽𝛿𝛿𝛼𝛾 𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇 + 𝜕𝛾𝐴𝛿 − 𝜕𝛿𝐴𝛾 𝛿𝛽𝜇𝛿𝛼𝜈 − 𝛿𝛽𝜈𝛿𝛼𝜇
= −1
4 𝜇0
𝜕 𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇 𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇
𝜕 𝜕𝛽𝐴𝛼
= −1
4 𝜇0𝑔𝜇𝛽 𝑔𝜈𝛼 − 𝑔𝜇𝛼 𝑔𝜈𝛽 𝜕𝜇𝐴𝜈 − 𝜕𝜈𝐴𝜇 + 𝜕𝛾𝐴𝛿 − 𝜕𝛿𝐴𝛾 𝑔𝛽𝛾 𝑔𝛼𝛿 − 𝑔𝛼𝛾 𝑔𝛽𝛿
= −1
4 𝜇0𝜕𝛽𝐴𝛼 − 𝜕𝛼𝐴𝛽 − 𝜕𝛼𝐴𝛽 − 𝜕𝛽𝐴𝛼 + 𝜕𝛽𝐴𝛼 − 𝜕𝛼𝐴𝛽 − 𝜕𝛼𝐴𝛽 − 𝜕𝛽𝐴𝛼
= −1
𝜇0𝐹𝛽𝛼
Proof:
Page 17
17Prof. Sergio B. MendesSpring 2018
Page 18
18Prof. Sergio B. MendesSpring 2018
Relativistic Hamiltonian of EM Fields on Specified Charge Motion
𝐻 =
𝑖
𝜕𝐿
𝜕 ሶ𝑞𝑖ሶ𝑞𝑖 − 𝐿𝑖 𝑞𝑖 , ሶ𝑞𝑖 , 𝑡
ℋ𝛼𝛽=
𝜕ℒ
𝜕 𝜕𝛼𝐴𝛾𝜕𝛽𝐴𝛾 − 𝑔𝛼
𝛽ℒ
= −1
𝜇0𝐹𝛼𝛾 𝜕
𝛽𝐴𝛾 − 𝑔𝛼𝛽
−1
4 𝜇0𝐹𝜇𝜈 𝐹
𝜇𝜈 − 𝐽𝜇 𝐴𝜇
Page 19
19Prof. Sergio B. MendesSpring 2018
= −1
𝜇0𝐹𝛼𝛾 𝐹𝛽𝛾 + 𝜕𝛾𝐴𝛽 +
1
4 𝜇0𝐹𝜇𝜈 𝐹
𝜇𝜈𝑔𝛼𝛽+ 𝐽𝜇 𝐴
𝜇𝑔𝛼𝛽
ℋ𝛼𝛽= −
1
𝜇0𝐹𝛼𝛾 𝜕
𝛽𝐴𝛾 − 𝑔𝛼𝛽
−1
4 𝜇0𝐹𝜇𝜈 𝐹
𝜇𝜈 − 𝐽𝜇 𝐴𝜇
= −1
𝜇0𝐹𝛼𝛾 𝐹
𝛽𝛾 −1
𝜇0𝐹𝛼𝛾 𝜕
𝛾𝐴𝛽 +1
4 𝜇0𝐹𝜇𝜈 𝐹
𝜇𝜈𝑔𝛼𝛽+ 𝐽𝜇 𝐴
𝜇𝑔𝛼𝛽
= −1
𝜇0𝐹𝛼𝛾 𝐹
𝛽𝛾 −1
𝜇0𝜕𝛾 𝐴𝛽 𝐹𝛼𝛾 − 𝐴𝛽 𝜕𝛾𝐹𝛼𝛾 +
1
4 𝜇0𝐹𝜇𝜈 𝐹
𝜇𝜈𝑔𝛼𝛽+ 𝐽𝜇 𝐴
𝜇𝑔𝛼𝛽
= −1
𝜇0𝐹𝛼𝛾 𝐹
𝛽𝛾 −1
𝜇0𝜕𝛾 𝐴𝛽 𝐹𝛼𝛾 − 𝐽𝛼 𝐴
𝛽 +1
4 𝜇0𝐹𝜇𝜈 𝐹
𝜇𝜈𝑔𝛼𝛽+ 𝐽𝜇 𝐴
𝜇𝑔𝛼𝛽
= −1
𝜇0𝐹𝛼𝛾 𝐹
𝛽𝛾 +1
4 𝜇0𝐹𝜇𝜈 𝐹
𝜇𝜈𝑔𝛼𝛽− 𝐽𝛼 𝐴
𝛽 + 𝐽𝜇 𝐴𝜇𝑔𝛼
𝛽−
1
𝜇0𝜕𝛾 𝐴𝛽 𝐹𝛼𝛾
Page 20
20Prof. Sergio B. MendesSpring 2018
= 𝑇𝛼𝛽− 𝐽𝛼 𝐴
𝛽 + 𝐽𝜇 𝐴𝜇𝑔𝛼
𝛽−
1
𝜇0𝜕𝛾 𝐴𝛽 𝐹𝛼𝛾
𝑇𝛼𝛽≡ −
1
𝜇0𝐹𝛼𝛾 𝐹
𝛽𝛾 +1
4 𝜇0𝐹𝜇𝜈 𝐹
𝜇𝜈𝑔𝛼𝛽
ℋ𝛼𝛽= −
1
𝜇0𝐹𝛼𝛾 𝐹
𝛽𝛾 +1
4 𝜇0𝐹𝜇𝜈 𝐹
𝜇𝜈𝑔𝛼𝛽− 𝐽𝛼 𝐴
𝛽 + 𝐽𝜇 𝐴𝜇𝑔𝛼
𝛽−
1
𝜇0𝜕𝛾 𝐴𝛽 𝐹𝛼𝛾
when: 𝐽𝜇 = 0
ℋ𝛼𝛽= 𝑇𝛼
𝛽−
1
𝜇0𝜕𝛾 𝐴𝛽 𝐹𝛼𝛾
𝜕𝛼ℋ𝛼𝛽= 𝜕𝛼𝑇𝛼
𝛽−
1
𝜇0𝜕𝛼𝜕𝛾 𝐴𝛽 𝐹𝛼𝛾 = 𝜕𝛼𝑇𝛼
𝛽
and
Page 21
21Prof. Sergio B. MendesSpring 2018
𝑇𝛼𝛽 =
𝑢𝑐𝑔𝑥𝑐𝑔𝑦𝑐𝑔𝑧
𝑐𝑔𝑥𝑇𝑥𝑥𝑇𝑦𝑥𝑇𝑧𝑥
𝑐𝑔𝑦𝑇𝑥𝑦𝑇𝑦𝑦𝑇𝑧𝑦
𝑐𝑔𝑧𝑇𝑥𝑧𝑇𝑦𝑧𝑇𝑧𝑧
𝛽 = 0
𝜕𝛼𝑇𝛼0 = 𝜕0 𝑢 + 𝜕1 𝑐𝑔𝑥 + 𝜕2 𝑐𝑔𝑦 +𝜕3 𝑐𝑔𝑧 =
1
𝑐
𝜕𝑢
𝜕𝑡+ 𝛁. 𝑺
𝛽 = 𝑖
𝜕𝛼𝑇𝛼𝑖 = 𝜕0 𝑐𝑔𝑖 + 𝜕𝑗𝑇𝑗𝑖 =
𝜕𝑔𝑖𝜕𝑡
+𝜕𝑇𝑥𝑖𝜕𝑥
+𝜕𝑇𝑦𝑖𝜕𝑦
+𝜕𝑇𝑦𝑖𝜕𝑧
