Relativistic Mechanics and Electromagnetic Field Theory 611 spring 20... · 2020. 3. 17. · • Special Relativity and Electrodynamics by Prof. Leonard Susskind, part of the Theoretical

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

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

3Prof. Sergio B. MendesSpring 2018

𝐻 = 𝑃𝑛 ሶ𝑞𝑛 − 𝐿

𝑃𝑖 ≡𝜕𝐿

𝜕 ሶ𝑞𝑖

Hamiltonian of Particles

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

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

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

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

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

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 =

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

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

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

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

14Prof. Sergio B. MendesSpring 2018

ℒ = ℒ 𝐴𝜇 , 𝜕𝛽𝐴𝛼

𝜕𝛽𝜕ℒ

𝜕 𝜕𝛽𝐴𝛼=

𝜕ℒ

𝜕𝐴𝛼

Euler-Lagrange Equations for Continuous Fields

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

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:

17Prof. Sergio B. MendesSpring 2018

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

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

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

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