Derivation of E=mc^2 and Lorentz force from relativistic Lagrangian

Classical Mechanics and Relativity: Lecture 4
0:00 Introduction
2:22 Invariants in 3d space and 4d spacetime
13:50 The Action as a relativistic invariant
15:15 Relativistic Action
22:14 Relativistic Lagrangian
31:23 Derivation of E=mc^2
35:31 Example: Electromagnetism
45:14 Electromagnetic Action
47:05 Derivation of Lorentz force

Theoretical physicist Dr Andrew Mitchell presents an undergraduate lecture course on Classical Mechanics and Relativity at University College Dublin. This is a complete and self-contained course in which everything is derived from scratch.

In this lecture we generalize the Lagrangian formalism to include Einstein's special theory of relativity, starting from the requirement that the action be the same in all reference frames, to all observers. From this we can quickly derive E=mc^2. We then apply the relativistic Lagrangian formulation to the classical theory of electromagnetism, as an example, and derive the Lorentz force for magnetism.

A more in-depth discussion of relativity in electromagnetism can be found here:

Course textbooks:
"Classical Mechanics" by Goldstein, Safko, and Poole
"Classical Mechanics" by Morin
"Relativity" by Rindler

[haniefsofi]

You missed 1/2.
Your lectures are nice.
Thanks

[amiralivanaki]

Very instructive and comprehensive. Thanks a lot.

[strippins]

Phenomenal lecture, if there was just one change I would have loved you to have explained where the minus sign came from in the four vector

[jolez_]

At around 30:00 you asserted that the Hamiltonian equals the total energy. I have watched the proof of this for nonreletavistic Hamiltonian, but I have not seen the proof for this specific Hamiltonian you are using in the video. Do you know where I can find one?

[aname]

At 26.09 is it supposed to be kc - 1/2k/cv^2; so there's a 1/2 missing. Then I can get the k = - mc.
Also, this is fantastic series so far. And is appreciated.

[JSHD]

Dr Mitchell, since the Euler Legrange equations can also be presented or morphed into the geodesic metric could this mean that for non relativistic particles the total rest mass energy E=mc2 can be made equivalent to some geodesic distance (e.g. Work Done = Force x Distance (joules))?
NB/ Where, the geodesic distance would be Lorentz invariant and the mass from Force = mass × acceleration would be relativistic (and a rest mass).

[oded]

On min.26:12 there is a half factor missing.

[Darthvanger]

Dark magic. Makes no sense at all, but at the same time in the end you see it kind of makes sense to introduce all these structures, they're pretty and give rise to all the other laws from such a simple and basic concept.

[abcdef]

at 21:56 ds^2 <0 how can this mathematically be possible?
what would be the reason not to use i ct (i= complex number)? if you use "ict" then you dont need metrix tensor, maybe not covariant and contravariant concepts too.

"...we have a problem, the action cant be negative..." so you switched the terms around and put - sign in front of it... which was still the same nagative action. i found this ilogical.