Euler-Lagrange equation: derivation and application

Classical Mechanics and Relativity: Lecture 3
0:00 Introduction
0:51 Principle of Least Action and the Lagrangian
6:01 Generalized Coordinates
11:38 Derivation of the Euler-Lagrange equation in generalized coordinates
25:29 Generalized momentum and generalized force
27:55 Polar coordinates
38:33 Example: pendulum
46:18 Newtonian vs Lagrangian mechanics
49:47 Global vs Local approach
57:47 The Hamiltonian
1:03:18 Legendre transformation
1:07:26 Hamilton's Equations
1:08:44 The Hamiltonian and Energy
1:12:27 Conservation of Energy

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 I use the Principle of Least Action to derive the Euler-Lagrange Equation of Motion in generalized coordinates and perform the Legendre transformation to obtain Hamilton's equations. We will explore the connection between the classical Hamiltonian and the energy, and show that it is conserved. The concepts are illustrated with simple examples.

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

[Snowmaners]

Dr. Mitchell's presentations are clear and fresh, giving a view that really enforces ones understanding if you have a earlier introduction.

[jolez_]

Great lectures! Sad to see that there are few views as soon as the physics get really interesting.

[shashanks.k]

This lecture was such a pleasure to follow. Thanks a ton, Dr Andrew Mitchell, ur the best.

[mtb]

I’ve been loving your channel since I came across it recently. Keep up the good work. One note: The armchair physicists in the comments of your videos are almost as entertaining as the videos. They are theoretical physicists, because their physics degrees are theoretical 😂

[christophertamina]

Simply a great teacher 👏🏿👏🏿👏🏿 I take my heart off t you Sir

[jamestseng]

37:14 EOM in theta: Left side should be - partial derivative of V w.r.t theta instead of w.r.t. r.
Great job, thank you.

[oded]

A couple of cirrections: 1. on Min. 37:13 - the derivative od the potential should be w.r.t. the angle \theta. 2. on min. 41.40 - z should probably be y, for consistency.

[khnahid]

Why aren't we considering ∂L/∂t*dt in the differential of L at 17:26 ?

[andrewwrobel]

Good lecture, but the application of EL EoMs to the pendulum around 43:30 is somewhat faulty: the constraint r(t)=l should be taken into account from the very start, but here it is used only after writing out the EoMs. This does not matter for the EoM in theta, but if the EoM in r is written out as on the preceding slide, then it will falsely say that mr-dbldot=centrifugal force + radial component of gravity, instead of r-dbldot=0 in reality (the radial gravity term comes from differentiating -V = mgrcos(theta) w.r.t. r, with V=0 at the pivot). This says that the bob flies off the rod. The mistake comes form leaving out the reaction force of the constraining rod, which in reality balances both the radial component of gravity and the centrifugal force (or, from the inertial frame PoV, provides the centripetal force). In other words, the EL EoMs are applied to the problem with two coordinates (r, theta) as though there were no constraints except at the endpoints. The simplest remedy is, of course, to eliminate r from the start and consider the problem with just a single coordinate, theta. (Or, if one insists on keeping both coordinates (r, theta) then one has to apply the version of EL EoMs with additional holonomic constraints (which here would be r(t)=l for each t).)

[gibbogle]

Nice clear derivation. But at 24:20 did you justify that the sum of the integrals equal to 0 implies that each individual integral = 0 and that each integrated term = 0? This is the hard part for me.

[wei-chihchen]

20:22 time derivative of (q) --> time derivative of (dq)

[JP-rebc]

This lecture would improve a lot IMHO if the speaker focused on the concept of "action" before stating the Lagrangian. Explain what is going on with all the trajectories, what is the problem mathematically. As it is action remains something arbitrary and mystical, while the Lagrangian pops up "deos ex machina" of sorts.

[bejitasansensei]

41:40 z is l*costheta, it's not dz. But anyway, with derivation it does not matter in the end.

[nupursarkar]

sir can you pls tell me in 19:55 you expanded q and q' from L=L(q, q', t) but why t was not written kindly can you answer this doubt??

[zeroUnknown]

Could you do a video explaining Euler Lagrange using Cosines? Also within an algebraic setting.

[danielkonstantinovsky]

since you're setting delta_A to be 0, is that enough to say you are deriving the principle of least action (as in there's a global minimum at the Euler-Lagrange equation). Isn't it more like the principle of stationary action? I've heard that phrase somewhere but I'm not sure.

[ronaldjorgensen]

is there still a cash prize for simplifying LaGrange you simplified it for me poetic fluid 1/137