# š§ Fun with lagrangians

My long outstanding post on some experiments with Lagrangians: They see me flowin' they hatin'.

## Table of Contents

- related: lagrangianphysicstoblog
- [A] * why lagrangian
- [B] * ideas lagrangian
- TODO [B] I made a web-based interactive simulator for electromagnetic waves /r/Physics lagrangiansimviz
- lagrangiansimviz wow, awesome simulation..

- TODO [B] apply some ideas from 'toward an exploratory medium' to my post on lagrangians? make it really interactive, with typing in arbitrary formula and numeric integration?? lagrangian
- STRT [C] Lagrangian 3d lagrangianviz
- DONE [B] name the post "they see me flowin' they hatin'" and add short summary below lagrangian
- TODO [C] plot actual trajectories?? lagrangian
- STRT [C] draw slope field!!! lagrangian
- TODO [D] Meditate what would be ideal form for my post about Lagrangians (formula is important unlike what Bret Victor says).pinching for phase plots, automatic numerical solving, tooltips, highlighted stuff lagrangian

- TODO [B] I made a web-based interactive simulator for electromagnetic waves /r/Physics lagrangiansimviz
- [C] * publicity lagrangian
- ------------------------------- lagrangian
- [C] hmm I understood quite a few things while writing that post about lagrangian lagrangianwritingstudy
- [C] The Feynman Lectures on Physics Vol. II Ch. 19: The Principle of Least Action lagrangiantoblogphysics
- TODO [D] Shit, legendre transform is not unique??? lagrangian
- TODO [C] Calculus of variations - Wikipedia lagrangianviz
- TODO [B] Letās take another example. Suppose the Hamiltonian is given by lagrangian
- TODO xx lagrangiantotweetoutboxlagrangians
- TODO [B] Simulating Quantum Mechanics Using Matplotlib and Tkinter /r/Python lagrangian
- lagrangiansim huh. apparently people in comments really love the fact that you can interact with the simulator. even though it's quite simple to implement

- lagrangianphysicsspace good lagrangian points intuition https://www.reddit.com/r/dataisbeautiful/comments/adttul/oc_peusdopotential_in_the_earthmoon_system/edk9f1j/
- [B] classical mechanics - What if the Euler Lagrange equation yields a 'trivial' answer - Mathematics Stack Exchange lagrangiantoblog
- [B] For Lagrangian stuff, talk out loud lagrangianwritingtostudyblog

## Ā¶related: lagrangianphysicstoblog

## Ā¶[A] * why lagrangian

### Ā¶TODO [B] write motivation for trying to understand; and also note that lagrangians do not necessarily describe fundamental physics (e.g. solid state etc) lagrangian

### Ā¶TODO [C] Motivation for investigating the shape lagrangian

One could just say it's pointless to try different shapes because the normal lagrangian works. Analogy is aaronsons speculations on l2 norm for quantum mechanics

## Ā¶[B] * ideas lagrangian

### Ā¶TODO [B] I made a web-based interactive simulator for electromagnetic waves /r/Physics lagrangiansimviz

#### Ā¶ wow, awesome simulation.. lagrangiansimviz

### Ā¶TODO [B] apply some ideas from 'toward an exploratory medium' to my post on lagrangians? make it really interactive, with typing in arbitrary formula and numeric integration?? lagrangian

### Ā¶STRT [C] Lagrangian 3d lagrangianviz

### Ā¶DONE [B] name the post "they see me flowin' they hatin'" and add short summary below lagrangian

### Ā¶TODO [C] plot actual trajectories?? lagrangian

### Ā¶STRT [C] draw slope field!!! lagrangian

### Ā¶TODO [D] Meditate what would be ideal form for my post about Lagrangians (formula is important unlike what Bret Victor says).pinching for phase plots, automatic numerical solving, tooltips, highlighted stuff lagrangian

## Ā¶[C] * publicity lagrangian

### Ā¶TODO [B] (1) Jacopo Bertolotti on Twitter: "#PhysicsFactlet (230) If the forces acting on your system can't be written as a potential, you can still write a Euler-Lagrange equation in terms of the kinetic energy T and the components of the forces tangent to the generalized coordinates q. https://t.co/Hd0xbUsHCw" / Twitter lagrangiantotweetphysics

post very short intro into lagrangians?

## Ā¶------------------------------- lagrangian

## Ā¶[C] hmm I understood quite a few things while writing that post about lagrangian lagrangianwritingstudy

## Ā¶[C] The Feynman Lectures on Physics Vol. II Ch. 19: The Principle of Least Action lagrangiantoblogphysics

http://www.feynmanlectures.caltech.edu/II_19.html

āHere is how it works: Suppose that for all paths, SS is very large compared toĀ āā. One path contributes a certain amplitude. For a nearby path, the phase is quite different, because with an enormousĀ SS even a small change inĀ SS means a completely different phaseābecause āā is so tiny. So nearby paths will normally cancel their effects out in taking the sumāexcept for one region, and that is when a path and a nearby path all give the same phase in the first approximation (more precisely, the same action withinĀ āā). Only those paths will be the important ones. So in the limiting case in which Planckās constantĀ āā goes to zero, the correct quantum-mechanical laws can be summarized by simply saying: āForget about all these probability amplitudes. The particle does go on a special path, namely, that one for which SS does not vary in the first approximation.ā Thatās the relation between the principle of least action and quantum mechanics. The fact that quantum mechanics can be formulated in this way was discovered in 1942 by a student of that same teacher, Bader, I spoke of at the beginning of this lecture. [Quantum mechanics was originally formulated by giving a differential equation for the amplitude (SchrĆ¶dinger) and also by some other matrix mathematics (Heisenberg).]

### Ā¶TODO this is for my legendre post? lagrangiantoblogphysics

### Ā¶TODO pun around Le? lagrangiantoblogphysics

## Ā¶TODO [D] Shit, legendre transform is not unique??? lagrangian

## Ā¶TODO [C] Calculus of variations - Wikipedia lagrangianviz

https://en.wikipedia.org/wiki/Calculus_of_variations#Lavrentiev_phenomenon L(t,x,x')=(x^{3}-t)^{2}x'^{6}, hmm, that would be pretty cool to visualise!!

also what about Euler-Lagrange??

## Ā¶TODO [B] Letās take another example. Suppose the Hamiltonian is given by lagrangian

Todo H=pq similar to my expriments

from The Theoretical Minimum by George Hrabovsky

## Ā¶TODO xx lagrangiantotweetoutboxlagrangians

in classical mechanics, lagrangian blablabla

but what if we use a different function, which is not included in the textbooks

ā¦

complete bullshit

well maybe they don't include it in the textbook for reasons

N/N

## Ā¶TODO [B] Simulating Quantum Mechanics Using Matplotlib and Tkinter /r/Python lagrangian

## Ā¶ good lagrangian points intuition https://www.reddit.com/r/dataisbeautiful/comments/adttul/oc_peusdopotential_in_the_earthmoon_system/edk9f1j/ lagrangianphysicsspace

This is a graph of potential, which also serves as an analogy that sort of helps you figure out what the overall force on an orbiting object is: if you placed a ball on this surface and it was going to roll "downhill", which way would it go? The five crosses mark points where the top surface is flat, so an object at those points will experience no overall force acting on it (if it's orbiting at the same speed as the moon). These are therefore pseudo-stable points where you could leave a satellite orbiting.

## Ā¶[B] classical mechanics - What if the Euler Lagrange equation yields a 'trivial' answer - Mathematics Stack Exchange lagrangiantoblog

https://math.stackexchange.com/questions/191935/what-if-the-euler-lagrange-equation-yields-a-trivial-answer

example lagrangian independent of path, all paths are stationary