📕 Node [[classical mechanics]]
📄 classical-mechanics.md by @karlicoss

Table of Contents

Definitions

A nonholonomic system – state depends on the path taken in order to achieve it.

Phase space vs configuration space

Lagrangian [[lagrangian]]

units of energy

no single expression for all physical systems

only applicable to systems with holonomic constraints

Why only first derivatives are appearing

Ostrogradsky instability
https://physics.stackexchange.com/questions/4102/why-are-there-only-derivatives-to-the-first-order-in-the-lagrangian

Independent position and velocity

https://physics.stackexchange.com/questions/885/calculus-of-variations-how-does-it-make-sense-to-vary-the-position-and-the-ve

in a sense these are initial conditions so both are necessary

Let me refer to this great book: "Applied Differential Geometry". By William L. Burke. The very first line of the book (where an author usually says to whom this book is devoted) is this: "To all those who like me have wondered how in the hell you can change q’ without changing q"

https://physics.stackexchange.com/questions/119992/what-do-the-derivatives-in-these-hamilton-equations-mean

q and q’ are just labels, treat them independently
good points about meaning in the very end

another explanation from the same guy https://physics.stackexchange.com/questions/60706/lagrangian-mechanics-and-time-derivative-on-general-coordinates

interesting point that var(q’) = d/dt var(q) (why?) https://physics.stackexchange.com/a/985/40624

might be insightful?… https://physics.stackexchange.com/a/2895/40624

https://physics.stackexchange.com/questions/168551/independence-of-position-and-velocity-in-lagrangian-from-the-point-of-view-of-ph – not sure if useful…

https://physics.stackexchange.com/questions/60706/lagrangian-mechanics-and-time-derivative-on-general-coordinates – not sure if useful..

For every symmetry, there is a conserved quantity

Einstein was not satisfied about GR until he derived it from lagrangian (as an indication how powerful is the concept) https://www.reddit.com/r/Physics/comments/3me1hr/explanation_of_lagrangian_mechanics/cveb611/

reddit recommends Taylor’s book

as an analogy: when you learn energy, dealing with forces is much easier; when you learn lagrangian, dealing with crazy coordinates and constraints much easier

When you find the Euler-Lagrange equations for your system, they will be written in terms of these generalized coordinates, and the terms in the equations are known as generalized forces. This is because usually the Euler-Lagrange equations have something that looks a lot like "ma" (mass times acceleration) on one side of the equations, and thus the other terms could be interpreted as "forces", but written in these general variables.

[2019-01-15] http://cp3-origins.dk/a/14332 [[toblog]]

When the action, and hence the phase, is stationary changing it by a small amount doesn’t change the phase by much. In a small region (compared to ℏ) these paths can add up coherently to give a significant contribution to the sum above. This is what we see in the cartoon above for a very small subset of paths.
Classical mechanics is quantum mechanics using the stationary phase approximation.

hmm, interesting about Wick rotation…

Paths far from the minimum hardly contribute anything and so it isn’t necessary to calculate the action arbitrarily accurately.

eh?

Galilean invariance forces classical lagrangian to depend on velocity quadratically

[2019-01-15] classical mechanics - Deriving the Lagrangian for a free particle - Physics Stack Exchange [[lagrangian]]

https://physics.stackexchange.com/questions/23098/deriving-the-lagrangian-for-a-free-particle
Comment:
justification of lagrangian for classical mechanics from Landau… weird, didn’t really get it

[2018-11-29] classical mechanics - Why does Lagrangian of free particle depend on the square of the velocity ? - Physics Stack Exchange

https://physics.stackexchange.com/questions/63370/why-does-lagrangian-of-free-particle-depend-on-the-square-of-the-velocity/92561

The Lagrangian should not only be independent of the direction of v⃗ v→ but it should also change correctly under a Galilean transformation. For instance, if KK and K′K′ are two frames of reference with a relative velocity V⃗ V→ then the two Lagrangians LL and L′L′ should differ only by a total time derivative.

[2018-11-29] newtonian mechanics - Galilean invariance of Lagrangian for non-relativistic free point particle? - Physics Stack Exchange [[lagrangian]]

https://physics.stackexchange.com/questions/14875/galilean-invariance-of-lagrangian-for-non-relativistic-free-point-particle

[2018-11-30] Degenerate Lagrangian? - My Math Forum

http://mymathforum.com/differential-equations/43493-degenerate-lagrangian.html
a degenerate Lagrangian is one who’s Hesse determinant is zero. It’s a condition on the second partial derivatives of the Lagrangian.

there is also a link to pdf, might be worth reading…

[2018-11-25] What does a Lagrangian of the form (L=m^2\dot x^4 +U(x)\dot x^2 -W(x)) represent? - Physics Stack Exchange

https://physics.stackexchange.com/questions/17406/what-does-a-lagrangian-of-the-form-l-m2-dot-x4-ux-dot-x2-wx-represent
eh, weird. complex expression for lagrangian that ends up looking same as classical. well ok

on Lagrangian being extreme value/minimum

[2018-12-04] lagrangian formalism - Confusion regarding the principle of least action in Landau & Lifshitz "The Classical Theory of Fields" - Physics Stack Exchange

https://physics.stackexchange.com/questions/122486/confusion-regarding-the-principle-of-least-action-in-landau-lifshitz-the-clas#comment249472_122504
conjugate points; about infinitesimal path, characteristic scale of the problem
conditions for lagrangian regularity and conjugate points

[2018-12-04] lagrangian formalism - Hamilton’s Principle - Physics Stack Exchange

https://physics.stackexchange.com/questions/9/hamiltons-principle

Basically, the whole thing is summarized in a nutshell in Richard P. Feynman, The Feynman Lectures on Physics (Addison–Wesley, Reading, MA, 1964), Vol. II, Chap. 19. (I think, please correct me if I'm wrong here). The fundamental idea is that the action integral defines the quantum mechanical amplitude for the position of the particle, and the amplitude is stable to interference effects (-->has nonzero probability of occurrence) only at extrema or saddle points of the action integral. The particle really does explore all alternative paths probabilistically.

[2018-12-02] http://www.scholarpedia.org/article/Principle_of_least_action#When_Action_is_a_Minimum

or some 1D potentials V(x) (those with ∂2V/∂x2≤0 everywhere), e.g. V(x)=0 , V(x)=mgx , and V(x)=−Cx2 , all true trajectories have minimum S . For most potentials, however, only sufficiently short true trajectories have minimum action; the others have an action saddle point. "Sufficiently short" means that the final space-time event occurs before the so-called kinetic focus event of the trajectory.

[2018-12-02] Even more trivial example when least action principle doesn’t work

Принцип наименьшего действия. Часть 2 / Хабр https://habr.com/ru/post/426253/

На рисунке нарисованы обе физически возможные траектории движения шара. Зеленая траектория соответствует покоящемуся шару, в то время как синяя соответствует шару, отскочившему от пружинящей стенки.
Однако минимальным действием обладает только одна из них, а именно первая! У второй траектории действие больше. Получается, что в данной задаче имеются две физически возможных траектории и всего одна с минимальным действием. Т.е. в данном случае принцип наименьшего действия не работает.

[2018-11-30] Лагранжиан L {\displaystyle L} L называется вырожденным, если его оператор Эйлера — Лагранжа удовлетворяет нетривиальным тождествам Нётер. В этом случае уравнения Эйлера — Лагранжа не являются независимыми

https://ru.wikipedia.org/wiki/%D0%A2%D0%BE%D0%B6%D0%B4%D0%B5%D1%81%D1%82%D0%B2%D0%B0_%D0%9D%D1%91%D1%82%D0%B5%D1%80

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