Classical mechanics

Table of Contents

Definitions physics

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

DONE Phase space vs configuration space physics

Lagrangian physicslagrangian

units of energy physicslagrangian

no single expression for all physical systems physicslagrangian

only applicable to systems with holonomic constraints physicslagrangian

Why only first derivatives are appearing physicslagrangian

Independent position and velocity physicslagrangian

in a sense these are initial conditions so both are necessary physicslagrangian

TODO 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" physicslagrangian physicslagrangian

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

TODO interesting point that var(q') = d/dt var(q) (why?) physicslagrangian

TODO might be insightful?… physicslagrangian

For every symmetry, there is a conserved quantity physicslagrangian

Einstein was not satisfied about GR until he derived it from lagrangian (as an indication how powerful is the concept) physicslagrangian

reddit recommends Taylor's book physicslagrangian

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 physicslagrangian

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. physicslagrangian

TODO [2019-01-15] physicslagrangiantoblog

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… physicslagrangiantoblog

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


Galilean invariance forces classical lagrangian to depend on velocity quadratically physicslagrangian

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

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.

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

TODO [2018-11-30] Degenerate Lagrangian? - My Math Forum physicslagrangian
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 physicslagrangian
eh, weird. complex expression for lagrangian that ends up looking same as classical. well ok

START on Lagrangian being extreme value/minimum physicslagrangian

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

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] physicslagrangian

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 physicslagrangian

Принцип наименьшего действия. Часть 2 / Хабр

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.

TODO [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 physicslagrangian

[2019-01-15] Legendre transform physics

[2019-01-15] nice intuition in terms of areas physics

[2018-11-29] Преобразование Лежандра — Википедия physicslagrangian

[2019-01-15] В том случае, когда лагранжиан не вырожден по скоростям, то есть physicslagrangian

{\displaystyle p=∇ _{u}L(q,u)≠ 0,} {\displaystyle p=∇ _{u}L(q,u)≠ 0,}
можно сделать преобразование Лежандра по скоростям и получить новую функцию, называемую гамильтонианом:

[2019-01-15] Making Sense of the Legendre Transform physics

nice pdf, basically they say it's just a different view, sometimes it's easier to control the derivative
they introduce generalised forces too
that's not surprising there is connection with thermodynamics, they show some stuff with Gibbs energy etc

So the Fourier transform and the Legendre transform may be interpreted as the same thing, just over different semirings. physics physics

(Legendre transformation is "zero temperature limit" of the Laplace Transformation) physics

Two convex functions f and g are Legendre transforms of each other when their first derivatives are inverse functions:
and another nice plot with areas intuition as well
All of the dynamical laws are constructed from derivatives of H and L, and we decline to specify an additive constant for the same reason we do so with conservative potentialsi   and, more generally, anti-derivatives.

TODO Hamiltonian physics

has some meaning in statistical physics physics

TODO something about Poisson brackets physics

configuration space with dimension n: 2n Hamilton equations of first order; n Euler-Lagrange of second order physics

Hamiltonians are easier to find transformations to canonical coordinates physics

[2019-01-16] is hamiltonian same thing as energy? physics

[2018-11-25] classical mechanics - Example where Hamiltonian \(H \neq T+V=E\), but \(E=T+V\) is conserved - Physics Stack Exchange physics

START [B] physics sim for phase space repos/physics-sim physicsstudyvizlagrangian

CREATED: [2018-11-19]

TODO [B] additional term depending on velocity is kinda like time transformation? physicsstudylagrangian

CREATED: [2018-11-19]

-----–— [2019-01-15] -------–— review later… physics

[C] [2018-11-29] homework and exercises - Lagrangian in a system with a specific velocity dependent potential - Physics Stack Exchange physicslagrangian

------–— [2019-01-15] ---------–— needs review physics

TODO [D] Griffith classical mechanics physics

[2018-07-24] Classical Mechanics physics

I guess I need to work out some simple classical system by myself

lagrangian (kinda + there was some intuition in baez notes?) physics

hamiltonian (bit more tricky) physics

poisson brackets: ??? physics

canonical coordinates and derivatives – why's that enough? or by definition of 'classical'? physics

???? physics

START [B] baez lagrangian mechanics physicsbaez

principle of minumum energy explanation 1.2.2 physicsbaez

p.33 special relativity physicsbaez

Many Lagrangiansdothis,butthe\best"oneshouldgive anactionthatisindependentoftheparameterizationofthepath|sincetheparameterizationis\unphysical":it can'tbe measured.Sotheaction

gauge symmetries physicsbaez

Thesesymmetriesgive conservedquantitiesthatworkouttoequalzero!
gauge symmetries result in conserved quantities… which are just equal to zero

p46 cool analogy between refraction and riemannian metric in GR physicsbaez

ham vs lagr physics

Furthermore, whereas in Lagrangian mechanics there is a dependence between the generalized coordinates q and their velocities (the latter being the time derivatives of the former), in Hamiltonian mechanics the momenta are to be regarded are independent from the generalized coordinates.
With these new coordinates, one proceeds to demand again that the action is minimized, and, instead of the Euler-Lagrange equations, one finds what are known as Hamilton's canonical equations. Again these are the equations of motion of the system, which are to be solved in order to find the trajectory. One key difference is that if your system required N generalized coordinates, and thus N Euler-Lagrange equations, there will be 2N Hamilton canonical equations but they are "half as difficult" to solve.

That's the best I can do without getting technical. Also, Hamiltonian mechanics is cooler, just saying.

I think this is sort of misleading, they talk about dependency again… physics

TODO How are symmetries precisely defined? - Physics Stack Exchange physics

[C] action principle for SR physicsrelativity

CREATED: [2018-07-30]

infer ansatz for action from dimensional analysis

Snonrel = int 1/2 m v2(t) dt
hamilton's equation: dv/dt = 0, hence constant velocity

doesn't work for sr, rationale: is not forbidding v > c.

require action to be Lorentz scalar

S = -mc int ds – in the nonrelativistic limit results in same physics ans nonrel lagrangian
also, that explains the fact that particle traces the path minimizing spacetime interval

momentum and hamiltonian – coincide with energy
reparameterisation: express invariant via square root of metric and coord. derivatives

right, and we get euler-lagrange equations as a result d2 xu/ds2 = 0 – basically 4-velocity is constant!

guessing electric charge lagrangian..

TODO also problems physicsrelativity

TODO nice book, read more from it? physicsrelativity physics

Generalized coordinates – like normal coordinates, but without redudancy in constraints. They are independent; basically it means that for any generalised coordinates [tuple] there must be a valid system?

benefit of generalised coordinates is most apparent when considering double pendulum

[2018-07-31] some random notes physicslagrangianhamiltonian

x'2 + x2 = C2 – energy conservation

Force F(x); potential energy U(x) as integral of force

Take 1/2 m v2 + U(x) – call it "total energy", it is conserved

TODO what if force depends on time explicitly?

Law of physics: there exists a three-dimenstional potential!

Principle of least action

Hamiltonian from Lagrangian

dH/dt = - ∂ L / ∂ t

Lagrangian -> Euler-Lagrange equations

Holonomic constraints take form: f(q1, … qn, t) = 0

Holonomic system => L = K - U

Nonholomonic system: rubber ball allowed to roll, but not slide/spin

Lagrange multipliers and forces of constraint, Taylor 278

If coordinate qi is ignorable (dL/dqi = 0), the corresponding generalized momentum pi = dL/dq'i is conserved

H(q1 … qn, p1 … pn)

q' = dH/dp
p' = -dH/dq

B = ∇ x A + ∇ S

A is defined up to the gradient of some scalar field, guage field

Poisson brackets

  • {A, B} = - {B, A}
  • {A + B, C} = {A, C} + {B, C}
  • {a A, B} = a {A, B}
  • {AB, C} = {A, C} B + A {B, C}
  • {qi, qj} = 0
  • {pi, pj} = 0
  • {qi, pj} = deltaij
  • Q' = {Q, H} – change of quantity over time
  • {Q, Ly} – change of quantity over rotation

TODO [C] [2018-11-25] Zero Hamiltonian and its energies | Physics Forums physics

First of all, you are not understanding what he Hamiltonian is. The Hamiltonian is not the value of the energy, it is a relationship between position and momentum for a particular system. If the Hamiltonian is p^2 + q^2, and the value of p^2 + q^2 is zero, then the Hamiltonian is p^2 + q^2, not zero. It is analogous to Bush being the president. Bush is the current VALUE of "president", but the concept of president is not synonymous with "Bush".

[2019-06-18] eh, they are talking about invariance by reparametrization, but I don't think I really understand what they mean… physics

STRT [B] [2018-12-02] Are the Hamiltonian and Lagrangian always convex functions? - Physics Stack Exchange physicslagrangian

[2019-01-16] also good answer, basically explaining that it's not great to impose convexity conditions on only one set of canonical coordinates physicslagrangian

In conclusion, convexity does not seem to be a first principle per se, but rather a consequence of the type of QFTs that we typically are able to make sense of. It might be that it is possible to give a non-perturbative definition of a non-convex (but unitary) theory.

[2019-01-16] physicslagrangian

hmm that's interesting, he got a reply about considering sheets of the hamiltonian, each sheet convex… so maybe it does make sense??

[B] [2018-11-15] Proof by Picture physicsviz

TODO [B] [2018-11-18] book: Structure and Interpretation of Classical Mechanics physics

TODO [C] hmm, to visualise phase trajectories, we can just do 3D plot, then we know that the particle is moving along isolines physicshamiltonianviz

CREATED: [2019-01-15]

TODO [C] Isotropic lagrangian velocity physicslagrangian

[2019-06-18] Now since space is isotropic, L should be independent of velocity v⃗ , and should in fact be a function of |v⃗ |2. physicslagrangian

STRT [C] discrete lagrangian? vary it on space of matrices?? physicsthink

CREATED: [2018-12-04]

TODO [C] [2018-11-30] Задачка на Лагранжиан : Помогите решить / разобраться (Ф) - Страница 3 physics

les в сообщении #552466 писал(а):
И как в таком случае вводят импульсы?
Связями. Если интеренсно, посмотрите книгу Дирак, "Принципы квантовой механики". Бонус-глава "Лекции по квантовой механике

бы очень рекомендовал замечательную книгу
Гитман Д.М., Тютин И.В. Каноническое квантование полей со связями.
Думаю, ТС хватит прочитать первые две главы, чтобы получить ответы на инересующие в

[B] [2020-08-09] Notes & HW for Section 6.1 physicslagrangian

classification of critical points

[B] [2020-08-09] Structure and Interpretation of Classical Mechanics: Chapter 7 physics

The Lagrangian L must be interpreted as a function of the position and velocity components qi and q˙i, so that the partial derivatives make sense, but then in order for the time derivative d/dt to make sense solution paths must have been inserted into the partial derivatives of the Lagrangian to make functions of time. The traditional use of ambiguous notation is convenient in simple situations, but in more complicated situations it can be a serious handicap to clear reasoning. In order that the reasoning be clear and unambiguous, we have adopted a more precise mathematical notation. Our notation is functional and follows that of modern mathematical presentations.2 An introduction to our functional notation is in an appendix.

[C] [2018-08-25] In classical mechanics, the state of a system is determined by a point in phase space physicslagrangian

It's unique! In the same way as quantum state is unique

[C] [2019-03-20] lagrangian formalism - What is the difference between a complex scalar field and two real scalar fields? - Physics Stack Exchange physics

They're identical. Typically, we use complex fields if we have a U(1)U(1) symmetry, or some more complicated gauge group with complex representations.

Incidentally, the same comment applies to whether we use Majorana spinors or Weyl spinors.

[C] [2018-07-31] – explanation why differential equations of orders higher than two do not appear in physics physicsmathdiffeq

DONE [A] [2020-01-15] Noether’s Theorem – A Quick Explanation (2019) physics

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