Classical mechanics

https://physics.stackexchange.com/questions/89035/whats-the-point-of-hamiltonian-mechanics#89036

symmetry is transformation of a (mathematical?) model that doesn't change the physics it predicts

https://physics.stackexchange.com/questions/103997/are-the-hamiltonian-and-lagrangian-always-convex-functions

• node
  all trajectories approach/receed point, and every trajectory is tangent to a straight line through the point
  
• saddle
  two trajectories approach point, all other recede
  
• spiral
  trajectories 'wind around' point
  same as node, but no approach direction? (hence no tangent)
  
• center
  all nearby trajectories are concentric closed curves
  

stability: a critical point is stable if trajectories remain nearby for t > 0; asymptitically stable if approach the point as t -> infty
stability determined by eigenvalues (complex, sign of real part etc)

The assumption that ∂2L∂x˙2≠0 is known as nondegeneracy.
If the Lagrangian is nondegenerate one can write (1) in the form Newton assumed so long ago for the laws of physics,

Derivation of one-dimensional Euler–Lagrange equation – this section is actually the one that makes sense of it

from https://www.math.tecnico.ulisboa.pt/~jnatar/MG-03/

infer ansatz for action from dimensional analysis

S_nonrel = 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 x^u/ds² = 0 – basically 4-velocity is constant!

guessing electric charge lagrangian..

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".

https://physics.stackexchange.com/questions/212909/lagrangian-is-isotropic-in-space

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

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

https://physics.stackexchange.com/questions/3503/what-is-the-difference-between-a-complex-scalar-field-and-two-real-scalar-fields

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.

shit.. some interesting mathematical details about metric from the very beginning

Although the action S may not generally have a meaningful interpretation, there is an alternative formulation of the EL-equations which gives an equation that gives the value of S directly but not the function that achieves the minimum value of S. This equation is called the Hamilton-Jacobi equation, and is also a very widely studied equation, particularly in the context of conservation laws and symplectic geometry. The standard method of solving the HJ-equation is by the method of characteristics. The characteristic equations are precisely the Hamiltonian equations also learned in a typical mechanics course.

(old link) http://www.physics-quest.org/Book_Chapter_Lagrangian.pdf
whoa. that's something interesitng

https://books.google.co.uk/books?id=9nL7ZX8Djp4C&pg=PA367&lpg=PA367&dq=%22euler+lagrange%22+solutions+%22global+minimum%22&source=bl&ots=oUmdX3lEHJ&sig=lWeYTxDvVFemkLadlfSg7rgc8_8&hl=en&sa=X&ved=2ahUKEwjF4c7ukoDfAhU9SxUIHatDDmYQ6AEwD3oECA8QAQ#v=onepage&q=%22euler%20lagrange%22%20solutions%20%22global%20minimum%22&f=false
sufficiently long segments cease to be the minimum..

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

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…

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?
https://en.wikipedia.org/wiki/Holonomic_constraints#Transformation_to_independent_generalized_coordinates

benefit of generalised coordinates is most apparent when considering double pendulum https://en.wikipedia.org/wiki/Generalized_coordinates#Double_pendulum

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, … q_n, 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 q_i is ignorable (dL/dq_i = 0), the corresponding generalized momentum p_i = dL/dq'_i is conserved

H(q₁ … q_n, p₁ … p_n)

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}
  

• {q_i, q_j} = 0
  
• {p_i, p_j} = 0
  
• {q_i, p_j} = delta_ij
  
• Q' = {Q, H} – change of quantity over time
  
• {Q, L_y} – change of quantity over rotation
  

https://quantum-friend-theory.tumblr.com/post/172814384897/noethers-theorem-a-quick-explanation

ok, this is kind of on the right track, deriving GR lagrangian