Does anyone know a good heuristic motivation for the Lagrangian Formalism? I think most physicist just accept at one point that it works and thats that. I think I understand the historic origin. Lagrange was fascinated by the Fermat Principle and searched for a similar principle for massive objects. By guessing he ended up with the classical Lagrangian that reproduces the Newtons second law.

I would be interested in a modern viewpoint. Can in be maybe understood, in some way, from a quantum field theoretic or relativistic point of view? I stumbled upon the book chapter by stackexchange member Hans de Vries: http://physics-quest.org/Book_Chapter_Lagrangian.pdf , but he never elobarted on this. For example in What is the physical meaning of the action in Lagrangian mechanics? he just repeats his idea of "least action = least proper time", but does not give any further explanations for this claim.

Does anyone know some elaboration on this or some other illuminating idea for why the Lagrangian formalism works (not in classical mechanics, but the more fundamental case of Quantum Field Theory)?

PS: I dont think this is a dublicate of What is the physical meaning of the action in Lagrangian mechanics? because the question there is specifically about the physical meaning of the action

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    $\begingroup$ On a related note: Through physics.SE, I once stumbled upon an amazing, relatvely simple motivation why $L=T-V$ that used some basic differential geometry to explain it in a few pages. Id love to reas it again, but have lost it. Would anyone happen to know what I'm talking about? $\endgroup$ – Danu Jun 3 '14 at 7:28
  • $\begingroup$ Lagrangian formalism continues on the work of Newton and the principle of D'Alembert. It provides a better way to handle arbitrary coordinate systems so that is why it is used especially in QFT and GR. Hamilton's formulation which follows after Langrange gives a physical interpretation as the principle of Least Action (or more correctly Stationary Action) and which is generalized into the Feynman Path Integral Formulation $\endgroup$ – Nikos M. Jun 3 '14 at 8:19

OP is asking several questions. To keep the answer simple, let us focus on point mechanics. The considerations below may be generalized to field theory.

I) Classically, OP's question (v2) has be asked many times before on Phys.SE in various forms, e.g. here, here, and links therein. This Phys.SE post discusses how to derive the form of the Lagrangian action

$$\tag{1} S~=~\int_{t_i}^{t_f} \! dt~L, \qquad L~=~T-V,$$

starting from just Newton's laws and D'Alembert's principle.

In turn, the Lagrangian formalism is formally$^1$ equivalent to the Hamiltonian formalism via a Legendre transformation, cf. e.g. this Phys.SE post.

II) Quantum mechanically, the Lagrangian path integral

$$ \tag{2} \langle q_f,t_f|q_i,t_i \rangle~\sim~\int_{q(t_i)=q_i}^{q(t_f)=q_f} \!{\cal D}q~~\exp\left[ \frac{i}{\hbar}S[q]\right],$$

can be formally$^2$ derived from the operator formalism via a time slicing procedure, as is done in many textbooks.

On the other hand, the operator formalism itself may be viewed as a quantum version of the Hamiltonian formalism. This is most easily seen in the Heisenberg picture, where Heisenberg's equations of motion and commutators correspond to Hamilton's equations of motion and Poisson brackets, respectively, at the classical level.


$^1$ A singular Legendre transformation leads to primary constraints.

$^2$ In physics we often ignore the important mathematical problem of how to define the path integral rigorously. For more on the correspondence between the operator formalism and the path integral formalism, see e.g. this Phys.SE post.


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