# What makes a Lagrangian a Lagrangian?

I just wanted to know what the characteristic property of a Lagrangian is?

How do you see without referring to Newtonian Mechanics that it has to be $L=T-V$?

People constructed a Lagrangian in Special Relativity and General Relativity. But is there a general recipe to find a Lagrangian for a theory or is a Lagrangian just chosen so that it works out?

Of course there are some symmetry and invariance properties that play an important role there, but is it possible to put the theory of Lagrangian to a more abstract and general level so that we can say a priori how a Lagrangian for a given theory must look like?

Which information are necessary to construct a Lagrangian for a physical theory?

• Jun 24 '13 at 22:45
• I want to say that this question there is only related to classical mechanics. I am interested in a general way how one rigorously constructs Lagrangians. Jun 24 '13 at 22:46
• "How do you see without referring to Newtonian Mechanics that it has to be $L=T−V$?" You may find Elisha Huggins' paper "Gravity, Time, and Lagrangians" enlightening. I did. It's not clear to me that it answers you questions as it assumes General Relativity in order to motivate the minus sign ... Jun 25 '13 at 0:52
• If you have a force like $F = F(q)$, it it trivial to construct $V(q)$ - to a constant -, so it is trivial to construct $L$. But if you have, for instance, a force like $F = \dot q$, you are not able to give an explicit Lagrangian. Jun 25 '13 at 10:05

As far as I know there is no way of rigorously constructing a Lagrangian for a new physical theory. The point is you just have to guess a Lagrangian (i.e. construct your own theory), check all the invariance/symmetry properties you want to have and hope that the predictions your theory makes will agree with the measurements.

The hard part is guessing 'correctly' - as my professor in QFT put it: So far, that 'correct guess' happenend twice, in both cases earning the authors a Nobel Prize (QED and electroweak Lagrangian).

• You can construct Lagrange's equations from d'Alembert's principle and Newton's F=ma, then assume the generalized forces Q can be constructed from a potential (i.e. Q=-grad(V)) to eventually obtain L=T-V. For details you can refer to Fetter and Walecka's Theoretical Mechanics of Particles and Continua. Mar 16 '18 at 1:25