# 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?

• – Qmechanic 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. – Xin Wang 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 ... – dmckee 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. – Trimok Jun 25 '13 at 10:05