Formulating the correct least action expression

[qtm912]

I am trying to understand how, for a given physical system it is possible to come up with the correct expression for the lagrangian and thence the least action formulation from which the governing equations can be derived by application of the euler lagrange equation. From what I have seen so far (and I may be mistaken) this seems to involve some intelligent guesswork. So I conclude as follows :

- the lagrangian (or if not the lagrangian then the least action) must be chosen so that it is gauge invariant

- the lagrangian should contain terms like (dx/dt) squared for single particles where x represents phase space and correspondingly for a field ∅, with terms in d∅/dt squared

- for relativistic systems the lagrangian (or the least action?) should also be lorenz invariant

Then through a process of experimentation the lagrangian and corresponding least action can be found.

This is my understanding so far and I am asking if this (ie any of the above) is correct.

Many thanks

 

[Matterwave]

The general Lagrangian is simply the Lagrangian which leads to the correct equations of motion (i.e. the one that agrees with experiment). There are some general "guidelines" that you can follow if you want your theory to obey some properties, but that's about it. You cannot, in general, find the correct Lagrangian of a theory from first principles.

If you want your theory to be Lorentz invariant, then the action must be Lorentz invariant. If you want your theory to be generally covariant, then the action should be a scalar (in the sense that it is coordinate system independent). If you want your field equations to be linear, then you need a quadratic term in the fields in your Lagrangian.

There are just some general guidelines; however.

 

[qtm912]

Thank you Matterwave for clarifying.