# Formulating the correct least action expression

• qtm912
In summary, the process of finding the correct Lagrangian and corresponding least action for a physical system involves experimentation and following general guidelines such as gauge invariance, Lorentz invariance, and scalar nature. It is not possible to derive the correct Lagrangian from first principles.

#### 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

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.

Thank you Matterwave for clarifying.

## 1. What is the principle of least action?

The principle of least action states that the path taken by a physical system between two points in time is the one that minimizes the action, which is a mathematical quantity that takes into account the system's energy and time. This principle is a fundamental concept in classical mechanics and is used to derive the equations of motion for a physical system.

## 2. Why is the least action principle important in physics?

The least action principle is important in physics because it provides a powerful and elegant way to describe the behavior of physical systems. It allows us to derive the equations of motion for a system without needing to know the underlying forces acting on it, making it a useful tool for solving complex problems in classical mechanics.

## 3. How do you formulate the correct least action expression?

To formulate the correct least action expression, you must first identify all the relevant variables and parameters of the system, such as position, velocity, and potential energy. Then, you can use the principle of least action to set up an integral that represents the action of the system. By varying this integral with respect to the system's variables, you can find the path that minimizes the action and derive the correct least action expression.

## 4. What are some common applications of the least action principle?

The least action principle has many applications in physics, including in classical mechanics, electromagnetism, and quantum mechanics. It is used to derive the equations of motion for particles and fields, to analyze the behavior of systems in equilibrium, and to understand the behavior of complex physical systems.

## 5. Are there any limitations to the least action principle?

While the least action principle is a powerful tool in physics, it does have some limitations. It is based on the assumption that the system's path is continuous and differentiable, which may not always be the case in real-world situations. Additionally, the principle only applies to systems that can be described by classical mechanics, and may not be applicable to systems described by other theories, such as quantum mechanics.