# Physical Interpretation of point transformation invariance of the Lagrangian

• anton01
In summary, the conversation discusses the invariance of Euler-Lagrange's equations under a point transformation and the physical interpretation of this concept. It is concluded that coordinates are not physical and are created for calculations, and the Lagrangian formalism is independent of coordinate systems, making it universal.
anton01

## Homework Statement

The problem asked us to show that the Euler-Lagrange's equations are invariant under a point transformation q$_{i}$=q$_{i}$(s$_{1}$,...,s$_{n}$,t), i=1...n. Give a physical interpretation.

## Homework Equations

$\frac{d}{dt}(\frac{\partial L}{\partial \dot{s_{j}}})$=$\frac{\partial L}{\partial s_{j}}$

## The Attempt at a Solution

I proved the invariance.
I am stumped with the physical interpretation. Except for the fact that the E-L equations are invariant when we change coordinates pointwise, I don't see any other physical interpretation. But this answer seems just repeating their question.

Are coordinates something physical to begin with?

voko said:
Are coordinates something physical to begin with?

No they are not. They are something we make in order to do the calculations.
It was just a wild guess really.

That's what one should expect physically. Now you have proved that the Lagrangian formalism does not require any special coordinates, they all work. What does that mean about the formalism itself?

This means that the Lagrangian is independant of the coordinate system. And this makes sense, because a coordinate system is nothing physical and is arbitrary.
So, this means that the Lagrangian is universal? In other words, it works for any coordinate system.

## 1. What is the significance of point transformation invariance in the Lagrangian?

Point transformation invariance is a fundamental principle in physics that states that the laws of nature should remain unchanged when the coordinates used to describe a system are changed. In the context of the Lagrangian, this means that the equations of motion derived from the Lagrangian should remain the same under different coordinate systems.

## 2. How does point transformation invariance relate to the principle of relativity?

The principle of relativity states that the laws of physics should be the same for all observers in uniform motion. Point transformation invariance is a mathematical expression of this principle, as it ensures that the equations of motion remain the same regardless of the chosen reference frame.

## 3. Can point transformation invariance be violated in certain cases?

Yes, there are certain scenarios where point transformation invariance can be violated. For example, in systems with non-conservative forces or non-inertial reference frames, the equations of motion derived from the Lagrangian may not be invariant under point transformations.

## 4. How does point transformation invariance affect the conservation laws in physics?

Point transformation invariance plays a crucial role in the conservation laws of physics. Invariance under translations in time and space leads to the conservation of energy and momentum, respectively. Invariance under rotations leads to the conservation of angular momentum. These conservation laws are essential for understanding the behavior of physical systems.

## 5. Are there any practical applications of point transformation invariance in physics?

Yes, point transformation invariance is a fundamental principle in the development of physical theories and has many practical applications. For example, it is used in the study of symmetries in quantum field theory and in the formulation of gauge theories, such as the Standard Model of particle physics. It also plays a crucial role in applications such as fluid dynamics and electromagnetism.

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