Understanding Action as a Scalar: Exploring the Concept and its Importance

In summary, the action is a scalar because it is invariant under Galilei transformations, but it is not invariant under boosts.
  • #1
Adams2020
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TL;DR Summary
Why is the action a scalar? Please explain.
Why is the action a scalar? Please explain.
 
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  • #3
Dale said:
Why would it be a vector?
I don't know. This is the question our professor asked us without explanation to think. What is the reason that it is a vector?
 
  • #4
Is energy a vector?
 
  • #5
Adams2020 said:
I don't know. This is the question our professor asked us without explanation to think. What is the reason that it is a vector?
If your professor wanted you to think then maybe you should think a little about it and write your thoughts. Sometimes it is easier to think what would happen if it were not a scalar. Action is the basis of "the principle of least action", so how would that work if action were a vector?
 
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  • #6
anorlunda said:
Is energy a vector?
no
 
  • #7
I understand what the professor means but for me the question still sounds strange or at least too philosophically.
A scalar is a kind of a tensor on a manifold. The scalar is a function of the manifold with values in ##\mathbb{R}##. In this sense the Action is not a scalar. The Action is a function of definite functional space with values in ##\mathbb{R}##. The Action is not defined on the manifold. It is defined on functions. Reasonable question could sound as "is the Lagrangian a scalar?" "on which manifold is the Lagrangian defined?"
 
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  • #8
Adams2020 said:
Summary:: Why is the action a scalar? Please explain.

Why is the action a scalar? Please explain.
I guess your professor gave a bit more of context before asking the question. I can only guess, what he might be after. First of all you need to specify in which sense the action is a scalar, i.e., with respect to which transformations it should be invariant.

If you describe Newtonian mechanics, the equations of motion for the full system must be Galilei invariant. For this to be true it's sufficient that the action is an invariant under all Galilei transformations.

That's of course NOT true for the standard Lagrangian,
$$L=T-V=\sum_k \frac{m_k}{2} \dot{\vec{x}}_k^2 + \sum_{j<k} V(|\vec{x}_j-\vec{x}_k|),$$
when considering Galilei boosts. On the other hand it's known that the equations of motion are Galilei invariant. So there must be a weaker condition on the action sufficient to yield Galilei-invariant equations of motion. What's this weaker condition also applying to Galilei boosts?
 
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  • #9
It is hard to think about the question posed by your professor without knowing the context in which he raised it. I have been looking at general relativity lately and I can begin to see the context that wrobel puts his answer in, but this may not be where your professor was going.

He may likely be asking, if a observer in a rotating or translating frame formulates the lagrangian, and solves for the equations of motion, will the observer get the same lagrangian and motion when the observer goes back to the original frame?
 

1. What is the concept of action as a scalar?

The concept of action as a scalar refers to the idea that actions can be measured and compared on a single scale, rather than being categorized into discrete categories. This allows for a more nuanced understanding of actions and their impact.

2. How is the concept of action as a scalar important in scientific research?

The concept of action as a scalar is important in scientific research because it allows for more precise and accurate measurement and analysis of actions. It also allows for a better understanding of the relationships between different actions and their effects.

3. Can you provide an example of action as a scalar?

One example of action as a scalar is measuring the impact of different types of exercise on physical fitness. Instead of categorizing exercises as "high intensity" or "low intensity," they can be measured on a scalar scale of intensity, allowing for a more comprehensive understanding of their effects.

4. How does the concept of action as a scalar relate to other scientific concepts?

The concept of action as a scalar is closely related to other scientific concepts such as measurement, causality, and variability. It also has implications for fields such as psychology, sociology, and economics, where actions and their effects are often studied.

5. What are the potential limitations of using action as a scalar in research?

One potential limitation of using action as a scalar is that it may oversimplify complex actions and their effects. Additionally, the scale used to measure actions may not capture all relevant factors or may not be universally applicable. It is important for researchers to carefully consider the limitations and potential biases when using this concept in their studies.

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