I Can configuration space be observer independent?

AI Thread Summary
The discussion centers on the formulation of configuration space in a way that is independent of observers and coordinates. It is noted that while spacetime can be described in an observer-independent manner, the configuration space often appears to depend on specific observers and coordinates. The example of a pendulum's configuration space being represented as S^1 raises questions about why configuration space would be observer dependent. The inquiry seeks a method to define configuration space that aligns with the observer-independent nature of spacetime, suggesting that a more universal definition could be crafted from the fundamental properties of the system. The conversation highlights the need for clarity on how to achieve this observer-independent formulation of configuration space.
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We can formulate the spacetime in an observer/coordinate independent way, i.e. a particle becomes a worldline in the 4d space. Then relative to each observer, the worldline can be casted to a function in R^3. However, I haven't found any reference on formulating configuration space in a coordinate independent way. It seems that the configuration space is formed in a way with a particular observer and some particular coordinates for the state of the system. Just want to ask if there's such a concept as observer independent configuration space?
 
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I don’t understand. Why would configuration space be observer dependent?
 
An example could clarify the question I guess.
Say, the configuration space of the pendulum is ##S^1##. Which observer does see anything else?
 
Dale said:
I don’t understand. Why would configuration space be observer dependent?
I just don't usually see coordinate independent formulation of the configuration space.
 
By definition a Lagrangian system is a pair (L,M), where M is a smooth manifold called the configuration space and ##L: TM\to \mathbb{R}## is a smooth function
ok?
 
wrobel said:
By definition a Lagrangian system is a pair (L,M), where M is a smooth manifold called the configuration space and ##L: TM\to \mathbb{R}## is a smooth function
ok?
Sorry, the question may sound a bit weird. I will try to give a more detailed explanation on what structure I have in mind before rasing the question.

So first, we have a 4d spacetime, there are n worldlines in the spacetime representing n particles forming a system with certain interation. There might be constraints at each spatial slice, such as the distance between two particles maintain the same.

These definitions are observer independent that fully describe the system, so if the configuration space is also observer independent, we should be able to derive the configuration space structure purely from the above. My question is how to craft such a definition?
 

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