Local Sampling vs Global Sampling

[FallenApple]

So from what I understand from my readings in Shankar's QM book, in the preface, that Newton's Laws give a particle some local sampling of the gravitational field around it and then decides where to move next based on the current step and that the sequential sum of all the path segments( determined from small delta t compared to current time at the current step) add up to the actual path traveled after all the steps are accounted for. Whereas in the Lagrangian view, we have the arbitrary future time point(which is completely assumed to exist a priori) and current time point, and can understand everything in one fell swoop by minimizing the action between the two time points. From a scientific/philosophical perspective, is the second view the correct one? I mean, it is sound to assume a global solution?

 

[AndyW]

I would say that both views have their own merits and are valid in their own ways. The Newtonian approach is based on the idea of local interactions and is more intuitive in understanding the behavior of particles in a gravitational field. On the other hand, the Lagrangian view is based on a more abstract concept of minimizing the action and provides a more elegant and comprehensive understanding of the system.

From a philosophical perspective, it is not about which view is "correct" but rather about which view is more useful in explaining and predicting the behavior of particles in a gravitational field. Both approaches have been extensively used in physics and have proven to be successful in different scenarios.

However, it is important to note that the Lagrangian view does not assume a global solution. It simply considers the behavior of the system between two specific time points and minimizes the action to find the most probable path. It is still a local approach, but with a more abstract and elegant mathematical formulation.

In conclusion, as a scientist, I would say that both views have their own value and are useful in understanding and predicting the behavior of particles in a gravitational field. It is not about which one is "correct" but rather about using the most appropriate approach for a particular scenario.