(I'm not commenting on the "All-at-once" angle, that is out of my league.)
You assert a contrast, with on one hand (traditional physics) tracking motion step by step, and on the other hand (Lagrangian) an approach that considers the overall path.
I will argue that in actual fact that contrast is far smaller than it appears to be.
In preparation I start with addressing the following: it is not the case that the true trajectory always coincides with a minimum of the action. There are also classes of cases such that the true trajectory coincides with a maximum of the action. Within the scope of Hamilton's stationary action there is an inversion: from classes of cases with minimum to classes of cases with maximum.
How can it be that within the single scope, here Hamilton's stationary action, both are viable?
The reason for that is: it is not about minimum nor maximum. The actual criterion is the property that the two have in common: as you sweep out variation: the point in variation space such that the derivative of the action is zero coincides with the true trajectory.
Next item in the preparation: the far reaching scope of differential equations.
When we solve a differential equation the solution that is obtained is a function. In that sense a differential equation is a higher level equation. A low level equation has a number as its solution. But a differential equation has an entire function as its solution. A differential equation states: this relation must be satisfied concurrently for all values of the domain. That is to say: when you solve a differential equation the solution that you obtain is for the entire path.
Now to the main point:
Calculus of variations has a particular mathematical property, I will use the catenary problem to showcase that property. The catenary problem: what is the shape of a chain that is suspended between two points? We consider the most general case: for any height difference between the two points of suspension. We have that the resting state is a state of minimal potential energy. That is to say: for the shape of the catenary the derivative of the potential energy wrt variation of the shape is zero.
Now divide the solution in subsections. Every subsection is an instance of the catenary problem. We can solve each of the subsections, and then concatenate those subsections. We can continue the subdividing; you can still concatenate the subsolutions. There is no lower limit to the size of the subsections; the reasoning remains valid down to infinitesimally short subsections.
Given that infinitesimal property: it follows that it should be possible to solve the catenary problem with a differential equation. I have on my website a demonstration of how to set up and solve the differential equation for the catenary problem. It's in an article titled: 'Calculus of Variations as applied in physics'.
http://cleonis.nl/physics/phys256/calculus_variations.php
More generally, this infinitesimal property explains why the Euler-Lagrange equation is a differential equation.
Action concepts are stated in the form of an integral, but here's the thing: the variational property obtains at the infinitesimal level, and from there it propagates to the level of the integral.
There is a note about the Euler-Lagrange equation (author: Preetum Nakkiran), in which the Euler-Lagrange equation is derived using differential reasoning only. That is: stating the integral is skipped altogether. That demonstrates that stating the integral is not necessary for deriving the Euler-Lagrange equation.
https://preetum.nakkiran.org/lagrange.html
At the start of this comment I announced: the suggested contrast between traditional approach (force-acceleration) and Lagrangian approach is only an apparent contrast. On closer examination we see the two formalisms are in fact very closely connected.
I have a comment about Lagrangian models.
(I'm not commenting on the "All-at-once" angle, that is out of my league.)
You assert a contrast, with on one hand (traditional physics) tracking motion step by step, and on the other hand (Lagrangian) an approach that considers the overall path.
I will argue that in actual fact that contrast is far smaller than it appears to be.
In preparation I start with addressing the following: it is not the case that the true trajectory always coincides with a minimum of the action. There are also classes of cases such that the true trajectory coincides with a maximum of the action. Within the scope of Hamilton's stationary action there is an inversion: from classes of cases with minimum to classes of cases with maximum.
How can it be that within the single scope, here Hamilton's stationary action, both are viable?
The reason for that is: it is not about minimum nor maximum. The actual criterion is the property that the two have in common: as you sweep out variation: the point in variation space such that the derivative of the action is zero coincides with the true trajectory.
Next item in the preparation: the far reaching scope of differential equations.
When we solve a differential equation the solution that is obtained is a function. In that sense a differential equation is a higher level equation. A low level equation has a number as its solution. But a differential equation has an entire function as its solution. A differential equation states: this relation must be satisfied concurrently for all values of the domain. That is to say: when you solve a differential equation the solution that you obtain is for the entire path.
Now to the main point: Calculus of variations has a particular mathematical property, I will use the catenary problem to showcase that property. The catenary problem: what is the shape of a chain that is suspended between two points? We consider the most general case: for any height difference between the two points of suspension. We have that the resting state is a state of minimal potential energy. That is to say: for the shape of the catenary the derivative of the potential energy wrt variation of the shape is zero.
Now divide the solution in subsections. Every subsection is an instance of the catenary problem. We can solve each of the subsections, and then concatenate those subsections. We can continue the subdividing; you can still concatenate the subsolutions. There is no lower limit to the size of the subsections; the reasoning remains valid down to infinitesimally short subsections.
Given that infinitesimal property: it follows that it should be possible to solve the catenary problem with a differential equation. I have on my website a demonstration of how to set up and solve the differential equation for the catenary problem. It's in an article titled: 'Calculus of Variations as applied in physics'. http://cleonis.nl/physics/phys256/calculus_variations.php
More generally, this infinitesimal property explains why the Euler-Lagrange equation is a differential equation.
Action concepts are stated in the form of an integral, but here's the thing: the variational property obtains at the infinitesimal level, and from there it propagates to the level of the integral.
There is a note about the Euler-Lagrange equation (author: Preetum Nakkiran), in which the Euler-Lagrange equation is derived using differential reasoning only. That is: stating the integral is skipped altogether. That demonstrates that stating the integral is not necessary for deriving the Euler-Lagrange equation. https://preetum.nakkiran.org/lagrange.html
At the start of this comment I announced: the suggested contrast between traditional approach (force-acceleration) and Lagrangian approach is only an apparent contrast. On closer examination we see the two formalisms are in fact very closely connected.