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To find the node arrangement with the lowest possible potential energy: interate to converge onto the extremum.
If there would be 4 nodes:
To the left of node 1 is the suspension point, to the right of node 1 is node 2.

To find the node arrangement with the lowest possible potential energy: iterate to converge onto the extremum.
If there would be 4 nodes:
To the left of node 1 is the suspension point, to the right of node 1 is node 2.

What it takes to have solid expectation that a single extremum exists

Some time ago I implemented what in effect is numerical analysis of the catenary problem. (The implementation is an interactive diagram, part of an article about calculus of variation that is available on my website.)

I divided the catenary in straight subsections, the subsections connect nodes. In the interactive diagram the nodes can be moved vertically. Moving a node changes the length of the total catenary. Counterweights exert a constant force.

When all the nodes are moved down the potential energy of the counterweights is the dominant term; pulling the nodes further down increases the total potential energy.

When all the nodes are up high the weight of the catenary has large mechanical advantage, so moving the nodes down lowers the total potential energy.

To find the node arrangement with the lowest possible potential energy: interate to converge onto the extremum.
If there would be 4 nodes:
To the left of node 1 is the suspension point, to the right of node 1 is node 2.

Move node 1, finding the extremum, relative to the current position of node 2.
Proceed to adjust node 2: find the extremum, relative to the current positions of node 1 and node 3.
Rinse and repeat.

Keep cycling the sequence of nodes.

This iterative process converges onto a global extremum. The fact that the solution space is subdividable ensures that there is a single, unique solution.

The larger the number of nodes, the better the approximation onto the analytical solution.

Porting the numerical analysis reasoning to the Brachistochrone problem

The crudest implementation would be a single node in between the two fixed end points. That would give a profile consisting of two straight sections, joined at the node. Obtain the gained velocity from the height difference. Changing the height of the node changes the length of each section. Under those circumstances the existence of an extremum is garanteed.

With a large number of nodes:
Then each triplet of nodes is converged to its extremum, iterating over all the noddes. In the converged end state: each triplet is at its extremum, therefore the convergeed state is a global extremum.

We have that the Euler-Lagrange equation is a differential equation. The explicit expression states a local condition, but of course the demand is that it is satisfied for the entire curve all at once, constituting a global demand. Since the explicit statement states a local condition it should be possible to derive the Euler-Lagrange equation using differential reasoning only.

We have that the Euler-Lagrange equation is a differential equation. The explicit expression states a local condition, but of course the demand is that it is satisfied for the entire curve all at once, constituting a global demand. Since the explicit statement states a local condition it should be possible to derive the Euler-Lagrange equation using differential reasoning only.