Question

With regards to the principle of least action: I read that from path of least action, on first approximation, there is little difference if you change the path a little bit. What, in this case, does 'on first approximation' mean?

Answer 1

From a standard path (not the least action one), changing the path "a little bit" means the new value will be the old one + something (already small) proportional to the "little bit" you mention.

In case of the least action path, the new value will be the old one + something proportional to the "little bit" squared, which will be one order of magnitude smaller the the previous change.

Answer 2

So does 'on first approximation' not add anything to what you described?

Answer 3

Had I written the book, I would even have written:
"from path of least action, on first approximation, there is NO difference if you change the path a little bit."
because in that case, the difference is only of the second order.

Answer 4

So approximation= order? Thank you!

Answer 5

I would say so:
first approximation = first order or lowest order
you can choose the way to interpret it.