Classic example

Prove that the straight line between two points is the shortest

How to solve

Solve the extreme value of a function by analogy

use $x^2$ Solve it

$$Ignore\; high-order\; small\; quantities\; in\; the\; process\; of\; Taylor\; expansion$$

nature ： Variational and differential operations can exchange orders

         For a curve and its “ Disturbance ”, Take a position x, The points of the two curves on it are A,B, Right position x The points of the disturbance results are C,D.
$$\begin{gathered} CThe\; point\; coordinates\; are\; determined\; byAPoint\; calculation \\ DThe\; point\; coordinates\; can\; be\; determined\; byBPoint\; orCPoint\; calculation \\ fromDProof\; of\; identity\; ：\; Variational\; and\; differential\; operations\; can\; exchange\; orders \end{gathered}$$

Continue with the previous first-order variation $\delta V$ The assignment is 0 The calculation of

$The\; distribution\; integral\; method\; is\; used\; to\; obtain\; the\; integral\; relation\; formula\; of\; the\; original\; function\; and\; the\; derivative\; function\; ：$

Euler equation ( from $\delta yThe\; arbitrariness\; of$+ The integral is 0 =》 The internal integral must be 0, It is Euler equation )

• The right side of the first arrow is the train of thought of disproof
  

summary