Extreme condition
Let's review the condition that the function value on the function is at the extreme value:
When the derivative of the function value relative to the independent variable is equal to zero, that is, the independent variable changes slightly (increases or decreases), the function value still does not tend to change, and the function value is at the extreme value. At this time, the independent variable is the independent variable that produces the extreme value of the function.
Function is a function in English, and it is usually lowercase. Functional is functional in English and is often capitalized. Then the extreme condition is extended to the condition that the function value in the functional set is at the extreme value: when one of the functions is slightly disturbed (variational) to make it change slightly, the function value mapped by the function is at the extreme value, which is the function that makes the function value at the extreme value. Considering the condition of function derivative under the extreme value of function, when the function value is at the extreme value, its derivative of perturbation (variation) of function is also equal to zero. In other words, it is a function that will not change the value of the function even after a small disturbance is applied.
Integral expression of functional
The expression of function value is the integral expression of the starting point and end point of a function. The infinitesimal value in each functional integral value is determined by the source function, including the value of independent variable, the value of function (the value of dependent variable) and the derivative of function. The mapping relationship between function and function value is flexible, which depends not only on the current function value, but also on the independent variable value and derivative value of the function, so its expression is:
There are two function values that are often used as examples. A function value is the length of the function curve from the starting point to the end point. An example is to prove that the function with the shortest length is a straight line between two points. Another example is the time required for a ball to fall from the starting point to the end point along the function curve. The example is to prove that the function that takes the shortest time is cycloid (steepest descent line). In the first example, the functional differential value is equal to the tiny "arc length" element; In the second example, the functional differential value is equal to the tiny unit of "duration" (the unit of "arc length" divided by the instantaneous velocity obtained by converting potential energy into kinetic energy).
Item-by-item derivation of function expression under extreme conditions
Through the partial differential formula, the derivative of the function value to the function variable value can be obtained as follows:
Look at this part: (partial derivative of function value to function value) times (variation)+(partial derivative of function value to function value derivative) times (derivative of variation). One multiplier is variation, and the other multiplier is the derivative of variation. It is necessary to unify the multipliers by methods to facilitate further derivation.
The latter is equal to ((partial derivative of function value to function value) multiplied by variation) the difference between the two ends minus (derivative of function value to function value) multiplied by (variation).
Because the two endpoints are fixed and the variation of the two endpoints is zero, the condition that the derivative of the functional to the variation is zero becomes the following form.
Change is an infinitesimal amount that tends to zero, so the required relationship becomes
This is the expression of Euler-Lagrange equation. In other words, a function is required to have a functional at the extreme value.
Can the formula be understood simply?
I don't feel so good. At first, I tried many ways to understand this formula by simple metaphor or intuitive way, but I didn't think clearly. Intuitively, it seems unnecessary to prove the conclusion that the straight line between two points is the shortest. If you need proof to think clearly, it's not intuition. For example, try the shortest straight line between two points. Imagine that the starting point is that one end of the rope is fixed with a nail and the end point is a pulley with a fixed position. When the pulley rotates in one direction, the rope is "tightened", and a part of the length of the rope is drawn from the pulley at the end point between AB points, just like a tape measure. When rotating in the opposite direction, the rope "loosens" and the rope hidden in the pulley is pushed out. Then the functional is the length of the rope between two points, and the function is the shape of the rope. Suppose there is a spring in the pulley, just like a tape measure, which tends to reduce the exposed length of the rope. The change of function is like pulling a rope by hand to change its shape. The functional change is the length by which the rope retracts/extends from the pulley when the shape of the rope changes. The extreme value of the function curve is when the rope is straightened, because the rope is plucked by hand at that time, just like the strings, the change of the rope length in the pulley tends to be constant. But it is not easy to compare further, because this analogy has more assumptions: the rope is forced by the pulley spring. At present, I haven't thought of an intuitive way to understand Lagrange equation.
However, the physical angle is easier. In the most common conservative system, the inertial force of an object (mass multiplied by acceleration) minus the force generated by the change of potential energy difference is equal to zero.
From Mathematics to Physics
Why is the Lagrangian quantity in the differential equation of motion of conservative system dynamics equal to T-V?
I understand it this way: in a conservative system, at every moment, an object is either on the way to increase kinetic energy and decrease potential energy, or on the way to increase potential energy and decrease kinetic energy. So the differential value at this time is defined as the difference of dynamic potential energy. Because of the conversion relationship between kinetic energy and potential energy, the functional (action) from the initial moment to the final moment is the smallest.
Finally, the basic steps of the differential equation of motion can be obtained by Euler-Lagrange formula:
1. Get the expression of total kinetic energy+total potential energy of the system, and get the expression of Lagrange L=T-V;
2. The Lagrangian quantity is expanded by Euler-Lagrange equation (derivative of velocity, acceleration and position), and the differential equation of motion based on force, velocity, acceleration and position is obtained;
3. If we want to analyze the stability of the system, we can get an equation of multiplying the characteristic matrix of y'= Ay by the vector by transforming the differential equation. At this time, by solving Det(A), the eigenvalue λ of the characteristic matrix can be obtained (when λ; 0, the system tends to be unstable. When the imaginary part is contained in λ, the system oscillates under a stable general trend. A system will solve more than one eigenvalue, and each eigenvalue corresponds to a eigenvector. Stability will be analyzed by eigenvalues, and the stable/unstable trend direction will be obtained by eigenvectors. )
So much for the derivation and understanding of Euler-Lagrange equation.