Euler-Lagrange equation

Differential equations are applied a lot in a higher level of mathematical calculations and in many fields of physics like quantum mechanics, aerodynamics, etc. One such differential equation is known as the Euler-Lagrange equation. When we talk about this equation, people generally only remember the Lagrange equation because according to Hamilton’s principle, the Euler equation can be used to solve a solid mechanics problem to get the solution. In this equation, energy is used instead of forces in the physical system. The evolution of a physical system can also be described using Newton’s law but even then the Euler-Lagrange equation is preferred and this is what we will be discussing in this article.

What is the Euler-Lagrange Equation?

According to Euler-Lagrange’s equation, an extremal curve must satisfy the necessary condition of the equation, which is a differential equation. This equation is preferable when determining or solving a problem of solid mechanics because we won’t always be able to obtain forces. Newton’s law cannot be applied to some problems due to its reliance on forces, and in these cases, the Euler-Lagrange equation comes in handy since it utilizes the energy of a system rather than its forces. For this equation, only the inertial velocities need to be solved, but not the accelerations. In mathematics, this equation is recognized as one of the classic proofs when we talk about derivations. It states that J can be defined as the integral of the following form:

J = a∫b L (x, f (x), f’(x)) dx

Where,

Ý = dy/dt,

To further simplify this equation, we will replace the time-derivative notation Ý with the space-derivative notation yx. This equation can then be further expressed as:

მf/მy – d/dt (მf/მÝ) = 0

The partial derivative of f with respect to x, referred to as fx, has a value of 0, resulting in a highly simplified version of Euler-Lagrange’s equation known as Beltrami’s identity, which is written in the following manner:

F – yx მf/მyx = C

Furthermore, we can simplify and generalize the equation for three independent variables:

მf/მu – მ/მx მf/მux – მ/მy მf/მuy – მ/მz მf/მuz = 0

It is usually possible to solve problems in the calculus of variations if one chooses the appropriate Euler-Lagrange equation.

How to derive the Euler-Lagrange Equation?

We can derive the Euler-Lagrange differential equation in the following way:

δJ = δ ∫ L (q, q’, t) dt

    = ∫ (მL/მq δq + მL/მq’ δq’) dt

    = ∫ [მL/მq δq + მL/მq’ d (δq)/dt] dt

As long as we know that δ q’ = d (δq)/dt, then we can integrate the second term by parts using:

u = მL/მq’ dv

= d (δq)

     du  = d/dt (მL/მq’) dt     v = δq,

So now,

 ∫ მL/მq’ d (δq)/dt dt = ∫ მL/მq’ d (δq) = [მL/მq’ δq]t2t1 – t1∫t2 (d/dt მL/მq’ dt) δq

As a result of combining these equations, we get:

        δ J = [მL/მq’ δq]t2t1 + t1∫t2 (მL/მq – d/dt მL/მq’) δq dt

 The path is not altered, but only the endpoints, so δq (t1) = δq (t2) = 0. Hence the other equation becomes,

        δ J = t1∫t2 (მL/მq – d/dt მL/მq’) δq dt

The purpose of simplifying the equation so far is to get the stationary value, such that δJ = 0 and it should disappear for any small change and hence the equation can be further expressed as:

      მ L/მq – d/dt (მL/მq’) = 0

The above equation we just derived is known as the Euler-Lagrange equation.

What are the variations of the Euler-Lagrange Equation?

 It is also possible to write variation J in terms of a parameter k as

δJ = ∫ [f (x, y + kv, y’ + kv’) – f (x, y, y’)] dt

    = kI1 + ½ k2 I2 + ⅙ k3 I3 + 1/24 k4 I4 + …

Where,

v = δy

v = δy’

Thus, the first, second, third, and fourth variations are as follows:

I1 = ∫ (v fy + v’ fy’) dt

I2 = ∫ (v2 f y y + 2 v v’ f y y’ + v2 f y y’) dt

I3 = ∫ (v3 f y y y + 3 v2 v’ f y y y’ + 3 v v’2 f y y’ y’ + v3 f y’ y’ y’) dt

I4 = ∫ (v4 yyyyy + 4 v3 v’ yyyyy’ + 6 v2 v’2 f y y y’ y’ + v’4 f y’ y’ y’ y’) dt

Conclusion

Mathematicians and physicists use differential equations a great deal. It is an integral part of many derivations, and formulas are derived using derivations. A differential equation such as the Euler-Lagrange equation plays an imperative role in the calculus of variations and classical mechanics since differential functions are stationary at extremes. This makes the Euler-Lagrange equation very useful for solving optimization problems, in which if some function is given, one seeks to minimize or maximize it. Taking an in-depth look at what Euler-Lagrange is and how it’s derived, we explored the Euler-Lagrange equation in this article. In addition, we saw the expression of its variations.