# Variational Calculus: Euler-Lagrange vs. Lagrange's Equation of Motion

• babtridge
In summary, the Euler-Lagrange equation and Lagrange's equation of motion are essentially the same equations, but with different names. The Lagrangian density is the volumic density of the Lagrangian and is introduced when dealing with classical fields.
babtridge
Could someone please direct me to a good web page or comment on the main difference between the euler lagrange eqn and lagranges eqn of motion. I'm struggling to differentiate between the two...
Also, I'm struggling to grasp the concept of Lagrange density - when does one introduce this into the action integral?

Cheers

Last edited:
Well,technically,they are the same bunch of equations.The names are confusing.It would be fair,if the general ones,which don't have anything to do with physics,Lagrangians and lagrangian actions be called :"Euler's equations".

And the ones who do have everything to do with physics:"Lagrange's equations".

Daniel.

P.S.The Lagrangian density is the volumic density of lagrangian.Thay appear once people introduce the concept of (classical) field.

The Euler-Lagrange equation and Lagrange's equation of motion are both important tools in the field of variational calculus. While they may seem similar, there are some key differences between the two.

The main difference lies in the types of problems they are used to solve. The Euler-Lagrange equation is used for finding the extremum of a functional, which is a mathematical object that takes in a function and outputs a single value. This is commonly used in physics to find the path that a system will take between two points in space and time, given certain constraints.

On the other hand, Lagrange's equation of motion is used for solving problems in classical mechanics, where the goal is to find the equations of motion for a system of particles. This equation is derived from the principle of least action, which states that the true path of a system is the one that minimizes the action, a quantity that combines the kinetic and potential energy of the system.

In terms of the Lagrange density, this is introduced in the action integral when dealing with systems that have continuous degrees of freedom, such as fields. The Lagrange density is a function that describes the dynamics of the system, and is integrated over space and time in the action integral to find the equations of motion.

A good resource for understanding the differences between these equations and their applications is the textbook "Classical Mechanics" by John R. Taylor. It provides a comprehensive explanation of variational calculus and its applications in physics. Another helpful resource is the website "Variational Principles in Classical Mechanics" by Douglas Cline, which provides a detailed overview of the topic with examples and exercises.

I hope this helps clarify the main differences between the Euler-Lagrange equation and Lagrange's equation of motion, as well as the role of the Lagrange density in the action integral. Keep practicing and seeking out resources, and you will surely grasp these concepts in no time. Best of luck!

## 1. What is variational calculus?

Variational calculus is a branch of mathematics that deals with finding the optimal solution to a problem involving a functional. It is used to find the path or function that minimizes or maximizes a given functional.

## 2. What is the difference between Euler-Lagrange and Lagrange's equation of motion?

Euler-Lagrange and Lagrange's equation of motion are two different methods used in variational calculus to find the optimal solution to a problem. Euler-Lagrange is used to find the stationary points of a functional, while Lagrange's equation of motion is used to find the equations of motion that describe the behavior of a physical system.

## 3. When is Euler-Lagrange method used?

Euler-Lagrange method is used when the problem involves finding the optimal path or function that minimizes or maximizes a given functional. This method is commonly used in physics and engineering to find the path that a system will follow in order to minimize its energy or maximize its efficiency.

## 4. When is Lagrange's equation of motion used?

Lagrange's equation of motion is used when the problem involves finding the equations that describe the motion of a physical system. It is commonly used in classical mechanics to find the equations of motion for a system of particles or a rigid body.

## 5. Are there any real-world applications of variational calculus?

Yes, variational calculus has many real-world applications in fields such as physics, engineering, economics, and biology. It is used to solve problems involving optimization, such as finding the optimal path for a spacecraft to travel to a distant planet, or the optimal shape for a bridge to withstand certain forces. It is also used in the study of optimal control theory, which involves finding the best control strategy for a system to achieve a desired outcome.

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