# Euler-Lagrange Equation Q: Where Does 2mr˙r˙θ Come From?

• I
• volican
In summary, the 2mr˙r˙θ term in the Euler-Lagrange equation for the variable-length pendulum comes from the time derivative of the partial derivative of the Lagrangian with respect to the angular velocity. This can be derived using the chain rule and is an important concept to understand when studying this topic further.
Hello Volican,

It comes from the time derivative of ##\partial {\mathcal L}\over \partial \dot\theta## $${\partial {\mathcal L}\over \partial \dot\theta }= mr^2\dot \theta$$

Thanks for writting back, much appreciated. I can see where that comes from and also the -mgrsinθ but can't see how they got the 2mr˙r˙θ. Any ideas? I am thinking that I can just completley ignore the dr/dt as it is not in the euler lagrange equation? Is this wrong?

$${d\over dt} \left ( {\partial {\mathcal L}\over \partial \dot\theta } \right ) = {d\over dt} mr^2\dot \theta = m {d\over dt} \left ( r^2\right ) \dot \theta + mr^2 {d\over dt} \dot \theta = 2mr\dot r\dot\theta + mr^2\ddot\theta$$

Aewsome! to differentiate r^2 did you use the chain rule?

Yes, the chain rule is awesome $${d\over dt} r^2 = {d\over dt} \left ( r \right ) r + r {d\over dt} \left ( r \right ) = 2 r\dot r$$

Make sure you speak this jargon fluently when pursuing further studies !

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## 1. What is the Euler-Lagrange equation?

The Euler-Lagrange equation is a mathematical formula that describes the motion of a system, such as a particle or a field, in terms of its energy and the forces acting upon it. It is commonly used in physics and engineering to analyze and solve problems involving motion and dynamics.

## 2. How is the Euler-Lagrange equation derived?

The Euler-Lagrange equation is derived from the principle of least action, which states that a physical system will follow the path that minimizes the action, a quantity that represents the integral of the system's Lagrangian over time. By varying the path of the system and setting the resulting change in action to zero, the Euler-Lagrange equation is obtained.

## 3. What does the term "r˙r˙θ" represent in the Euler-Lagrange equation?

The term "r˙r˙θ" represents the kinetic energy of a particle moving in a circular path. It is a combination of the particle's mass (m), its velocity in the radial direction (r˙), and its velocity in the tangential direction (r˙θ). This term is often used in problems involving rotational motion or circular orbits.

## 4. Where does the term "2mr˙r˙θ" come from in the Euler-Lagrange equation?

The term "2mr˙r˙θ" comes from the Lagrangian, which is the difference between the kinetic and potential energies of a system. In this case, the Lagrangian is equal to the kinetic energy, which is half the product of the particle's mass (m) and the square of its speed (r˙r˙θ). The factor of 2 is included to account for the fact that the particle is moving in two dimensions.

## 5. How is the Euler-Lagrange equation used in scientific research?

The Euler-Lagrange equation is used in a wide range of scientific fields, including physics, engineering, and mathematics. It is a powerful tool for analyzing and solving problems involving motion and dynamics, and is often used to study the behavior of complex systems. Its applications include classical mechanics, quantum mechanics, electromagnetism, and relativity.

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