What Does Stationarity Mean in the Context of the Euler-Lagrange Equations?

[TheDoorsOfMe]

What does it mean when it says "the integral of the Lagrange equation is stationary for the path followed by the particle"?

 

[TheDoorsOfMe]

Is it just saying that the integral is a constant?

 

[MaceLee]

I would assume it means that the action $$s = \int Ldt$$ is a stationary point (i.e. a min most likely as the action is minimised in real systems).

You might want to wait for some confirmation however as I haven't studied Lagrangian mechanics in too much depth.

 

[Cyosis]

A stationary point is a point where the derivative of a function is 0. To obtain the Euler-Lagrange equations we set the variation of the action to 0.

 

[paranormal]

The Euler-Lagrange equations are a set of equations used in classical mechanics to describe the motion of a particle. They are derived from the principle of least action, which states that the path a particle takes between two points is the one that minimizes the action, a quantity that combines the particle's kinetic and potential energies.

When it is said that "the integral of the Lagrange equation is stationary for the path followed by the particle," it means that the path the particle takes is the one that results in the minimum value for the integral of the Lagrange equation. In other words, the particle follows a path that minimizes the action, and therefore, the path is considered "stationary" or "unchanging."

This concept is important because it allows us to determine the path a particle will take without having to know the specific forces acting on the particle. Instead, we can use the Euler-Lagrange equations to find the path that minimizes the action and describes the particle's motion. This method is powerful and widely used in physics and engineering to solve a variety of problems involving the motion of particles.