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

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In the context of the Euler-Lagrange equations, stationarity refers to the condition where the action integral, defined as s = ∫ L dt, is at a stationary point along the path of a particle. This does not imply that the integral is constant, but rather that its variation is zero, indicating that the action is minimized or maximized for the actual path taken. A stationary point occurs when the derivative of the action with respect to the path is zero. The Euler-Lagrange equations are derived by applying this principle of stationarity to the action. Understanding this concept is crucial for analyzing the dynamics of systems in Lagrangian mechanics.
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What does it mean when it says "the integral of the Lagrange equation is stationary for the path followed by the particle"?
 
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Is it just saying that the integral is a constant?
 
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.
 
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.
 
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