Is action-reaction contained in the Lagrangian

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The discussion centers on the relationship between the Lagrangian formulation of mechanics and the principle of action-reaction. While conservation laws can be derived from the Lagrangian, the connection to action-reaction is not immediately clear. The principle of stationary action does not explicitly assume that action must have a reaction, leading to confusion. In classical physics, certain assumptions about the Lagrangian's shape can demonstrate this relationship, but it does not universally apply. The example of a pendulum illustrates that total momentum is not always conserved in every scenario.
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A lot of conservation laws are derived from the lagrangian in my book. However, I fail to see why the Lagrangian incorporates action-reaction. Since it works for an arbitrary amount of particles and linear momentum can be show to be conserved from translational invariance it must do so.
But deriving it from the principle of stationary action, doesn't really have any assumption that action must have a reaction as far as I can see it.
What do I fail to see?
 
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With some assumptions about the shape of the Lagrangian and in classical physics, it can be shown similar to that.

In the general case, you do not have a conserved total momentum - just look at a pendulum as example.
 

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