Is action-reaction contained in the Lagrangian

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SUMMARY

The discussion centers on the relationship between the Lagrangian formulation of mechanics and the principle of action-reaction. Participants highlight that while conservation laws, such as linear momentum, can be derived from the Lagrangian, the principle of stationary action does not inherently imply action-reaction. The conversation emphasizes that in classical physics, specific assumptions about the Lagrangian's structure are necessary to demonstrate these relationships, particularly in systems like a pendulum where total momentum is not conserved.

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  • Understanding of Lagrangian mechanics
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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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