I Designing an Invariant Lagrangian: Rules and Considerations

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A good Lagrangian should be a function of position and its first-order derivatives, as only two initial conditions are needed to predict a particle's future. The action must be a scalar to ensure consensus on the particle's trajectory, and high-order derivatives should be avoided to prevent non-locality. Generalized coordinates must be appropriately included, along with constraints that relate them. Additionally, the action must remain invariant under Galilei or Poincaré transformations, depending on whether Newtonian or relativistic mechanics is being considered. These principles are essential for designing an effective invariant Lagrangian.
accdd
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What are the rules for writing a good Lagrangian?
I know that it should be a function of the position and its first order derivatives, because we know that we only need 2 initial conditions (position and velocity) to uniquely determine the future of the particle.
I know that the action has to be a scalar because everyone has to agree on the trajectory the particle travels.
I know that high-order derivatives must be avoided to avoid non-locality.
What else?
 
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Be sure to include the appropriate generalized coordinates and list the constraints relating them.
 
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In addition to the very general properties you listed, you should make the action (or more precisely the first variation of the action) invariant under Galilei (Newtonian mechanics) or Poincare (special relativistic mechanics) transformations.
 

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