Designing an Invariant Lagrangian: Rules and Considerations

In summary, the rules for writing a good Lagrangian include it being a function of position and its first order derivatives, the action being a scalar, and avoiding high-order derivatives to avoid non-locality. The action should also be invariant under Galilei or Poincare transformations, and it should include the appropriate generalized coordinates and constraints relating them.
  • #1
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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  • #3
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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1. What is a Lagrangian?

A Lagrangian is a mathematical function that describes the dynamics of a physical system. It is used in the field of physics to determine the equations of motion for a system based on its potential and kinetic energy.

2. Why is it important to write a good Lagrangian?

A good Lagrangian accurately represents the physical system and allows for the derivation of accurate equations of motion. It also simplifies the mathematical calculations and makes it easier to analyze the behavior of the system.

3. What are the key components of a good Lagrangian?

A good Lagrangian should include all relevant variables, such as position, velocity, and time, and accurately represent the potential and kinetic energy of the system. It should also be written in a way that is mathematically consistent and easy to manipulate.

4. How do I know if my Lagrangian is correct?

One way to check the correctness of a Lagrangian is to compare the derived equations of motion with known physical laws and experimental data. Additionally, double-checking the mathematical consistency of the Lagrangian can help ensure its accuracy.

5. Are there any tips for writing a good Lagrangian?

Some tips for writing a good Lagrangian include starting with the simplest possible form and gradually adding complexity, making sure all terms are dimensionally consistent, and checking for symmetries in the system that can simplify the Lagrangian. It is also helpful to have a good understanding of the physical system and its dynamics.

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