I Designing an Invariant Lagrangian: Rules and Considerations

AI Thread Summary
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
Messages
95
Reaction score
20
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?
 
Physics news on Phys.org
Be sure to include the appropriate generalized coordinates and list the constraints relating them.
 
  • Like
  • Informative
Likes vanhees71, Delta2 and accdd
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.
 
Consider an extremely long and perfectly calibrated scale. A car with a mass of 1000 kg is placed on it, and the scale registers this weight accurately. Now, suppose the car begins to move, reaching very high speeds. Neglecting air resistance and rolling friction, if the car attains, for example, a velocity of 500 km/h, will the scale still indicate a weight corresponding to 1000 kg, or will the measured value decrease as a result of the motion? In a second scenario, imagine a person with a...
Dear all, in an encounter of an infamous claim by Gerlich and Tscheuschner that the Greenhouse effect is inconsistent with the 2nd law of thermodynamics I came to a simple thought experiment which I wanted to share with you to check my understanding and brush up my knowledge. The thought experiment I tried to calculate through is as follows. I have a sphere (1) with radius ##r##, acting like a black body at a temperature of exactly ##T_1 = 500 K##. With Stefan-Boltzmann you can calculate...
Thread 'Griffith, Electrodynamics, 4th Edition, Example 4.8. (First part)'
I am reading the Griffith, Electrodynamics book, 4th edition, Example 4.8 and stuck at some statements. It's little bit confused. > Example 4.8. Suppose the entire region below the plane ##z=0## in Fig. 4.28 is filled with uniform linear dielectric material of susceptibility ##\chi_e##. Calculate the force on a point charge ##q## situated a distance ##d## above the origin. Solution : The surface bound charge on the ##xy## plane is of opposite sign to ##q##, so the force will be...
Back
Top