Designing an Invariant Lagrangian: Rules and Considerations

  • Context: Undergrad 
  • Thread starter Thread starter accdd
  • Start date Start date
  • Tags Tags
    Lagrangian Writing
Click For Summary
SUMMARY

The discussion centers on the essential rules for designing an invariant Lagrangian in classical mechanics. Key requirements include the Lagrangian being a function of position and its first-order derivatives, ensuring the action is a scalar for trajectory agreement, and avoiding high-order derivatives to prevent non-locality. Additionally, the action must maintain invariance under Galilei transformations for Newtonian mechanics or Poincaré transformations for special relativity. Proper generalized coordinates and their constraints must also be incorporated into the formulation.

PREREQUISITES
  • Understanding of Lagrangian mechanics
  • Familiarity with Galilei and Poincaré transformations
  • Knowledge of generalized coordinates and constraints
  • Basic principles of classical mechanics
NEXT STEPS
  • Study the formulation of Lagrangians in classical mechanics
  • Explore the implications of invariance under Galilei transformations
  • Investigate the role of generalized coordinates in Lagrangian mechanics
  • Learn about the first variation of the action and its significance
USEFUL FOR

The discussion is beneficial for physicists, particularly those specializing in classical mechanics, theoretical physicists designing models, and students studying advanced mechanics concepts.

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?
 
Reply
  • Like
Likes   Reactions: vanhees71
Physics news on Phys.org
Be sure to include the appropriate generalized coordinates and list the constraints relating them.
 
Reply
  • Like
  • Informative
Likes   Reactions: 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.
 
Reply
  • Like
Likes   Reactions: accdd

Similar threads

Undergrad Why isn't the Lagrangian invariant under ##\theta## rotations?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Proving that the Lagrangian of a free particle is independent of q
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Taylor expansion about lagrangian in noether
  • · Replies 1 ·
Replies
1
Views
1K
Undergrad Classical Field Theory - Something isn't clicking
  • · Replies 12 ·
Replies
12
Views
2K
Undergrad Is there a way in Lagrangian mechanics to incorporate the inertial response from initial momentum when using initial conditions instead of BCs?
  • · Replies 9 ·
Replies
9
Views
1K
Graduate Show Lagrangian is invariant under a Lorentz transformation without using generators
  • · Replies 3 ·
Replies
3
Views
1K
Undergrad Types of symmetries in Lagrangian mechanics
  • · Replies 3 ·
Replies
3
Views
2K
Graduate Lagrangian Definition and Summary
  • · Replies 1 ·
Replies
1
Views
2K
Undergrad End point information in lagrangain variation principle
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad Lagrangian coordinate transformation (Noether)
  • · Replies 11 ·
Replies
11
Views
2K