Lagrangians and conserved quantities

Click For Summary
SUMMARY

The discussion centers on the relationship between Lagrangians and the conservation of angular momentum. It establishes that if a Lagrangian depends solely on time and position coordinates (and their derivatives) without explicit dependence on angular coordinates (theta or phi), angular momentum is conserved. The key conclusion is that conservation laws are linked to symmetries in spatial coordinates, affirming that any symmetry in the system correlates with a conservation law.

PREREQUISITES
  • Understanding of Lagrangian mechanics
  • Familiarity with angular momentum concepts
  • Knowledge of conservation laws in physics
  • Basic grasp of symmetries in physical systems
NEXT STEPS
  • Study the principles of Noether's theorem
  • Explore examples of Lagrangians with and without angular dependence
  • Investigate the implications of symmetries on conservation laws
  • Learn about the role of generalized coordinates in Lagrangian mechanics
USEFUL FOR

Students and professionals in physics, particularly those studying classical mechanics, Lagrangian dynamics, and conservation laws. This discussion is beneficial for anyone looking to deepen their understanding of the interplay between symmetries and conservation principles in physical systems.

quasar_4
Messages
273
Reaction score
0
Hi,

I have a relatively straight forward question. If we have a Lagrangian that only depends on time and the position coordinate (and its derivative), how can I decide whether angular momentum is conserved?

That is, if the Lagrangian specifically does not have theta or phi dependence, does that mean that angular momentum is always conserved?
 
Physics news on Phys.org
I think this is a really good question! I haven't thought of this before until now.

I imagine it would be a little weird for you to be interested in a situation with angular symmetry and not using \theta or \phi and their derivatives for your q and q dot things, but i guess its possible.

Here is my best stab, and I am pretty sure of the strength of this statement: Anytime there is a conservation law, it means there is a symmetry in anyone of the 4 spatial coordinates. Conservation laws are geometrically based, so look at your system, and if there is a symmetry in one of the coordinates, then there is conservation of something.


I hope this helps...
 

Similar threads

Undergrad Why isn't the Lagrangian invariant under ##\theta## rotations?
  • · Replies 5 ·
Replies
5
Views
2K
Undergrad Mathematical proof for the Lagrangian function
  • · Replies 4 ·
Replies
4
Views
2K
Graduate Identifying conserved quantities using Noether's theorem
  • · Replies 5 ·
Replies
5
Views
2K
Graduate Flaw in Lagrangian Mechanics for a rotary inverted pendulum system? (Simulation vs. Measured)
  • · Replies 11 ·
Replies
11
Views
2K
Undergrad Lagrangian for pendulum
  • · Replies 11 ·
Replies
11
Views
2K
Graduate The tautological 1-form: Lagrange vs. Hamilton formalism
  • · Replies 27 ·
Replies
27
Views
3K
Undergrad Is Information a conserved quantity or not?
  • · Replies 3 ·
Replies
3
Views
2K
Undergrad What does Noether's theorem actually say?
  • · Replies 7 ·
Replies
7
Views
3K
Undergrad Lacking intuition with partial derivatives
  • · Replies 18 ·
Replies
18
Views
2K
Graduate Cyclic coordinates in a two body central force problem
  • · Replies 30 ·
2
Replies
30
Views
3K