What are symmetries in a Lagrangian?

In summary, the Lagrangian of a particle moving in a potential field can be expressed in terms of cylindrical coordinates, and angular momentum and momentum in the z direction are conserved quantities due to rotational and translation symmetries in the Lagrangian. These symmetries can be used to derive conserved quantities of motion through Noether's theorem.
  • #1
Pyroadept
89
0

Homework Statement


Consider the Lagrangian of a particle moving in a potential field L = m/2( [tex]\dot{x}[/tex]2 + [tex]\dot{y}[/tex]2 + [tex]\dot{z}[/tex]2) - U(r), r = sqrt(x^2 + y^2)



(a) Introduce the cylindrical coordinates and derive an expression for the Lagrangian in terms of the coordinates.
(b) Identify the cyclic coordinates, and find the corresponding conserved charges. What is their physical meaning? What symmetries do they correspond?

Homework Equations





The Attempt at a Solution


Hi everyone,
I can do it all fine apart from the last part, the question in bold. I found angular momentum and momentum in the z direction to be conserved. I just don't know what they mean by symmetries in the last question. Is the question implying these are symmetries caused by the conserved quantities? Can someone please help?
 
Physics news on Phys.org
  • #2
maybe showing what you have done will help

Noethers theorem shows that any symmetry in the lagrangian can be related to a "conserved current". In effect any symmetry can be used to derive a conserved quantity of the motion

Translation symmetries relate to conservation of momentum & rotational to conservation of angular momentum
 

1. What is a Lagrangian?

A Lagrangian is a function that describes the dynamics of a physical system. It is used in the field of mechanics to determine the equations of motion of a system.

2. What are symmetries in a Lagrangian?

Symmetries in a Lagrangian refer to the invariance of the Lagrangian under certain transformations. These transformations can include changes in time, space, or other physical parameters. They represent fundamental properties of a system that do not change even when the system undergoes a transformation.

3. Why are symmetries important in a Lagrangian?

Symmetries play a crucial role in the study of physical systems because they provide insights into the underlying structure and behavior of a system. They also allow for the use of powerful mathematical tools, such as Noether's theorem, which relates symmetries to conserved quantities.

4. How do symmetries affect the equations of motion in a Lagrangian?

Symmetries in a Lagrangian lead to conserved quantities, which in turn affect the equations of motion. For example, a symmetry in time leads to the conservation of energy, while a symmetry in space leads to the conservation of momentum.

5. Can symmetries be broken in a Lagrangian?

Yes, symmetries can be broken in a Lagrangian. This can happen due to external forces or constraints acting on a system, which may cause the system to deviate from its symmetrical behavior. Broken symmetries can also provide insights into the behavior of a system under different conditions.

Similar threads

I Types of symmetries in Lagrangian mechanics
    • Mechanics
Replies
3
Views
919
I Why isn't the Lagrangian invariant under ##\theta## rotations?
    • Classical Physics
Replies
5
Views
722
A Symmetry in terms of Lagrangian
    • High Energy, Nuclear, Particle Physics
Replies
2
Views
607
I Lagrangian symmetry for the bullet and gun
    • Mechanics
Replies
7
Views
766
Stationary points classification using definiteness of the Lagrangian
    • Calculus and Beyond Homework Help
Replies
4
Views
823
I Does mass conservation correspond to a pseudo-symmetry?
    • Classical Physics
Replies
6
Views
737
Center of mass of spherical shell inside of cone (Apostol Problem).
    • Calculus and Beyond Homework Help
Replies
3
Views
547
I Calculus Question within Lagrangian mechanics
    • Mechanics
Replies
2
Views
177
I Could the Lorentz symmetry be theoretically broken in vacuum?
    • Beyond the Standard Models
Replies
3
Views
2K
Determine the Lagrangian for the particle moving in this 3-D cos^2 well
    • Advanced Physics Homework Help
Replies
6
Views
964

Hot Threads

Recent Insights

Back
Top