What are symmetries in a Lagrangian?

[Pyroadept]

Homework Statement

Consider the Lagrangian of a particle moving in a potential field L = m/2( $$\dot{x}$$² + $$\dot{y}$$² + $$\dot{z}$$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?

 

[lanedance]

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