Lagrangians and conserved quantities

In summary, the conversation discusses whether angular momentum is conserved in a Lagrangian that only depends on time and the position coordinate. The question is whether the absence of theta or phi dependence in the Lagrangian means that angular momentum is always conserved. It is stated that conservation laws are based on symmetry in spatial coordinates, so if there is a symmetry in one of the coordinates, then there is conservation of something.
  • #1
quasar_4
290
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?
 
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  • #2
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 [tex]\theta[/tex] or [tex]\phi[/tex] 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...
 

1. What is a Lagrangian?

A Lagrangian is a mathematical function that describes the dynamics of a physical system. It takes into account the positions and velocities of all particles in the system and is used to derive the equations of motion.

2. What are conserved quantities?

Conserved quantities are physical quantities that remain constant over time in a system. This means that their values do not change even as the system evolves. Examples of conserved quantities include energy, momentum, and angular momentum.

3. How are Lagrangians and conserved quantities related?

Lagrangians are used to derive the equations of motion for a system, and these equations can reveal which quantities are conserved. If a Lagrangian has a certain symmetry, such as time or space invariance, then the corresponding conserved quantity can be identified.

4. What is the principle of least action?

The principle of least action states that a physical system will always follow a path that minimizes the action, which is the integral of the Lagrangian over time. This means that the system takes the path of least resistance and follows the path of least energy.

5. How are Lagrangians and conserved quantities used in physics?

Lagrangians and conserved quantities are fundamental concepts in physics and are used in various fields, such as classical mechanics, quantum mechanics, and field theory. They help us understand the behavior of physical systems and make predictions about their future states.

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