Understanding Conservation and Symmetry in Physics

In summary: Euler-Lagrange or Hamiltonian?The symmetry in question is translational or rotational, not spatial. So the theorem still holds for systems that cannot be modeled with a Lagrangian. However, because angular momentum is conserved, it can still be used to describe such systems.
  • #1
Delta2
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We say that conservation of linear momentum follows from the translational symmetry while conservation of angular momentum from directional (rotational) symmetry. Can anyone explain what exactly do we mean by these kind of symmetries and how they imply conservation of certain quantities?
 
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  • #2
Delta² said:
We say that conservation of linear momentum follows from the translational symmetry while conservation of angular momentum from directional (rotational) symmetry. Can anyone explain what exactly do we mean by these kind of symmetries and how they imply conservation of certain quantities?

As the universe gets bigger, it increases in entropy and asymmetry. Linear momentum appears to always be conserved, always traveling from one place to another. Angular momentum isn't really conserved like linear momentum is because you can completely get rid of angular momentum with some external torque.
 
  • #3
Delta² said:
We say that conservation of linear momentum follows from the translational symmetry while conservation of angular momentum from directional (rotational) symmetry. Can anyone explain what exactly do we mean by these kind of symmetries and how they imply conservation of certain quantities?

Let me first make the statements more precise. It goes back to a theorem about the mathematics of theories of motion. That is, the theorem says something about the math of the theory, not necessarily about the physics.

There is a correspondence: when there is a symmetry in the theory then there is a correspondong conserved entity in that theory. For instance, given the definition of kinetic energy there is a correspondence between conservation of kinetic energy and symmetry with respect to time translation.

So if you're curious whether some attempt at formulating a new theory will lead to a theory that implies conservation of energy it suffices to figure out whether the theory is symmetrical with respect to time translation.

Our theories of motion have in common that for any system going through its motions (for example the solar system) the orientation in space is not a factor; there is no dependence on orientation; there is symmetry under shifts of orientation. According to the theorem there must be a corresponding conserved entity, and as we know that's angular momentum.

In any derivation of conservation of angular momentum from the laws of motion the independence on orientation in space is part of it. For example, there is http://www.cleonis.nl/physics/phys256/angular_momentum.php" [Broken]. Among the elements used there is the fact that the same reasoning applies for all orientations in space.
 
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  • #4
zeromodz said:
Angular momentum isn't really conserved like linear momentum is because you can completely get rid of angular momentum with some external torque.
Why do you allow external torques but not external forces?
 
  • #5
Doc Al said:
Why do you allow external torques but not external forces?

I say torque because the force has to be perpendicular for angular momentum to be lost. If I were to apply a force directly parallel to the Earth relative to the sun, the angular momentum is still conserved because sin90 = 1. Its very important to understand that it must be external torque which is:

t = FRsin = Iα
 
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  • #6
Cleonis said:
Let me first make the statements more precise. It goes back to a theorem about the mathematics of theories of motion. That is, the theorem says something about the math of the theory, not necessarily about the physics.

There is a correspondence: when there is a symmetry in the theory then there is a correspondong conserved entity in that theory. For instance, given the definition of kinetic energy there is a correspondence between conservation of kinetic energy and symmetry with respect to time translation.

So if you're curious whether some attempt at formulating a new theory will lead to a theory that implies conservation of energy it suffices to figure out whether the theory is symmetrical with respect to time translation.

Our theories of motion have in common that for any system going through its motions (for example the solar system) the orientation in space is not a factor; there is no dependence on orientation; there is symmetry under shifts of orientation. According to the theorem there must be a corresponding conserved entity, and as we know that's angular momentum.

In any derivation of conservation of angular momentum from the laws of motion the independence on orientation in space is part of it. For example, there is http://www.cleonis.nl/physics/phys256/angular_momentum.php" [Broken]. Among the elements used there is the fact that the same reasoning applies for all orientations in space.
In Lagrangian mechanics why the symmetry is always mentioned as invariance of Lagrangian with respect to time or space and why we always take small perturbations of the time and space variables and consider Langrangian to be invariant for these small changes? (what about big changes?)

From Wikipedia Noether's theorem it states that the theorem doesn't hold for systems that can not be modeled with a Lagrangian. Isnt it possible that these systems can be modeled with some other way which still has some symmetry with time or space?
 
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  • #7
zeromodz said:
Doc Al said:
zeromodz said:
Angular momentum isn't really conserved like linear momentum is because you can completely get rid of angular momentum with some external torque.

Why do you allow external torques but not external forces?

I say torque because the force has to be perpendicular for angular momentum to be lost. If I were to apply a force directly parallel to the Earth relative to the sun, the angular momentum is still conserved because sin90 = 1. Its very important to understand that it must be external torque which is:

t = FRsin = Iα
I think Doc Al's point was if you say "you can completely get rid of angular momentum with some external torque", why don't you also say "you can completely get rid of linear momentum with some external force"?
 
  • #8
DrGreg said:
I think Doc Al's point was if you say "you can completely get rid of angular momentum with some external torque", why don't you also say "you can completely get rid of linear momentum with some external force"?
Exactly. (Thanks, DrGreg. :wink:)
 

What is conservation?

Conservation is the practice of protecting and preserving natural resources and the environment for future generations. It involves managing and using resources in a sustainable way to prevent depletion or destruction.

What is symmetry?

Symmetry is a concept in mathematics and science that refers to balance and proportion in an object or system. It can also refer to the exact correspondence of form and configuration on opposite sides of a dividing line or plane.

How are conservation and symmetry related?

Conservation and symmetry are related in that both involve maintaining balance and order. In conservation, this means protecting the natural balance of ecosystems and the environment. In symmetry, this means maintaining balance and proportion in objects and systems.

What are some examples of symmetry in nature?

There are many examples of symmetry in nature, such as the bilateral symmetry of animals, where the left and right sides of the body are mirror images of each other. Other examples include the radial symmetry of starfish and the spiral symmetry of seashells.

How can we promote conservation and symmetry?

We can promote conservation and symmetry by being mindful of our actions and their impact on the environment. This can include reducing our carbon footprint, using sustainable resources, and supporting conservation efforts. In terms of symmetry, we can appreciate and celebrate the beauty of symmetry in nature and incorporate it into our designs and structures.

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