Conservation law for any potential field?

In summary, the conversation discusses the symmetry in the Lagrangian of a free particle moving in a time-dependent scalar potential. This symmetry, where the velocity only appears as its square, allows for rotations of the velocity without affecting the value of L. This results in conservation of angular momentum, which is a continuous symmetry even for an arbitrary potential. However, in order for this symmetry to exist, the potential must also have rotational symmetry, leading to conservation of angular momentum.
  • #1
maline
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Consider a free particle moving in a general time-dependent scalar potential. Energy & momentum are not conserved. However, there is a symmetry in the lagrangian: the velocity appears only as its square, so we can rotate it without affecting the value of L. What conservation law results from this symmetry?
 
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  • #3
DaleSpam said:
Conservation of angular momentum.
No, that's in the case of a central potential, where the symmetry is a rotation of both the position & velocity. I am asking about a completely arbitrary potential, and noting that rotation of the velocity alone should still be a symmetry.
 
  • #4
Hmm, that is a good point that I missed. You would still calculate the conserved quantity using Noether's theorem, but I don't know what it would be offhand.
 
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  • #5
One bump...
 
  • #6
Isn't this just time reversal symmetry?
 
  • #7
Jilang said:
Isn't this just time reversal symmetry?
No, I'm talking about rotating the velocity by a general angle in any direction. This should be a continuous symmetry.
 
  • #8
You cannot just rotate the velocity without rotating the coordinate system itself. The transformations covered by Noether's theorem are of the form ##t \to t +ks## and ##\vec x \to \vec X(t,s,\vec x)##, not transformations of the velocities.

You can do more general canonical transformations in Hamiltonian mechanics, but based on the symmetries of the Lagrangian this is not the case. In order to have a symmetry of the Lagrangian you therefore need to have rotational symmetry of the potential as well, resulting in conservation of angular momentum.
 
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1. What is conservation law for any potential field?

Conservation law for any potential field is a fundamental principle in physics that states that the total amount of a certain quantity remains constant over time, even as it may change form or move through space. This concept is often applied to fields, which are regions in space that have a certain value or potential at every point.

2. What is the significance of conservation law in physics?

Conservation laws are crucial in understanding the behavior of physical systems and predicting their future states. They provide a framework for understanding the fundamental principles of energy, momentum, and mass, and have been proven to hold true in a wide range of physical phenomena.

3. Are there different types of conservation laws for potential fields?

Yes, there are different types of conservation laws for potential fields, depending on the type of field and the type of quantity being conserved. For example, the conservation of energy is a common principle in many potential fields, while the conservation of electric charge is specific to electric fields.

4. How are conservation laws applied in real-world scenarios?

Conservation laws are used extensively in various branches of physics, such as mechanics, electromagnetism, and thermodynamics. They are also applied in fields like engineering, where they help in designing efficient systems and predicting their behavior.

5. Can conservation laws be violated?

In general, conservation laws are considered universal principles and are believed to hold true in all physical systems. However, there have been some rare cases where violations of these laws have been observed, usually at the quantum level. These cases are still being studied and are not fully understood.

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