WHat is more fundamental - conservation of momentum or conservation of energy?

In summary, the conversation discusses the concepts of momentum and energy conservation in closed systems. Momentum is often considered more fundamental as it is spatially invariant, while energy is time invariant. The conversation also explores the idea of time and spatial invariance and discusses an example of a system where energy is conserved but momentum is not. Ultimately, the conversation highlights the importance of considering both energy and momentum conservation in understanding and describing physical systems.
  • #1
Cassidon
2
0
Can someone explain this please.

From what I understand momentum is often the more fundamental as it is spatially invariant, whereas energy is time invariant and as more real world cases fall into the former category momentum is often more fundamental.

What is meant by spatial invariance? Is it that it is independent of position - like the acceleration of a mass in a uniform gravitational field in a polar vector space at a constant radius r.

Is time invariance then the same for time? Can someone provide an example of a system that is time invariant where energy is conserved and momentum is not and why?

Thanks in advance.
 
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  • #2
Before one can answer this question, one needs an empirical measurement of "fundamentalness". Otherwise we're just going to go around in circles as people argue according to taste.
 
  • #3
Cassidon said:
Can someone provide an example of a system that is time invariant where energy is conserved and momentum is not and why?

Sure, a ball on a hill. The ball begins with zero momentum, then gains it as it falls down the hill; momentum is clearly not conserved. Energy is however conserved.
 
  • #4
Energy conservation and momentum conservation both apply to closed systems. For the system made up of Earth and ball, both momentum and energy are conserved.

As to the original question: energy is the fourth (time-like) component of the momentum-energy 4-vector and it is this vector quantity which is conserved for a closed system. Therefore I wouldn't give primacy to either momentum conservation or energy conservation.
 
  • #5
Nabeshin said:
Sure, a ball on a hill. The ball begins with zero momentum, then gains it as it falls down the hill; momentum is clearly not conserved. Energy is however conserved.
What about the momentum of the Earth toward the ball?

AM
 
  • #6
Andrew Mason said:
What about the momentum of the Earth toward the ball?

AM

Obviously I have not included this in the system. It's a slightly artificial example, sure, since I have to define my system in a way which obviously leaves something important out. Nevertheless, it represents a situation described by the original poster.

As far as a fundamental interaction which conserves E but not p, I cannot think of one. As my example rightly illustrates, if you come up with such a scenario it is likely you have left something out of your system (perhaps you missed a neutrino, for example). The same is true if you get a time-dependent Hamiltonian.

Worth noting that these conservation laws are preserved only locally when we pass from special into general relativity.
 
  • #7
the wave nature of particles is a fundamental result that tells you there is a dispersion relation for any object in the universe. the dispersion relation equates the frequency and the wave number through some parameters, essentially then the energy and momentum are also equivalent, and classically both functions of velocity. conservation of one implies the conservation of the other.
 
  • #8
I should have said Kinetic energy in my OP as opposed to simply energy. Apologies.

Vanadium 50 said:
Before one can answer this question, one needs an empirical measurement of "fundamentalness". Otherwise we're just going to go around in circles as people argue according to taste.

Fundamental as in one superseeds the other.

i.e. Momentum is always conserved within a closed system but KE is not. Why is this in terms of space and time invariance.

Nabeshin said:
Sure, a ball on a hill. The ball begins with zero momentum, then gains it as it falls down the hill; momentum is clearly not conserved. Energy is however conserved.

How does that relate to time invariance if at all?

Philip Wood said:
Energy conservation and momentum conservation both apply to closed systems. For the system made up of Earth and ball, both momentum and energy are conserved.

As to the original question: energy is the fourth (time-like) component of the momentum-energy 4-vector and it is this vector quantity which is conserved for a closed system. Therefore I wouldn't give primacy to either momentum conservation or energy conservation.

sorry what is the energy momentum four vector. Energy is a scalar quantity?

What about momentum vs kinetic energy conservation.
 
  • #9
you should focus more on the Hamiltonian of the system, i.e. the total energy
 
  • #10
In Special Relativity time can usefully be regarded as a fourth component of a space-time 4-vector (vector with 4 components), the other three components being x, y and z. There are other, analogous, 4 vectors, the most important of which is the momentum-energy 4-vector. Thus instead of saying that in a closed system, the 3 momentum components are conserved and energy is conserved, we can just say that the momentum-enegy 4-vector is conserved.

You're entitled to argue that this is just an unnecessarily obscure way of saying that momentum and energy are conserved, but once you've bought into the space-time way of looking at things (see for example Taylor & Wheeler's classic introduction) it's hard not to see it as giving real insight.

To take up another point: there is no law of conservation of kinetic energy. It isn't usually a conserved quantity.
 
  • #11
Cassidon said:
How does that relate to time invariance if at all?

Well the point was that the Lagrangian for the system (consisting only of the ball) is time-invariant but not spatially invariant.
 
  • #12
Cassidon said:
Can someone explain this please.

From what I understand momentum is often the more fundamental as it is spatially invariant, whereas energy is time invariant and as more real world cases fall into the former category momentum is often more fundamental.

What is meant by spatial invariance? Is it that it is independent of position - like the acceleration of a mass in a uniform gravitational field in a polar vector space at a constant radius r.

Is time invariance then the same for time? Can someone provide an example of a system that is time invariant where energy is conserved and momentum is not and why?

Thanks in advance.

In an isolated system both are constant. If the system is not isolated both E and p can vary.

Therefore you are more or less asking what is more fundamental conservation or conservation?

However, if you are asking what is more fundamental energy or momentum? Well, momentum is more fundamental in the sense that momentum is one of the components of the basic phase space of any mechanical system, with energy and other mechanical properties being a function of momentum, e.g., E=E(p)
 
  • #13
There may be situations where the linear momentum of a particle is not conserved, more specifically in the case of a particle in an eletromagnetic field, where the generalised momentum is a sum of two terms, one being the linear momentum. In this case the generalised momentum is conserved. the total energy of a system, namely the Hamiltonian however, is always conserved. Energy is not created nor destroyed, but converted or transferred.
If you now imagine an inertial frame with no time dimension, energy still exists but momentum does not. Introduction of time means the energy will undergo some form of change, and this change is described through spatial variations, i.e. the momentum.
 

1. What does conservation of momentum mean?

Conservation of momentum is a fundamental principle in physics which states that the total momentum of a closed system remains constant over time, unless acted upon by external forces. In simpler terms, this means that the amount of motion in a system will stay the same unless an outside force changes it.

2. How is conservation of momentum related to Newton's laws of motion?

Conservation of momentum is an extension of Newton's third law of motion, which states that for every action, there is an equal and opposite reaction. In other words, the total momentum of a system remains constant because any change in momentum in one direction is matched by an equal and opposite change in momentum in the other direction.

3. What is the difference between conservation of momentum and conservation of energy?

Conservation of momentum and conservation of energy are both fundamental principles in physics, but they refer to different quantities. Conservation of momentum deals with the total amount of motion in a system, while conservation of energy deals with the total amount of energy in a system. Although they are separate principles, they are often interrelated and can both be used to describe the behavior of a system.

4. Which is more fundamental - conservation of momentum or conservation of energy?

This is a debated topic among scientists and there is no clear answer. Both conservation of momentum and conservation of energy are fundamental principles in physics and have been proven to hold true in countless experiments. Some argue that conservation of energy is more fundamental because it is a more general principle that applies to all forms of energy, while conservation of momentum only applies to motion. However, others argue that conservation of momentum is more fundamental because it is a more fundamental law of nature and can be derived from more basic principles. Ultimately, both principles are equally important and necessary to fully understand the behavior of physical systems.

5. Are there any exceptions to the principles of conservation of momentum and energy?

In general, conservation of momentum and conservation of energy are considered to be universal laws that hold true in all physical systems. However, there are some scenarios in which these principles may not seem to apply, such as in the case of quantum mechanics or extreme conditions like the Big Bang. In these cases, the concepts of conservation of energy and momentum may need to be modified to accurately describe the behavior of the system. Nevertheless, these principles remain fundamental and essential for understanding the behavior of the physical world.

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