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

by Cassidon
Tags: conservation, energy, fundamental, momentum
 P: 2 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.
 Mentor P: 15,403 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.
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 Quote by Cassidon 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.

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

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.
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 Quote by Nabeshin 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
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 Quote by Andrew Mason 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.
 P: 156 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.
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I should have said Kinetic energy in my OP as opposed to simply energy. Apologies.

 Quote by Vanadium 50 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.

 Quote by Nabeshin 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?

 Quote by Philip Wood 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.
 P: 156 you should focus more on the Hamiltonian of the system, i.e. the total energy
 P: 853 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.
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 Quote by Cassidon 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.
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 Quote by Cassidon 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)
 P: 156 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 spacial variations, i.e. the momentum.

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