Independence of Energy and Momentum Conservation

[nayanm]

Suppose we take the three Newton’s Laws as axioms.

1. Existence of inertial reference frames
2. F = ma
3. F(A on B) = -F(B on A)

Also suppose also we are considering purely classical mechanical processes on point particles (no heat transfer, etc.).

It is clear to me that the conservation of momentum directly follows from Newton's Laws. (If a force is the time rate of momentum transfer and forces come in pairs, then the momentum increase caused by one force is balanced by the momentum decrease caused by its reaction force.)

But here are my questions:

• If I take Newton's Laws as axioms, is the conservation of energy a derivable theorem or an independent axiom? In other words, does the conservation of energy directly follow from Newton's Laws just like the conservation of momentum does?
• Are equations written for the conservation of energy and conservation of momentum independent? (I.E. are they merely two representations of the same thing, or does one give additional information over the other?)

From what little I know about more advanced physics, I have a feeling that the two are independent (since energy and linear momentum conservations are consequences of spatial and time invariance). I was hoping someone could give me more insight on the matter.

Thanks!

 

[Abtinnn]

My knowledge is limited, but from what I think I know, the concept of work can be derived from Newton's laws. It can then be said that when a work is done on a system, the energy of the system changes.

Wapp= ΔE

The conservation of energy is very specific. In the case of an object falling on the surface of Earth, the work done on the object changes its kinetic energy:

W= -ΔU = ΔK

Where U is the potential energy (gravity is a conservative force) and K is the kinetic energy. Therefore

ΔE_system = ΔU + ΔK = 0

(mgh₂ - mgh₁) + (½mv₂² - ½mv₁²) = 0

∴ mgh₁ + ½mv₁² = mgh₂ + ½mv₂²

The conservation of energy for the falling object can then be derived from there.

The thing with momentum is that it is a much more "fundamental" concept than conservation of energy. Momentum can be applied in many broader situations. Conservation of energy can be applied to very specific ones. It's just that for a certain problem, you can solve it in an easier way by, say, using the concept of momentum, and another problem can be solved faster when you use conservation of energy.

 

[A.T.]

Are equations written for the conservation of energy and conservation of momentum independent?

Yes.

 

[nayanm]

Thanks for the replies!

The thing with momentum is that it is a much more "fundamental" concept than conservation of energy. Momentum can be applied in many broader situations.

Well, if both are a direct consequence of Newton's Laws, then both should be applicable at least whenever Newton's Laws are. I don't see how, in the consideration of only classical "slow-moving" point particles undergoing solely mechanical interactions, one can be more fundamental than the other.

Yes.

Then, assuming the invariance of both space and time, why can we derive one from the other? As in here: http://www.cs.utep.edu/vladik/2013/tr13-53.pdf

 

[Abtinnn]

Well, if both are a direct consequence of Newton's Laws, then both should be applicable at least whenever Newton's Laws are. I don't see how, in the consideration of only classical "slow-moving" point particles undergoing solely mechanical interactions, one can be more fundamental than the other.

Let's take the case of the inelastic collision of two balls.
Ball one is moving to the right and ball two is moving to the left. After the collision, ball one is moving to the left with a different velocity and ball two is moving to the right with yet another velocity.

Now it is wrong to say that the conservation of energy does not apply here, because the system is closed and no external work is being done on it (also energy cannot be created or destroyed). HOWEVER, in the collision, some of the energy transforms into other forms of energy (such as heat, sound and sometimes light). The difficulty with these types of energy is that they are hard to calculate. Sure it is possible, but it is most likely a long and inefficient thing to do. So energy is conserved and conservation of energy CAN be applied, but it just is not worth it when there is a better way. And that better way is the concept of conservation of momentum, which is kinda the broader and bigger picture.

 

[nayanm]

Let's take the case of the inelastic collision of two balls.
Ball one is moving to the right and ball two is moving to the left. After the collision, ball one is moving to the left with a different velocity and ball two is moving to the right with yet another velocity.

Now it is wrong to say that the conservation of energy does not apply here, because the system is closed and no external work is being done on it (also energy cannot be created or destroyed). HOWEVER, in the collision, some of the energy transforms into other forms of energy (such as heat, sound and sometimes light). The difficulty with these types of energy is that they are hard to calculate. Sure it is possible, but it is most likely a long and inefficient thing to do. So energy is conserved and conservation of energy CAN be applied, but it just is not worth it when there is a better way. And that better way is the concept of conservation of momentum, which is kinda the broader and bigger picture.

Everything you said is right, of course. Sure it may not be "worth it" to consider energy conservation in an elastic collision, but that is not a purely mechanical process (energy is dissipated as heat).

I'm really asking this as an exercise in mathematical physics than in application. You are pre-supposing the conservation of energy by saying energy cannot be created nor destroyed. True. Also, momentum cannot be created nor destroyed. Also true. But my question is: are these independent axioms or consequences of each other.

I have a feeling that the answer involves more advanced physics concepts such as Noether's Theorem and the symmetry of space and time, but I do not know enough to address this.

 

[anorlunda]

I have a feeling that the answer involves more advanced physics concepts such as Noether's Theorem and the symmetry of space and time, but I do not know enough to address this.

Your feelings are correct. If classical physics interests you, then you may find study of Noether's theorum very rewarding, and perhaps not as diffcult as you imagine. An excellent way to do that is through the series of video lectures by Professor Leonard Susskind, available on youtube and itunesU.

links to lecture 1 of 10.

 

[A.T.]

I have a feeling that the answer involves more advanced physics concepts such as Noether's Theorem and the symmetry of space and time, but I do not know enough to address this.

Yes you can derive both conservation laws from more general assumptions. But that doesn't mean that one conservation law implies the other.

 

[stevendaryl]

Newton's equations of motion don't imply conservation of energy by themselves. You can see that by considering dissipative forces (such as friction or viscous drag). Of course, in those cases, the loss of energy due to dissipative forces all goes into heat energy, but you can't prove that from Newton's laws alone.

 

[andresB]

Stevendaryl is right. You can have a mechanical system where the mechanical energy is not conserved. The same happen for the momentum (and angular momentum) and the electromagnetic interaction , the momentum will not be conserved if you only take into consideration the Newton equations for the charges.

 

[nayanm]

Your feelings are correct. If classical physics interests you, then you may find study of Noether's theorum very rewarding, and perhaps not as diffcult as you imagine. An excellent way to do that is through the series of video lectures by Professor Leonard Susskind, available on youtube and itunesU.

Thanks for the suggestion! I will definitely take a look at the videos when I get a little bit of time. Is there perhaps a "short answer" to my momentum and energy independence question, or does it maybe depend upon the system under consideration?

Yes you can derive both conservation laws from more general assumptions. But that doesn't mean that one conservation law implies the other.

Yes, but in the traditional derivation of the work-energy theorem, I don't see any "more general assumptions" other than mass is constant.

∫(F)dr = ∫(ma)dr = m∫(dv/dt)dr = m∫(dv/dr)(dr/dt)dr = m∫v dv = 1/2*m*Δ(v^2)

It is merely plugging in Newton's second law into the integral an rearranging using the Calculus Chain Rule and definitions from kinematics. How then can the result be independent?

Newton's equations of motion don't imply conservation of energy by themselves. You can see that by considering dissipative forces (such as friction or viscous drag). Of course, in those cases, the loss of energy due to dissipative forces all goes into heat energy, but you can't prove that from Newton's laws alone.

Stevendaryl is right. You can have a mechanical system where the mechanical energy is not conserved. The same happen for the momentum (and angular momentum) and the electromagnetic interaction , the momentum will not be conserved if you only take into consideration the Newton equations for the charges.

That's definitely a good point. But considering the derivation above, how can it simultaneously be true that Newton's Laws hold, yet the conservation of mechanical energy (which, considering the derivation above, is nothing but Newton's Second Law combined with kinematics manipulations) does not?

 

[A.T.]

Yes, but in the traditional derivation of the work-energy theorem, I don't see any "more general assumptions" other than mass is constant.

∫(F)dr = ∫(ma)dr = m∫(dv/dt)dr = m∫(dv/dr)(dr/dt)dr = m∫v dv = 1/2*m*Δ(v^2)

It is merely plugging in Newton's second law into the integral an rearranging using the Calculus Chain Rule and definitions from kinematics. How then can the result be independent?

That is not a derivation of general Energy Conservation.

 

[nayanm]

That is not a derivation of general Energy Conservation.

So then can it be said that mechanical energy conservation (when it applies) is a direct consequence of Newton's Laws, whereas total energy conservation is a more fundamental and independent postulate?

 

[stevendaryl]

So then can it be said that mechanical energy conservation (when it applies) is a direct consequence of Newton's Laws, whereas total energy conservation is a more fundamental and independent postulate?

I want you to think about this situation: You throw a ball straight up into the air. It reaches a maximum height, then drops again. Obviously, the kinetic energy at the top of the motion is zero. So kinetic energy is not conserved.

Of course, we say that the total energy is conserved, because we add the kinetic energy to the potential energy, and the potential energy is greatest at the top of the motion. But in order to have a "total energy" that is conserved, there must be a "potential energy". But only certain forces (conservative forces) have an associated potential energy. In particular, there is no potential energy associated with the friction force.

 

[stevendaryl]

Yes, but in the traditional derivation of the work-energy theorem, I don't see any "more general assumptions" other than mass is constant.

∫(F)dr = ∫(ma)dr = m∫(dv/dt)dr = m∫(dv/dr)(dr/dt)dr = m∫v dv = 1/2*m*Δ(v^2)

That's an argument that the change in kinetic energy is due to the work done by forces. That is pretty general. But it doesn't imply conservation of energy, except in the special case where F is a conservative force.

 

[mac_alleb]

Excuse me, could you explicitly derive momentum conservation equation (don't forget, momentum is vector value). Many thanks before.

 

[stevendaryl]

Excuse me, could you explicitly derive momentum conservation equation (don't forget, momentum is vector value). Many thanks before.

Well, if $\vec{P}$ is the total momentum, then, in Newtonian mechanics,

$\vec{P} = \sum_j m_j \vec{v}_j$ (where $j$ is the index over all particles)

So

$\frac{d\vec{P}}{dt} = \sum_j m_j \frac{d\vec{v}_j}{dt}$

By Newton's laws,
$m_j \frac{d\vec{v}_j}{dt} = \vec{F}_j = \sum_k \vec{F}_{jk}$

where $\vec{F}_{jk}$ is the force on particle $j$ due to particle $k$. So we have:

$\frac{d\vec{P}}{dt} = \sum_j \sum_k \vec{F}_{jk}$

At this point, we use that $\vec{F}_{jk} = - \vec{F}_{kj}$. The force on particle $j$ due to particle $k$ is equal and opposite to the force on particle $k$ due to particle $j$. So in the sum over all $j$ and $k$, we can add them in pairs $\vec{F}_{jk} + \vec{F}_{kj}$. Those pairs add up to zero. So the right-hand side is 0:

$\frac{d\vec{P}}{dt} = 0$

This proof only works for point-particles exerting instantaneous forces on each other, with no external forces (that is, the only forces are between particles). But the proof generalizes (Noether's theorem) to more complicated interactions involving fields and extended bodies.

 

[mac_alleb]

1) Strange tricking:
R = r1 + r2
dR / dt = r1/dt + r2/dt
V = v1 + v2
dV / dt = dv1/dt + dv2/dt
A = a1 + a2
Proceed?
2) How the j sum become jk sum?

 

[stevendaryl]

1) Strange tricking:
R = r1 + r2
dR / dt = r1/dt + r2/dt
V = v1 + v2
dV / dt = dv1/dt + dv2/dt
A = a1 + a2
Proceed?
2) How the j sum become jk sum?

Let's consider just two particles, to make things simple. Then let $\vec{F_{12}}$ be the force on particle #1 caused by particle #2. Let $\vec{F_{21}}$ be the force on particle #2 caused by particle #1. By Newton's 3rd law,

$\vec{F_{21}} = - \vec{F_{12}}$

So we have two equations:

$\vec{F_{12}} = \frac{d \vec{P_1}}{dt}$
$\vec{F_{21}} = \frac{d \vec{P_2}}{dt}$

where $\vec{P_1}$ is the momentum of the first particle, and $\vec{P_2}$ is the momentum of the second particle.

So if the total momentum is given by:

$\vec{P} = \vec{P_1} + \vec{P_2}$, then

$\frac{d \vec{P}}{dt} = \frac{d \vec{P_1}}{dt} + \frac{d \vec{P_2}}{dt}$
$= \vec{F_{12}} + \vec{F_{21}}$
$= 0$

That generalizes to any number of particles.

 

[mac_alleb]

Now you simply took from the very beginning that
P1 = F12 P2= F21
I.e. P1 = -P2
and then easily concluded
P1 + P2 = 0
Seems like it's NOT quite correct.

 

[Tosh5457]

Everything you said is right, of course. Sure it may not be "worth it" to consider energy conservation in an elastic collision, but that is not a purely mechanical process (energy is dissipated as heat).

I'm really asking this as an exercise in mathematical physics than in application. You are pre-supposing the conservation of energy by saying energy cannot be created nor destroyed. True. Also, momentum cannot be created nor destroyed. Also true. But my question is: are these independent axioms or consequences of each other.

I have a feeling that the answer involves more advanced physics concepts such as Noether's Theorem and the symmetry of space and time, but I do not know enough to address this.

Yes, in classical mechanics they're separate theorems. Momentum conservation has to do with space homogeneity and isotropy (which is true in the absence of external forces to the system), and energy conservation has to do with time homogeneity (true in the absence of non-conservative forces). It's not straightforward at all to understand why this implies conservation, only using Lagrangian Mechanics this becomes clearer, because it enables to connect space and time variables (or in general, other coordinates named generalized coordinates) to momentum and energy more clearly.

 

[vanhees71]

Momentum has to do with spatial homogeneity (symmetry under spatial translations). The conserved quantity according to isotropy (symmetry under rotations) is angular momentum.

 

[Khashishi]

Conservation of energy and momentum are two parts of the same law, as your link shows. That calculation uses Gallilean relativity, but the link between the two is even stronger in special relativity, where energy and momentum form a 4-vector, and Lorentz invariance implies that conservation of one is conservation of the other.

Treating them separately is almost like having a separate law for conservation of momentum in the x direction and conservation of momentum in the y direction. Of course, these are completely valid and correct laws, but it's obvious that the laws of physics don't change when we rotate the coordinate system, so it's unnecessary.

 

[vanhees71]

From the viewpoint of Noether's theorems, the two laws are separate: Any one-parameter Lie-subgroup of the symmetry group of the (variation of the action) definies a conserved quantity; any conserved quantity defines a one-parameter Lie-subgroup. Translation in a spatial direction defines momentum, and translation in time defines energy as these conserved quantities.

Of course, taken all space-time symmetries of non-relativistic or special relativistic space-time together, the 10 (classical) conserved quantities build the Lie algebra of the Galilei or Poincare group, respectively.

 

[Jano L.]

Thanks for the replies!
Well, if both are a direct consequence of Newton's Laws, then both should be applicable at least whenever Newton's Laws are. I don't see how, in the consideration of only classical "slow-moving" point particles undergoing solely mechanical interactions, one can be more fundamental than the other.
Then, assuming the invariance of both space and time, why can we derive one from the other? As in here: http://www.cs.utep.edu/vladik/2013/tr13-53.pdf

The authors of the paper use the assumption that

When we change to a moving frame, potential and thermal energy does not change

without explanation. Presumably by "thermal energy" they mean internal energy (no heat or temperature were discussed). They also say in the comment on page 1


From the microscopic viewpoint, thermal energy is nothing else but kinetic energy of molecules,
but from the macroscopic viewpoint, it is convenient to consider it separately.

From this it is clear the authors assume that translational kinetic energy changes when we change to a moving frame, but the internal energy does not. There is absolutely no reason why this should be intuitive to children who do not understand the law of conservation of momentum. It is not intuitive at all, because there is no way to even explain what internal energy is without law of conservation of momentum.

The proper way to explain these things is to present evidence for conservation of momentum, let kids experiment and only when they get accustomed to it and learn the necessary mathematics, use it to derive invariance of the internal energy (total energy - mechanical energy).

The law of conservation of momentum is logically independent of the law of conservation of energy. The former is part of Newton's laws, the latter is not.

Most importantly, laws are not derived from another laws, they are inferred from experience. If something was derived from something else already known, it is not a law, but a theorem.

 

[Khashishi]

Most importantly, laws are not derived from another laws, they are inferred from experience. If something was derived from something else already known, it is not a law, but a theorem.

Now that's just wrong. Laws are derived from laws all the time. There's plenty of redundancy in what we call laws of physics.

 

[Delta2]

So it all boils down to symmetry?. Symmetry of space for conservation of momentum and symmetry of time for conservation of energy?

Can we derive symmetry of time from symmetry of space or vice versa?

 

[nayanm]

Now that's just wrong. Laws are derived from laws all the time. There's plenty of redundancy in what we call laws of physics.

I think what Jano is trying to say is that, in a truly axiomatic system, if some claim about nature is derivable from some other claim about nature, they should not both be considered laws (i.e. axioms or postulates). However, in practice they often are because the choice of which is a law and which is a theorem is arbitrary. The choice also is often dependent upon which one was historically discovered first.

 

[nayanm]

Thank you all for the contributions above. I didn't bother to quote everyone, but I think the general conclusions I have drawn from this discussion are these:

• Assuming that we are dealing with an isolated system and that all interactions are two-body, Newton's Third Law and the Conservation of Linear Momentum are interchangeable axioms.
• Assuming no non-conservative (i.e. time-dependent) interactions, the Conservation of Energy is a direct consequence of Newton's Laws (or, equivalently, of the Conservation of Momentum). In other words, the Conservation of Mechanical Energy, which applies when exclusively conservative interactions occur in a system, IS just a theorem and not an independent axiom under these restricted conditions.
• If we want to generalize the above principles, we may instead take space and time invariance as two axioms. Then, the more general conservation of momentum and conservation of energy principles directly follow as theorems.

Please correct me if any of these don't seem quite right.

 

[Jano L.]

...Laws are derived from laws all the time. There's plenty of redundancy in what we call laws of physics.

You're right, I phrased that badly. I should have said "independent law".

Kepler's laws can be logically deduced from Newton's laws. But then they have redundancy as you say. They are not independent laws.

What I was trying to say is this:

If the law of conservation of momentum could be logically deduced from the law of conservation of energy, it would not be an independent law. Independent laws are not deduced with logic. They are induced from experience.

...in a truly axiomatic system, if some claim about nature is derivable from some other claim about nature, they should not both be considered laws (i.e. axioms or postulates). However, in practice they often are because the choice of which is a law and which is a theorem is arbitrary. The choice also is often dependent upon which one was historically discovered first.

Indeed. Physics is not an axiomatic system, the word law is used for broader set of ideas than "independent law" or "axiom" stand for.

 

[Apogee]

Momentum conservation is an immediate corollary of Newton's Laws. If we define a system such that no net external force acts on it, then:

By Newton's Second Law, force is equal to the product of mass and acceleration. Since mass is constant, integrating over force gives us the product of mass and the integral of acceleration, or velocity. This product is defined to be momentum and intuitively is the quantity of force through time required to bring an object from rest to a certain velocity (or equivalently from a certain velocity back to rest). Therefore:

The total momentum of the system remains conserved. As one can see, this is a direct result of Newton's Second Law.

Energy conservation is entirely different. In theory, it could technically still hold without Newton's Laws, or equivalently with different definitions of forces. Energy conservation is more of a mathematical construct grounded in line integrals and the notion of work.

First, we shall define energy as work through some force vector field. The scalar work W done on an object by a force along some directional vector is given by:

By letting r become small and summing all increments of work along a path, work can be defined along nonlinear paths by:

A conservative force is defined as any force for which this line integral is independent of the path taken. A good example of this is the electric force. The electric force can be defined by the charge of the object acted upon and the electric field magnitude, analogous to Newton's Second Law. The electric field is defined as the gradient of electric potential, literally potential energy per unit charge. Therefore:

Thus, the electric field is a conservative vector field:

Because the line integral is independent of path, the product of the charge and the line integral is no different, for the curl of the vector field in either case still equals 0.

The purpose of this example is to illustrate that energy conservation is more of a mathematical construct and acts independently of the definition of force itself, rather depends on the nature of the particular force in question. Momentum conservation on the other hand relies entirely on Newton's Second Law, the way that we've commonly defined forces. If we redefined force to be the product of the mass and jerk instead of mass and acceleration, assuming we kept the definition for momentum, momentum conservation would no longer hold, but energy would still be conserved in systems undergoing conservative forces alone. Energy conservation only does not hold when nonconservative forces are at play.