Haven't got space for conservation
Consider a system with N particles whose coordinates are r i = { r i x , r i y , r i z } , and whose velocities are r ˙ i . The energy of the system is described by the kinetic energy 2 1 m ∑ r ˙ 2 and an effective potential energy term V ( r 1 , … , r N ) .
All we know about V is that it depends on the positions only through their differences r 1 − r 2 , i.e. V ( { r i } , t ) = V ( r 1 − r 2 , r 1 − r 3 , … , t ) . As this system evolves in time, which of the following bulk quantities must be conserved?
Assumptions
- All interactions of the system with the outside world are described by V .
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4 solutions
Well, that answer is insufficient .. I would have thought that all 3 quantities must be conserved .. I am pretty sure those are fundamental physical laws. Can you please explain how energy and angular momentum FAIL to be conserved in this system?
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I would agree, this is a simple central force problem. All three would be conserved because we are considering the entire system.
Consider a potential like V ( r 1 , r 2 , t ) = ( r 1 − r 2 ) 2 + sin t . Momentum is conserved, but energy isn't since ∂ V / ∂ t = cos t .
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Conservation of energy is the first law of thermodynamics. In order for the potential to have a time-dependent component, the system would need to be moving through a field, like a box being dropped from the sky. Even then, total energy would be conserved because the acceleration of the system would be balanced by the change in potential energy of moving through the field. If you say the question was limited to the defined system, then a reference frame argument should be applied and the movement of the system through the field should be ignored. Donald Duck is correct. Not well done.
A question: Potentials, by definition require their total energy to be preserved. How can the total energy of the system not be preserved?
Moreover, in order for momentum to be preserved one must be in a preservatory field, or potential well, which require their total energy preserved.
What is more, you say that δV/δr1=-δV/δr1 which means that δV/δr1=0 => dp1/dt=0 => du1/dt=0=> the body in question remains at rest or is in fact in no potential well which would have exert a force and thus an generate an acceleration. So, ok, what you say sounds Greek to me. So, could you please explain it to me again ?
It should read partial V/partial r1 = - partial V/partial r2. Probably a typo ...
And yes, total energy should be conserved as well! Not well done, putting an exclusive multiple choice answer button there.
It has bugged me for a long time that since momentum is a function of "v" and energy is a function of "v sqr" it is impossible that both law be satisfied at the same time; until I finally admitted that momentum is about time and energy is about space. Basicly this is what Newton's 2nd is saying with the added benefit of a precise relation.
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You've got it backwards; momentum is about spacial homogeneity, and enegery is about temporal homogeneity. A handwaving way to see this is the there are three components of momentum and three spacial coordinates, and only one time and one energy.
Even the total momentum of the system can't be conserved as it is interacting with the outside world
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Yes, you are right, but I think the hint here is that the potential energy depends only upon the pairs of particles. The total momentum will conserve. If the system fully interacted with the outside world, the potential would have also depended on the positions of individual particles.
The confusion is caused by "total energy" for me that includes also thermal or other form of energy related to anelastic collisions. If the answer choice would have been: kinetic+potential or total momentum only the second would have been correct but since it was "total" energy or total momentum they are both true
Momentum is conserved in collisions. Energy is only consrved in collision if they are perfectly elastic
Energy is always conserved in a Newtonian system, even if some of it is sound and heat.
Momentum is not conserved during the collision.
Entropy must increase. Unless you view heat and sound as having momentum. Energy is conserved and will tend towards the it’s highest entropic form of heat. Mass/energy cannot be created or destroyed except by God.
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Very good answer, James. How to say that energy isn’t conserved?? Wtf.
Any system of particles would tend to reach a lower potential value, irrespective of the potential field. Here since the potential is proportional to the difference between the coordinates, the particles would tend to come together and clump into one solid mass. Thus all the particles would come together starting from rest, so the net momentum vector for each particle will cancel out. So they speed up until the collide and clump thus losing out all the kinetic energy in the form of heat or sound. Thus momentum is conserved but not energy.
Ameya Patil says . . . "So they speed up until the collide and clump thus losing out all the kinetic energy in the form of heat or sound. Thus momentum is conserved but not energy."
BUT . . . the energy "lost" in the form of heat and sound is STILL energy, so how possibly could there be ANY loss of energy?
I think BOTH energy AND momentum are conserved.
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^Thats what i thought too
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Time dependent potential generally does not conserve energy.
yes! how can energy not be conserved in a closed system? (since there aren't external forces)
Whether any physical quantity's measure is conserved depends on the system that we define ourselves. Hence in his explanation the particles are considered as the system and not the whole universe because of which we are able to conclude that energy in our system is not conserved.
Using Symmetry arguments: The whole system is invariant at translation, thus momentum is conserved. It is symmetric under rotation, thus angular momentum is conserved. Since the Interaction potential might be dependant on an absolute time t the whole system is not symmetric in time translation. Thus energy might not be conserved.
Noether's theorem in a nutshell.
For simplicity, consider the potential V ( r 1 , r 2 ) = V ( r 1 − r 2 ) for two particles in 1 dimension.
Newton's second law state that the time derivative of momentum is given by minus the space derivative of the potential energy. Thus, the rate of change of each particle's momentum is given by
d t d p 1 d t d p 2 = − ∂ r 1 ∂ V = − ∂ r 2 ∂ V
However, since V ( r 1 − r 2 ) is a function of only the difference between the two coordinates, we have
∂ r 1 ∂ V = − ∂ r 2 ∂ V
Thus, the rate of change of the total momentum is equal to zero, and momentum is conserved in the system.