Is the energy conserved FOR EACH BODY in a two-body central force problem?

In summary, the total energy of a two-body gravitational problem is conserved, but it doesn't make sense to separate the energy into individual bodies. The potential energy belongs to the system as a whole and cannot be attributed to one specific body. In central force motion, the problem is simplified to just one particle in a potential field, but in the two-particle system, the energy cannot be partitioned between the two bodies.
  • #1
Kamikaze_951
7
0
Hi everyone,

I would like to know if the energy of each body of a two body gravitationnal problem is separately conserved. I know that the individual angular momentum are separately conserved and that the TOTAL energy of the two bodies is conserved. However, I don't know if there could be energy transfer between the two bodies or if it's forbidden by symmetry considerations.

In my mechanics textbook, the two-body central force problem is only treated as a one-dimensional problem of the motion of a reduced mass in an effective potential, where only the energy of a reduced mass orbiting around the center of mass is considered. This confuses me when I try to think of the energy of ONE of the two bodies. Is the energy of ONE of the two bodies conserved? Is there a way to see this?

Thank you a lot for considering my request.

Kami
 
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  • #2
The angular momentum about any point of the two body system is conserved. For each body separately the angular momentum about the centre of mass is conserved, but not, I think, about other points.

Regarding energy, I don't think it makes sense to partition it into body A's and body B's. This is because the potential energy belongs to the system as a whole.
 
  • #3
Philip is correct. As an example, think of a system of two point particles which only interact via gravity. The only relevant energies here are kinetic and potential. If we pick some given reference frame, then each body has some kinetic energy (which could be zero) but the potential energy is just the gravitational potential energy GMm/r. Which body does this potential energy belong to? The energy only exists as an interaction between the two bodies.

Imagine we had one particle very massive (and thus roughly stationary), and the other very light and in a highly elliptical orbit--let's call this a Sun-comet system. When the comet is very close to the Sun, it's moving very fast and thus has high kinetic energy, but when the comet is far from the Sun, it moves much slower. The comet's kinetic energy is not conserved. But the potential energy has changed, and thus the system's total energy is conserved. But the potential energy does not necessarily "belong" to the comet, so it doesn't make sense to talk about the comet's total energy.

When we study central force motion, we do what you said: move to the center of mass frame and consider the motion of a single particle (with a mass equal to the reduced mass) inside a time-independent potential field. The problem has been changed: now there is only a field and a particle. So we can just say that all the energy belongs to the particle. But if we go back to the two-particle system, it's ambiguous which particle the energy belongs to.
 
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1. What is a two-body central force problem?

A two-body central force problem is a type of physics problem that involves analyzing the motion of two objects that are attracted to each other by a central force, such as gravity. This type of problem is commonly used in celestial mechanics and orbital dynamics.

2. Is the energy conserved for each body in a two-body central force problem?

Yes, according to the law of conservation of energy, the total energy of a closed system (such as the two bodies in a central force problem) remains constant over time. This means that the energy is conserved for each individual body, even as they interact with each other.

3. How is the total energy calculated in a two-body central force problem?

The total energy in a two-body central force problem is equal to the sum of the kinetic and potential energies of the two bodies. The kinetic energy is calculated using the masses and velocities of the bodies, while the potential energy is determined by the distance between the bodies and the strength of the central force acting between them.

4. Can the energy change in a two-body central force problem?

In a theoretical scenario where there are no external forces acting on the two bodies, the energy would remain constant. However, in real-world situations, there may be other forces present (such as friction) that can cause a change in the energy of the system.

5. What are some real-life examples of two-body central force problems?

One example of a two-body central force problem is the motion of a planet around a star, where the gravitational force between the two objects is the central force. Another example is the motion of the moon around the Earth, where the Earth's gravitational force is the central force. Additionally, the motion of satellites or spacecraft orbiting a planet or moon can also be modeled as a two-body central force problem.

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