Gravitational Lagrangian PE term

[Dale]

I was just doing a simple 2-body Newtonian gravitation problem. The force on each mass is:

$$f=\frac{G m_1 m_2}{r^2}$$

and the integral of the force wrt r is:

$$\int \! f \, dr = -\frac{G m_1 m_2}{r}$$

So, since there are two forces in the system, one on each object, I had assumed that there would be two potential energy terms so the total potential energy would be twice the above integral, or:

$$U = -2\frac{G m_1 m_2}{r}$$

but I checked my work using Newtonian mechanics it turns out that it gives the wrong equation of motion and the correct potential energy is only one times the integral.

So, my question is, can anyone explain how I should have known to only include one times the energy even though there were two objects, and how I can generalize to N-body problems.

 

[Valerie Park]

The key is to remember that the potential energy of a system is the sum of the potential energies of the individual objects. In the two-body case, each object exerts a force on the other, and the total potential energy of the system is equal to the potential energy of one object due to the other minus the potential energy of the other object due to the first. This means that even though there are two forces acting on the system, there is only one potential energy term, since they cancel each other out.For N-body problems, the same logic applies. The total potential energy of the system is equal to the sum of the potential energy of one object due to the other N-1 objects, minus the sum of the potential energy of the other N-1 objects due to the first. In this way, the total potential energy is always one times the integral.

 

[charng]

I can provide an explanation for the discrepancy you have observed in your calculations. The equation you have used for the potential energy (U) is derived from the work-energy theorem, which states that the work done by a force is equal to the change in kinetic energy (K) of an object. In the case of a two-body gravitational system, the work done by the force of gravity is equal to the change in the total kinetic energy of the system, which is the sum of the kinetic energies of both objects.

However, when we consider the potential energy of the system, we are looking at the energy stored in the system due to its configuration or arrangement. In this case, the potential energy is only dependent on the relative position of the two objects, not their individual kinetic energies. Therefore, we only need to consider the work done by the force on one object at a time, not both simultaneously. This is why the correct potential energy term for a two-body gravitational system is only one times the integral, not twice.

To generalize this to N-body problems, we can use the principle of superposition. This principle states that the total force on an object due to multiple forces acting on it is equal to the vector sum of the individual forces. Similarly, the total potential energy of an N-body system is equal to the sum of the potential energies of each individual pair of objects. Therefore, for an N-body gravitational system, the potential energy term would be the sum of N(N-1)/2 terms, each representing the potential energy of one pair of objects.

In summary, the discrepancy in your calculations is due to the difference between the work-energy theorem, which considers the total kinetic energy of the system, and the principle of superposition, which allows us to calculate the potential energy of an N-body system by considering the energy of each individual pair of objects.