Gravitational Lagrangian PE term

In summary, the potential energy of a system with N objects is equal to one times the integral of the force over r, as each object's potential energy is affected by the other objects in the system.
  • #1
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I was just doing a simple 2-body Newtonian gravitation problem. The force on each mass is:

[tex]f=\frac{G m_1 m_2}{r^2}[/tex]

and the integral of the force wrt r is:

[tex]\int \! f \, dr = -\frac{G m_1 m_2}{r}[/tex]

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:

[tex]U = -2\frac{G m_1 m_2}{r}[/tex]

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.
 
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  • #2
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.
 
  • #3


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.
 

Related to Gravitational Lagrangian PE term

1. What is the Gravitational Lagrangian PE term?

The Gravitational Lagrangian PE term is a mathematical concept used in physics to describe the potential energy associated with the gravitational field of a system of objects. It is a term in the Lagrangian, which is a mathematical function used to describe the dynamics of a system.

2. How is the Gravitational Lagrangian PE term calculated?

The Gravitational Lagrangian PE term is calculated using the formula U = -GmM/r, where G is the gravitational constant, m and M are the masses of the two objects, and r is the distance between them. This formula is based on Newton's Law of Universal Gravitation.

3. What is the significance of the Gravitational Lagrangian PE term?

The Gravitational Lagrangian PE term is significant because it allows us to mathematically describe the potential energy of a system due to the gravitational force. This is important in understanding the dynamics of celestial bodies and other objects in space.

4. How does the Gravitational Lagrangian PE term relate to other forms of energy?

The Gravitational Lagrangian PE term is a type of potential energy, meaning it is energy that is stored in a system and has the potential to be converted into other forms of energy. This potential energy can be converted into kinetic energy, for example, as objects move closer together due to the force of gravity.

5. Can the Gravitational Lagrangian PE term be negative?

Yes, the Gravitational Lagrangian PE term can be negative. This occurs when the two objects in the system are moving away from each other, resulting in a decrease in potential energy. However, in most cases, the term is positive as objects tend to move towards each other due to the attractive force of gravity.

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