# Dependence of force on relative position and velocity

• Felipe Lincoln
In summary, the force between two bodies can only depend on their relative position and relative velocity. This statement is based on the translational and rotational symmetry of space, which means that the potential (V) of one body due to the other only depends on their relative position (r). The force (F) is then the gradient of V, represented as -∇V. This principle applies to various types of forces, including gravitational, electrostatic, magnetostatic, electromagnetic, and nuclear forces. To learn more about these symmetries, you can study the continuous symmetries of spacetime, which can be described by Lie groups.

#### Felipe Lincoln

Gold Member
I read that a force between two bodies can only depend on their relative position and relative velocity. But I can't understand in what is this statement leaning on and what it means.

It means that it can't be dependent on their position of velocity relative to anything else.
In other words, if we moved planet Earth to another galaxy, but you were in the same position relative to the Earth, then you would still have the same weight.

• Felipe Lincoln
I think this statement is leaning (at least partially) on the translational and rotational symmetry of space. Due to those symmetries the potential V of one body due to the presence of the other body, depends only on their relative position so it is ##V(r)## where ##r=|r1-r2|##. The force is the gradient of V, ##\vec{F}=-\nabla V##.

What it means is that whether we consider gravitational forces or electrostatic forces (between electrically charged bodies), or magnetostatic forces (between magnetized bodies), or electromagnetic forces (for example the Lorenz force and the Laplace force), or nuclear forces (for example between quarks and gluons), all of these types of forces have something in common, that they can only depend on the relative position of the bodies and their relative velocities.

• Ibix and Felipe Lincoln

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• Felipe Lincoln

## 1. What is the concept of "Dependence of force on relative position and velocity"?

The dependence of force on relative position and velocity is a fundamental principle in classical mechanics that explains how the motion of an object is affected by the position and velocity of another object it is interacting with. This concept is often referred to as the law of action and reaction, or simply Newton's third law.

## 2. How does relative position and velocity affect the force between two objects?

The force between two objects is directly proportional to their relative position and velocity. This means that the closer two objects are and the faster they are moving relative to each other, the stronger the force between them will be. Conversely, if the two objects are far apart or have small relative velocities, the force between them will be weaker.

## 3. Can the dependence of force on relative position and velocity be applied to all types of forces?

Yes, the dependence of force on relative position and velocity is a universal principle that applies to all types of forces, including gravitational, electromagnetic, and nuclear forces. This is because all forces can be described by mathematical equations that take into account the relative positions and velocities of the interacting objects.

## 4. How is the concept of "Dependence of force on relative position and velocity" used in real-world applications?

The dependence of force on relative position and velocity is used in a wide range of real-world applications, such as designing spacecraft trajectories, predicting the motion of celestial bodies, and understanding the behavior of atoms and molecules. It is also essential in engineering and designing structures that can withstand external forces.

## 5. Is the dependence of force on relative position and velocity affected by the mass of the objects?

No, the mass of the objects does not affect the dependence of force on relative position and velocity. This is because the mass of an object only determines its inertia, or resistance to changes in motion, while the relative position and velocity of two objects determine the strength of the force between them. Therefore, the dependence of force on relative position and velocity is independent of the mass of the objects.