Interaction between two material points

[kreeb]

Hi,
there are two material points with unitary mass that can move on a stright line. They are subject to a potential of mutual interaction V which depends on x and y and it's not symmetric, where x and y are the coordinates of the two points.
I can choose in the istant 0 v1,v2,x and y. How can I find, in t=0, the force that x and y suffer because of the potential?

 

[ENgez]

I may have misunderstood the question, but if you have the potential field at every point, the force felt should be:

$F=-∇U(x,y)$.

F is the negative of the gradient of U

to simplify even further, if you say the masses move on straight lies only, this becomes a one dimensional problem, as you can express y as a function of x.

 

[kreeb]

Yes, but x e y are coordinates along the same straight line.
I might get the two forces by the derivative of the potential respect to x and y with changed sign, but if you look at how the system evolves at regular intervals, the mechanical energy is not conserved.

 

[kreeb]

up..

 

[rezeew]

To find the force on x and y at t=0, you will need to use the equations of motion for each point. These equations are derived from Newton's second law, which states that the net force acting on an object is equal to its mass times its acceleration.

In this case, the net force acting on each point will be the sum of the force due to the potential and any other external forces present. Since the potential depends on x and y, you will need to use the partial derivatives of the potential with respect to x and y to find the force on each point.

Once you have the equations of motion for each point, you can plug in the initial values of v1, v2, x, and y at t=0 to solve for the forces on x and y. It is important to note that the forces on x and y will depend on the initial conditions you have chosen, so if you change these values, the forces will also change.

In summary, to find the force on x and y at t=0, you will need to use the equations of motion and the partial derivatives of the potential with respect to x and y. Then, plug in the initial values of v1, v2, x, and y to solve for the forces.

 

[DeeNos]

I would first like to clarify that the term "material points" is typically used to refer to objects with mass, but the description states that these points have unitary mass, meaning they have a mass of 1. This may not be a realistic scenario, but for the sake of this response, I will assume that the points have a mass of 1.

To determine the force experienced by the points at t=0, we can use the equations of motion for particles under the influence of a potential. The force experienced by each point can be calculated using the following equation:

F = -∇V(x,y)

Where F is the force, ∇ is the gradient operator, and V(x,y) is the potential that the points are subject to. The negative sign indicates that the force is in the opposite direction of the gradient of the potential.

To find the force at t=0, we need to know the initial positions (x and y) and velocities (v1 and v2) of the points, as well as the functional form of the potential V(x,y). Once we have this information, we can plug it into the equation above to calculate the force experienced by each point.

It is important to note that the potential V(x,y) is not symmetric, which means that the force experienced by each point will be different. This is because the potential depends on both the x and y coordinates of the points, and their positions relative to each other will affect the magnitude and direction of the force.

In summary, to find the force experienced by the points at t=0, we need to know the initial conditions and the potential function, and then use the equation F = -∇V(x,y) to calculate the force for each point. This will give us a better understanding of the interaction between the two material points.