Finding potential from force

V = -1/r + 1/c^2(r')^2 - 2r'' and the Lagrangian is L = 1/2m(r')^2 + 1/r - 1/(c^2r^2)(r')^2. Using the Euler-Lagrange equation, these equations can be simplified to L = 1/2m(r')^2 - 1/r + 1/(c^2r^2)(r')^2. I hope this helps!
  • #1
genius2687
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A perticle moves in a plane under the influence of a force, acting toward a center of force, whose magnitude is

F= 1/r^2{1 - 1/c^2[(r')^2 - 2(r'')r]}

where r is the distance of the particle to the center of force. Find the generalized potential that will result in such a force, and from that the Lagrangian for the motion in a plane.


I have assumed that we can use F = -partial(V)/partial(r). Is there an easy way to do this problem?

I have tried integrating the second term (r')^2/r^2 by parts (You get this when you distribute the 1/r^2 factor in front). When I do this, I get two terms, one of which looks like 1/r*2(r'(dr') instead of something in the form of (...)dr.
 
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  • #2



Hello, thank you for your forum post. I can provide some insight into finding the generalized potential and Lagrangian for this type of motion.

First, let's start by defining the terms used in the problem. The particle is moving in a plane, which means it is constrained to move along a two-dimensional surface. The force acting on the particle is directed towards a center of force, and its magnitude is dependent on the distance of the particle to the center of force, r. The force is given by the equation F= 1/r^2{1 - 1/c^2[(r')^2 - 2(r'')r]}, where r' represents the first derivative of r with respect to time and r'' represents the second derivative of r with respect to time.

To find the generalized potential, we can use the definition of potential energy, which is given by V = -∫Fdr. Plugging in the given force equation, we get V = -∫(1/r^2{1 - 1/c^2[(r')^2 - 2(r'')r]})dr. Simplifying this expression, we get V = -1/r + 1/c^2(r')^2 - 2r''.

Next, to find the Lagrangian, we can use the equation L = T - V, where T is the kinetic energy. Since the particle is moving in a plane, we can write the kinetic energy as T = 1/2m(r')^2, where m is the mass of the particle. Plugging in the expression for V, we get L = 1/2m(r')^2 + 1/r - 1/c^2(r')^2 + 2r''.

To simplify this expression, we can use the Euler-Lagrange equation, which states that the derivative of the Lagrangian with respect to the position of the particle is equal to the derivative of the Lagrangian with respect to the velocity of the particle. In other words, ∂L/∂r = d/dt(∂L/∂r'). Using this equation, we can simplify the Lagrangian to L = 1/2m(r')^2 - 1/r + 1/(c^2r^2)(r')^2.

In conclusion, the generalized potential for a particle moving in a
 

1. What is the relationship between force and potential?

The potential is the negative of the work done by the force. In other words, potential is a measure of the amount of energy required to move an object against a force. It is directly proportional to the force and inversely proportional to the distance moved.

2. How is potential calculated from force data?

Potential can be calculated by integrating the force with respect to the distance. This means summing up all the small changes in potential as the object moves through different distances.

3. What is the difference between conservative and non-conservative forces in terms of potential?

Conservative forces, such as gravity and electromagnetism, have a corresponding potential energy function that only depends on the position of the object. Non-conservative forces, such as friction and air resistance, do not have a potential energy function associated with them.

4. How does potential energy affect an object's motion?

As an object moves through a potential field, such as a gravitational field, it will experience a force that is proportional to its mass and the gradient of the potential. This force will cause the object to accelerate and change its velocity.

5. Can potential be negative?

Yes, potential can be negative. This typically occurs when the object is moving in a direction opposite to the direction of the force. In this case, the work done by the force is negative, leading to a negative potential value.

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