Introduction to Conservative Vector Fields and Potential Energy

A vector field F is said to be conservative if the work performed in moving a particle between two points depends solely on the endpoints and not on the path taken. This property allows us to define a potential energy function U that is related to F. Understanding conservative vector fields is crucial in many areas of physics and engineering, particularly in the study of mechanical systems.

Definition and Properties of a Conservative Vector Field

If the work for an applied force is independent of the path, then the work done by the force can be uniquely defined by a potential function U. This function is evaluated at the start and end of the trajectory of the point of application. Mathematically, F is conservative if there exists a potential function U(x, y) such that:

Along any path C, the work done is given by:

W ∫CF · dx U(xt_1) - U(xt_2)

This can be expressed as:

W U(x2) - U(x1)

The force F derived from such a potential function is said to be conservative:

?W -?U -F

Example and Verification of a Conservative Vector Field

Consider the vector field F [2x3y?, 2x?y3] as given in the problem statement. We need to determine if this vector field is conservative.

First, we check if the partial derivatives of the vector field components are equal:

For F [2x3y?, 2x?y3]:

Fx 2x3y?

Fy 2x?y3

Now, calculate the second partial derivatives:

?Fx/?y 8x3y3

?Fy/?x 8x3y3

Since ?Fx/?y ?Fy/?x, the vector field is conservative.

Determine the potential function U(x, y) that is related to F.

Given:

Fx 2x3y?

Fy 2x?y3

Integrate with respect to x and y to find U(x, y):

U(x, y) ∫2x3y? dx x?y?/2 g(y)

U(x, y) ∫2x?y3 dy x?y?/2 f(x)

To find the integration constants, we use the condition that the partial derivatives of U(x, y) should match F(x, y):

?U/?x 2x3y? Fx

?U/?y 2x?y3 Fy

This gives us the potential function:

U(x, y) 0.5x?y? C

Note: Here we can see that the potential function U(x, y) 0.5x?y? is a valid function that satisfies the conditions.

Note that the constants of integration were chosen to be zero for simplicity in this example, but in a more general scenario, they could be non-zero.

Conclusion

In this article, we explored the concept of conservative vector fields and their relation to the potential energy function U(x, y). By verifying the conditions and integrating the given vector field, we confirmed that the vector field is conservative. Understanding these concepts is essential for analyzing and solving problems in physics and engineering, particularly in systems where the work performed is dependent on the potential energy difference between two points.