Answer to Question #230794 in Classical Mechanics for alwande hadebe

Question

*Conservative force and its potential energy function Consider the components of a position dependent force F(r): Fx = ayz + bx + c Fy = axz + bz Fz = axy + by. (a) Show that this force is conservative. (b) Find its corresponding potential energy function, V (r)

Answer 1

Given:

$F_x=ayz + bx + c \\ F_y=axz + bz\\ F_z=axy + by$

(a) For conservative field

$$\bf curl\: F=0$$

In our case

$$\bf curl\: F=\begin{vmatrix} \hat i & \hat j & \hat k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ F_x&F_y&F_z \end{vmatrix}$$

$$=\begin{vmatrix} \hat i & \hat j & \hat k \\ \frac{\partial}{\partial x} & \frac{\partial}{\partial y} & \frac{\partial}{\partial z}\\ ayz + bx + c&axz + bz&axy + by \end{vmatrix}$$

$$={\hat i}(ax+b-ax-b)+{\hat j}(ay-ay)+{\hat k}(az-az)\\=\bf 0$$

(b) The potential energy function satisfies equation

$${\bf F}=-\nabla V$$

So

$$V(x,y,z)=-(axyz+b(x^2+y^2+z^2)/2+cx)$$