24 Mathematical Background for Conservative Forces
24.1 Gradient
In coordinate-free terms, the gradient of a function \(f({\bf r})\) is defined by \[\begin{align} df = \nabla f\cdot d{\bf r}, \end{align}\] where \(df\) is the total infinitesimal change in \(f\) for an infinitesimal displacement \(d{\bf r}\).
24.1.1 Cartesian Coordinates
For \(f = f(x,y,z)\) with \(d{\bf r} = dx{\bf E}_x+dy{\bf E}_y+dz{\bf E}_z\): \[\begin{align} df = \frac{\partial f}{\partial x}dx+\frac{\partial f}{\partial y}dy+\frac{\partial f}{\partial z}dz. \end{align}\] Since this must hold for all independent variations \(dx\), \(dy\), \(dz\): \[\begin{align} \nabla f = \frac{\partial f}{\partial x}{\bf E}_x+\frac{\partial f}{\partial y}{\bf E}_y+\frac{\partial f}{\partial z}{\bf E}_z. \end{align}\]
24.1.2 Cylindrical-Polar Coordinates
For \(f = f(r,\theta,z)\) with \(d{\bf r} = dr{\bf e}_r+r\,d\theta{\bf e}_\theta+dz{\bf E}_z\): \[\begin{align} df = \frac{\partial f}{\partial r}dr+\frac{\partial f}{\partial \theta}d\theta+\frac{\partial f}{\partial z}dz = \nabla f\cdot(dr{\bf e}_r+r\,d\theta{\bf e}_\theta+dz{\bf E}_z). \end{align}\] Matching coefficients: \[\begin{align} \nabla f = \frac{\partial f}{\partial r}{\bf e}_r+\frac{1}{r}\frac{\partial f}{\partial \theta}{\bf e}_\theta+\frac{\partial f}{\partial z}{\bf E}_z. \end{align}\]
24.1.3 Curvilinear Coordinates
This section is beyond the scope of this course. See O. M. O’Reilly’s Intermediate Dynamics Section 1.6 for more information.
Consider a curvilinear coordinate system \(\{q^1,q^2,q^3\}\) defined by \(q^i = \hat{q}^i(x_1,x_2,x_3)\), assumed locally invertible. The covariant basis vectors are: \[\begin{align} {\bf a}_i = \frac{\partial{\bf r}}{\partial q^i} = \sum_{k=1}^3\frac{\partial\hat{x}^k}{\partial q^i}{\bf E}_k. \end{align}\] The contravariant basis vectors are the gradients of the curvilinear coordinates: \[\begin{align} {\bf a}^k = \nabla q^k = \sum_{i=1}^3\frac{\partial\hat{q}^k}{\partial x_i}{\bf E}_i. \end{align}\]
24.2 Condition for a Vector Field to Be Conservative
A force \({\bf F} = F_x{\bf E}_x+F_y{\bf E}_y+F_z{\bf E}_z\) is derivable from a potential if \(F_x = \partial f/\partial x\), \(F_y = \partial f/\partial y\), \(F_z = \partial f/\partial z\). This requires: \[\begin{align} \frac{\partial F_x}{\partial y} = \frac{\partial F_y}{\partial x}, \qquad \frac{\partial F_x}{\partial z} = \frac{\partial F_z}{\partial x}, \qquad \frac{\partial F_y}{\partial z} = \frac{\partial F_z}{\partial y}, \end{align}\] summarized by \(\text{curl}\,{\bf F} = {\bf 0}\).
This condition is both necessary and sufficient for \({\bf F}\) to be derivable from a potential (see this proof).
24.3 Conservative Forces
A conservative force \({\bf F}_c\) is derivable from a potential \(U\): \[\begin{align} {\bf F}_c = -\nabla U. \end{align}\] The work of a conservative force depends only on the endpoints: \[\begin{align} W_{{\bf F}_c,AB} = \int_{{\bf r}(t_A)}^{{\bf r}(t_B)}{\bf F}_c\cdot d{\bf r} = -(U_B-U_A) = U_A-U_B. \end{align}\]
24.3.1 Constant Force
For a constant force \({\bf C}\), the potential is \(U = -{\bf C}\cdot{\bf r}\). In Cartesians: \[\begin{align} \nabla({\bf C}\cdot{\bf r}) = \frac{\partial{\bf C}\cdot{\bf r}}{\partial x}{\bf E}_x+\frac{\partial{\bf C}\cdot{\bf r}}{\partial y}{\bf E}_y+\frac{\partial{\bf C}\cdot{\bf r}}{\partial z}{\bf E}_z = {\bf C}. \end{align}\] Hence \({\bf C} = -\nabla(-{\bf C}\cdot{\bf r})\). ✓
24.3.2 Spring Force
For the spring force \({\bf F}_s = -K(r-\ell_0){\bf e}_r\) (taking the anchor at the origin), the potential is \(U = \frac{1}{2}K(r-\ell_0)^2\). Using the polar gradient: \[\begin{align} \nabla\lp\frac{1}{2}K(r-\ell_0)^2\rp = K(r-\ell_0){\bf e}_r, \end{align}\] so \({\bf F}_s = -\nabla U\). ✓
24.3.3 Gravitational Force
For Newton’s law of gravitation, the force on particle 1 due to particle 2 is \[\begin{align} {\bf F}_1 = G\frac{m_1 m_2}{\lnorm{\bf r}_{2/1}\rnorm^3}{\bf r}_{2/1}. \end{align}\] Using the polar gradient with \({\bf r}_{2/1} = r{\bf e}_r\), one can show that \(U = -Gm_1m_2/r\) is the corresponding potential.