Question

Using cylindrical coordinates write the Lagrange equations for the motion of a particle acted on by a force, where V is the potential energy. Divide each Lagrange equation by the corresponding scale factor so that the components of F (that is, of) appear in the equations. Thus write the equations as the component equations of, and so find the components of the acceleration a. Compare the results with Problem.

Short answer

Answer

The components of accelerations are mentioned below.

.

Step-by-step solution

Given Information

The motion of as particle acted on by force is.

Definition of cylindrical coordinates.

The coordinate system primarily utilized in three-dimensional systems is the cylindrical coordinates of the system. The cylindrical coordinate system is used to find the surface area in three-dimensional space.

Find the value.

Formula for the Lagrange of the system in cylindrical coordinates is mentioned below.

$$\begin{gathered} L=\frac{1}{2}m{r}^2-V\left(r,\varphi ,z\right) \\ r=r{\stackrel{\wedge }{e}}_r+r\varphi {\stackrel{\wedge }{e}}_{\varphi }+z{\stackrel{\wedge }{e}}_z \\ L=\frac{1}{2}m\left(r^2+r^2+z^2\right)-V\left(r,\varphi ,z\right) \end{gathered}$$

Apply Euler Lagrange’s on the equation mentioned above

$$\begin{gathered} \frac{d}{dt}\left(\frac{\partial L}{\partial r}\right)=mr{\varphi }^2-\frac{\partial \mathit{V}}{\partial r} \\ \frac{d}{dt}\left(\frac{\partial L}{\partial \varphi }\right)=\frac{\partial L}{\partial \varphi }m{r}^2\varphi +2mrr\varphi \\ =-\frac{\partial V}{\partial \varphi } \\ \frac{d}{dt}\left(\frac{\partial L}{\partial z}\right)=-\frac{\partial V}{\partial z} \end{gathered}$$

Divide the equation with respective scale factors given below.

$$\begin{gathered} h_{\varphi }=r \\ {h}_r=1 \\ {h}_z=1 \end{gathered}$$

The equation becomes as follows.

$$\begin{gathered} m\left(r-r{\varphi }^2\right)=-\frac{\partial V}{\partial r} \\ m\left(r\varphi +2r\varphi \right)=-\frac{1\partial V}{1\partial \varphi } \\ mz=-\frac{\partial V}{\partial z} \end{gathered}$$

The component of acceleration are mentioned below.

$$\begin{gathered} a_r=\ddot{r}-\dot{r}{\varphi }^2 \\ {a}_{\varphi }=\ddot{r}\varphi +\dot{2}\dot{r}\varphi \\ {a}_z=\ddot{z} \end{gathered}$$