91Ó°ÊÓ

An ordinary first-order differential equation that is linear in the independent variable and unknown function but not solved for the derivative is termed as Lagrange's equation.

A function that equals the difference between potential and kinetic energy and characterises the state of a dynamic system in terms of position coordinates and their time derivatives is termed as Lagrangian.

Given in the problem 2, Potential field is$\mathrm{V}\left(\mathrm{r},\mathrm{θ},\mathrm{ϕ}\right)$ and mass is .

To obtain the kinetic energy, the velocity in spherical coordinate should be mentioned.

The velocity vector will be given by:

Then the kinetic energy will be given by:

According to the definition of the Lagrangian,

So, the lagrangian is

Since the lagrangian has three degrees of freedom:

$\frac{\mathrm{d}}{\mathrm{dt}}\frac{∂\mathrm{L}}{∂\stackrel{˙}{\mathrm{r}}}-\frac{∂\mathrm{L}}{∂\mathrm{r}}=0$

Then the calculation of the required derivation will be:

Then the Euler equation will be:

The first Lagrange equation will be:

The Euler equation of $\mathrm{θ}$degree of freedom will be given by:

$\frac{\mathrm{d}}{\mathrm{dt}}\frac{∂\mathrm{L}}{∂\stackrel{˙}{\mathrm{θ}}}-\frac{∂\mathrm{L}}{∂\mathrm{θ}}=0$

Then the calculation of the required derivation will be:

Then the Euler equation will be:

Now dividing the above Euler equation by localid="1664365691504" $\mathrm{r}$, then the result will be:

So, the second Lagrange equation is

Now for the $\mathrm{Ï•}$degree of freedom, the Euler equation will be given by:

$\frac{\mathrm{d}}{\mathrm{dt}}\frac{∂\mathrm{L}}{∂\stackrel{˙}{\mathrm{ϕ}}}-\frac{∂\mathrm{L}}{∂\mathrm{ϕ}}=0$

Then the required derivative will be:

Then the Euler equation will be:

Divide the above Euler equation by$\mathrm{r}\mathrm{and}\sin \mathrm{θ}$, the result will be

So, the third Lagrange’s equation is