# Period of a particle in a given potential (ex. from Mechanics Landau Lifshitz))

• alex.dranoel
In summary, the task is to determine the period of oscillation, as a function of the energy, for a particle moving in a field with potential energy U=U0tan^2αx. The relevant equation is T(E)=√(2m)∫x1(E)x2(E)√(E-U(x))dx, where x1(E) and x2(E) are roots of U(x)=E. To solve the problem, the integral is simplified by substituting y=αx and factoring out E under the square root. This leads to the definite integral ∫0(π/2)dz/(U0/E+sin^2z), which can be solved using the residue theorem
alex.dranoel

## Homework Statement

Determine the period of oscillation, as a function of the energy, when a particle of mass ##m## moves in a field for which the potential energy is
$$U = U_0 \tan^2\alpha x.$$

## Homework Equations

The relevant equation is given by the general formula for the period ##T## of the oscillations:
$$T(E) = \sqrt{2m} \int_{x_1(E)}^{x_2(E)} \frac{dx}{\sqrt{E-U(x)}}$$
where ##x_1(E)## and ##x_2(E)## are roots of ##U(x)=E##, giving the limit of the motion.

## The Attempt at a Solution

This problem is found in Mechanics by Landau and Lifshtiz at \$11 in the 3rd edition. It is basically a problem of integration. The first thing I did is to find ##x_2(E)##, which is not complicated:
$$E = U_0 tan^2\alpha x \rightarrow x = \frac{1}{\alpha}\arctan \sqrt{\frac{E}{U_0}}$$
Then given the symmetry of the problem, it is clear that
$$T(E) = 2\sqrt{2m} \int_{0}^{x_2(E)} \frac{dx}{\sqrt{E-U_0\tan^2\alpha x}}$$
Now I am left with an integral that I didn't manage to compute while the result in the book looks very simple.

Thanks for help

I found the way to compute the integral. Firstly, subsitute ##y=\alpha x## to get ride of the ##\alpha## term and then factorize ##E## under the square root. Then substitute ##\sin z = \frac{U_0}{E}\tan y##. If you do everything correctly you will end up with the definite integral:
$$\int_0^{\pi/2} \frac{dz}{\frac{U_0}{E} + \sin^2 z}$$
that you can compute by using the residue theorem.

If any of want to try, feel free :)

## 1. What is the period of a particle in a given potential?

The period of a particle in a given potential refers to the amount of time it takes for the particle to complete one full cycle of motion in the potential. This is typically measured in seconds.

## 2. How is the period of a particle in a given potential calculated?

The period of a particle in a given potential can be calculated using the formula T = 2π√(m/k), where T is the period, m is the mass of the particle, and k is the potential constant. This formula is derived from the equations of motion in classical mechanics.

## 3. What factors affect the period of a particle in a given potential?

The period of a particle in a given potential is affected by the mass of the particle, the strength of the potential, and the initial conditions of the particle's motion (such as its initial position and velocity).

## 4. Can the period of a particle in a given potential change over time?

Yes, the period of a particle in a given potential can change over time if there are external forces acting on the particle, or if the potential itself changes. For example, in a system with friction, the period of the particle's motion will decrease over time.

## 5. How is the period of a particle in a given potential related to its energy?

The period of a particle in a given potential is inversely proportional to the square root of its energy. This means that as the energy of the particle increases, its period decreases. This relationship is described by the principle of conservation of energy in classical mechanics.

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