Are Oscillation Periods in a Quartic Potential Always Equal?

[PhyConnected]

Homework Statement

Consider a quartic potential,
i.e. $$V(x) \equiv ax^4 + bx^3 + cx^2 + dx + e$$
s.t. there are two local minimums for the potential.

For a given particle with energy E, prove that the period of oscillation around the two minimums are the same.

Homework Equations

$$dt \equiv \frac{dx}{\sqrt{(\frac{2} {m}) E-V(x)}}$$
I suppose?

The Attempt at a Solution

No clue at all, seems impossible to evaluate the integral above directly?

P.S. This is not a homework/coursework question but rather "challenge" type question.
Thanks

 

[gabbagabbahey]

Homework Statement

Consider a quartic potential,
i.e. $$V(x) \equiv ax^4 + bx^3 + cx^2 + dx + e$$
s.t. there are two local minimums for the potential.

For a given particle with energy E, prove that the period of oscillation around the two minimums are the same.

Homework Equations

$$dt \equiv \frac{dx}{\sqrt{(\frac{2} {m}) E-V(x)}}$$
I suppose?

The Attempt at a Solution

No clue at all, seems impossible to evaluate the integral above directly?

P.S. This is not a homework/coursework question but rather "challenge" type question.
Thanks

I think the easiest way to do this would be to use the Lagrangian (have you learned Lagrangian mechanics yet?) to get the equations of motion. Then since you want to look at (small) oscillations around the local minima of the potential, assume one of your two local minima is at $x=x_1$ and see what your equations of motion look like for $x(t)=x_1+\epsilon(t)$.

 

[PhyConnected]

I think the easiest way to do this would be to use the Lagrangian (have you learned Lagrangian mechanics yet?) to get the equations of motion. Then since you want to look at (small) oscillations around the local minima of the potential, assume one of your two local minima is at $x=x_1$ and see what your equations of motion look like for $x(t)=x_1+\epsilon(t)$.

Hi, thanks for the quick respond!
No, unfortunately I haven't learned Lagrangian mechanics.
I've tried using linear approximation by Taylor's Theorem to find a expression for SHM, but that doesn't seem to help since there's no simple expression to the root for a general case.
More importantly, the question does not say that it has to be a small oscillation and the actual figure that comes with the question (not this one) indicates that the energy level is well above the potential minimum.
I suppose the only constrain is that the energy doesn't exceed the local maximum so that there is a turning point.

Thanks again!

 

[gabbagabbahey]

No, unfortunately I haven't learned Lagrangian mechanics.

I don't think that will be a problem. Newtonian mechanics should be fine here. What is the force on the particle in terms of the potential?

I've tried using linear approximation by Taylor's Theorem to find a expression for SHM, but that doesn't seem to help since there's no simple expression to the root for a general case.

Taylor expansion will be useful here.

More importantly, the question does not say that it has to be a small oscillation and the actual figure that comes with the question (not this one) indicates that the energy level is well above the potential minimum.
I suppose the only constrain is that the energy doesn't exceed the local maximum so that there is a turning point.

Certainly, the oscillations must not take the particle past the local maximum in between the two minima or the particle will no longer be oscillating around the same minima, so they can't be too large.

 

[PhyConnected]

I don't think that will be a problem. Newtonian mechanics should be fine here. What is the force on the particle in terms of the potential?
Taylor expansion will be useful here.
Certainly, the oscillations must not take the particle past the local maximum in between the two minima or the particle will no longer be oscillating around the same minima, so they can't be too large.

So $$F(x) \equiv 4ax^3 + 3bx^2 + 2cx + d$$
and so around for small x around the equilibrium,
By Taylor Expansion,
$$F(x) \equiv F'(eq)x + H.O.T.$$
How should I continue from here?
It doesn't seem to be quite true that the two roots of the F(x) is symmetric with respect to the local/global minimum of the F'(x).
(s.t. F'(x) is the same at for the two equilibrium points.)
Thanks.

 

[gabbagabbahey]

So F(x) is simply 4x^3 + 3bx^2 + 2cx + d

I think you are off by a negative sign, and I would rewrite this as

$$m \ddot{x} = -\frac{d V}{ d x}$$

and so around for small x around the equilibrium,
F(x) = F'(x_equilibrium) * x
How should I continue from here?

I would actually start by writing out the full Taylor expansion of the right hand side of the above equation around one of your local minima $x(t)=x_1+\epsilon(t)$ (your potential is a quartic, so you only need to go up to the $\frac{d^5 V}{ d x^5}$ term). If $x_1$ is a local minimum, what can you say about $\left. \frac{d V}{ d x} \right|_{x = x_1}$ and $\left. \frac{d^2 V}{ d x^2} \right|_{x = x_1}$?

 

[PhyConnected]

I think you are off by a negative sign, and I would rewrite this as

$$m \ddot{x} = -\frac{d V}{ d x}$$
I would actually start by writing out the full Taylor expansion of the right hand side of the above equation around one of your local minima $x(t)=x_1+\epsilon(t)$ (your potential is a quartic, so you only need to go up to the $\frac{d^5 V}{ d x^5}$ term). If $x_1$ is a local minimum, what can you say about $\left. \frac{d V}{ d x} \right|_{x = x_1}$ and $\left. \frac{d^2 V}{ d x^2} \right|_{x = x_1}$?

I'm aware how to find the period for a S.H.M., but how would I find the period for a non-linear differential equation after the full Taylor Expansion?
So obviously $$\left. \frac{d V}{ d x} \right|_{x = x_1}$$ is zero at x= x1
but I don't really see any significance of $$\left. \frac{d^2 V}{ d x^2} \right|_{x = x_1}$$

 

[gabbagabbahey]

I'm aware how to find the period for a S.H.M., but how would I find the period for a non-linear differential equation after the full Taylor Expansion?
So obviously $$\left. \frac{d V}{ d x} \right|_{x = x_1}$$ is zero at x= x1
but I don't really see any significance of $$\left. \frac{d^2 V}{ d x^2} \right|_{x = x_1}$$

After thinking about his problem a little more, I don't see why it would be true in general that the oscillations about the 2 minima must have the same period. Is there more to the problem than what you've posted? (If it is taken from a textbook, please give the problem number and name of the text).

 

[PhyConnected]

After thinking about his problem a little more, I don't see why it would be true in general that the oscillations about the 2 minima must have the same period. Is there more to the problem than what you've posted? (If it is taken from a textbook, please give the problem number and name of the text).

It also confuses me since this result certainly seems non-trivial. (somewhat counter-intuitive by considering extreme cases)
I found the question from some local university physics exam.
"www.cap.ca/" + "sites/cap.ca/files/UPrize/cap_2008.pdf"
Thanks