# Show Peculiar Solution for Classical Equation of Motion for Harmonic Oscillator

• alfredblase
In summary, the author found a solution to the equation of motion for a harmonic oscillator that includes two constants of integration, q' and q''.
alfredblase
Show that

$$q_{c}(t)=\frac{1}{\sinh \omega \tau} }[ q' \sinh{\omega(t''-t)} + q'' \sinh{\omega(t-t')} ]$$

is the solution to the classical equation of motion for the harmonic oscillator

$$\ddot {q}_{c}(t)-\omega^{2}q_{c}(t)=0$$

where $$q_{c}(t)$$ is the position vector, $$\tau = t'' - t'$$, $$q_{c}(t')=q'$$, $$q_{c}(t'')=q''$$ and $$\ddot {q}_{c}(t)$$ is the double time derivative of $$q_{c}(t)$$

I can differentiate $$q_{c}(t)$$ twice with respect to time easily enough to show that that the first equation is a solution to the equation of motion but there are many functions that satisfy this.

How do I show that it's the solution?

I've tried solving the equation of motion directly but I can't find any way to solve it so that I end up with $$q_{c}(t)$$ as a funciton of t, t', t'', q' and q''. (For example the way everyone is usually taught to solve it, you simply end up with a sin function of t multiplied by an amplitude..)

Any help/suggestions would be very much appreciated. Thank you for taking the time to read this :)

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alfredblase said:
How do I show that it's the solution?

Do you really think that is what it is asking you to do?

I think that if you have shown that q_c(t) is a valid solution, then you are done.

hmm.. the book I'm following (Jean Zinn Justins QFT and critical phenomena) is "calculating the classical action explicitly" and one of the steps along the way is finding that $$q_{c}(t)$$ is as given.

perhaps you are right.. strictly speaking I have found that this form of $$q_{c}(t)$$ is valid but shouldn't I really understand the motivation behind using that particular solution? To be honest I think the book should be telling me that, but there's no reason given other than we find the solution to be thus.. I can't help feeling I'm missing something..

EDIT: perhaps the motivation is that this form has all the bits that define the path integral? (In the overall scheme of things we're applying a path integral with the action for a harmonic oscillator)

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Hi alfred, it is appropriate to say "the" instead of "a" because there are boundary conditions specified. In other words, you require that q_c satisfy the differential equation and have the value q' at t' and q'' at t''. As you no doubt recall from the theory of linear second order differential equations, solutions to such equations contain two constants of integration. These are fixed by the two boundary conditions. If you had not specified initial data or boundary conditions, then it would only make sense to say you had found a solution to the differential equation. The application of boundary conditions turns the "a" into a "the".

Hope this helps.

ohh i see now! how stupid of me. I should have at least thought of putting the values t' and t'' into the solution and thus seen that q_c satisfies those boundary conditions. doh!

grr i really need to develop these basic instincts when looking at equations and learn to play with them, and understand what they are telling me :D

oh well, thank you physics monkey!

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## 1. What is the classical equation of motion for a harmonic oscillator?

The classical equation of motion for a harmonic oscillator is m¨ + kx = 0, where m is the mass of the oscillator, ¨ represents the second derivative with respect to time, k is the spring constant, and x is the displacement of the oscillator from its equilibrium position.

## 2. What is a peculiar solution for the classical equation of motion for a harmonic oscillator?

A peculiar solution for the classical equation of motion for a harmonic oscillator is a solution that satisfies the equation but does not exhibit simple harmonic motion. These solutions can arise when there are external forces acting on the oscillator, such as friction or a driving force.

## 3. How do you find the peculiar solutions for the classical equation of motion for a harmonic oscillator?

To find the peculiar solutions, you can first solve the classical equation of motion for a general solution. Then, you can add in any additional forces and solve for the particular solution. The sum of the general and particular solutions will give the overall solution, including the peculiar solutions.

## 4. How do peculiar solutions affect the behavior of a harmonic oscillator?

Peculiar solutions can significantly alter the behavior of a harmonic oscillator. They can introduce damping, causing the oscillator to lose energy and eventually come to rest. They can also introduce a driving force, which can result in the oscillator having a different frequency or amplitude than it would have with just the spring force.

## 5. Can peculiar solutions occur in real-life harmonic oscillators?

Yes, peculiar solutions can occur in real-life harmonic oscillators. In fact, most oscillators in the real world will experience some form of damping or external forces, resulting in peculiar solutions. This is why most real-world oscillators do not exhibit perfect simple harmonic motion, but instead exhibit some deviations from it.

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