- #1

alfredblase

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Show that

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

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

[tex]\ddot {q}_{c}(t)-\omega^{2}q_{c}(t)=0[/tex]

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

I can differentiate [tex]q_{c}(t)[/tex] twice with respect to time easily enough to show that that the first equation is

How do I show that it's

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 [tex]q_{c}(t)[/tex] 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 :)

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

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

[tex]\ddot {q}_{c}(t)-\omega^{2}q_{c}(t)=0[/tex]

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

I can differentiate [tex]q_{c}(t)[/tex] 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 [tex]q_{c}(t)[/tex] 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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