- #1

spacetimedude

- 88

- 1

## Homework Statement

Show that the classical path satisfying ##\bar{x}(t_a) = x_a##, ##\bar{x}(t_b) = x_b## and ##T = t_b-t_a## is

$$\bar{x}(t) = x_b\frac{\sin\omega (t-t_a)}{\sin\omega T} + x_a\frac{\sin\omega (t_b-t)}{\sin\omega T}$$

## Homework Equations

The Lagrangian: ##L = \frac{1}{2}m(\dot{x}^2-\omega^2x^2)##

The EOM: ##\ddot{\bar{x}}+\omega^2\bar{x}=0##

## The Attempt at a Solution

The initial step to this problem is confusing me. I have only been exposed to SHM problems in which the solution to its differential equation is ##x(t) = A\cos(\omega t) + B\sin(\omega t)##. But in this particular question, the solution starts with

##\bar{x}(t) = A\sin\omega (t-t_a) + B\sin\omega (t-t_b)##.

My original thought was that it had something to do with the trig identities and rearrangements, but I could not get to the solution.

It would be great if someone can lead me to the process of solving this differential equation or linking me to a site with an explanation. I have only taken a course in differential equation that was integrated into my physics course and have not taken a standalone maths course.

Last edited: