I am currently attempting to understand different systems of telling the time: particularly, distinguishing local time (LT) from local sidereal time (LST). I understand that they often differ; despite this, is it possible to tell on what day of the year the two will be the same? How would I go about doing so?
Local siderean time is the time it takes for the stars to return to the same position in the sky. This is a different time as for the sun to return to her position. The sun takes 24 hours the stars a bit less. This is always the case. The Earth has moved a bit (on a circular orbit) so the stars will retain their position already a bit sooner. If the Earth didn't rotate around its own axis the sun would revolve around it (in one year) while the stars wouldn't. This is the reason that both times will always be different (the motion of the fixed stars will always be different from the motion of the sun as the sun seems to move relative to the stars). There is no way to make the sun and stars rotate at different speeds. You can express the times (or the angles the sun and the stars make) in one another but they will always differ. Of course will both speeds of time be the same. If you base your local time system on the stars (24 hours being the time it takes for the stars to be in the same position again) you have a somewhat different clock as the one which is based on the motion of the sun. The two clocks will never show the same time (if the days are not included). The speeds of time will be the same though (if one hour has passes on the sun-based clock less time will be seen to have elapsed on the sidereal clock but the amount of time passed will be the same, i.e. the time covered by a sun-based clock equals the amount covered by the siderean clock).
Will the two clocks ever show the same time? If both clocks show 24 hours the it it is the same as looking at two clocks with hands going at different speeds. If both start at twelve o'clock, when will both clocks show the same time again? I'll leave that up to you.