 #1
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Hello everyone :)
Not too long ago, I was thinking about planetary motion around a sun, both with circular orbits and elliptic orbits. However, when thinking a little longer about these two cases in a broader sense, I spotted a big difference which I found quite odd (assume purely classical mechanics  no relativity):
In the case of a perfectly circular orbit, there is circular symmetry both in the shape of the orbit AND the gravitational field, and the centres of symmetry align at exactly the same point. In other words, the symmetry of the orbit trajectory "aligns" perfectly with the symmetry of the field that is governing it. For this particular example, let us consider only linear symmetry in two perpendicular axes, x and y i.e. In the circular orbit, the x and y lines of symmetry for the orbit are the same as the x and y lines of symmetry for the gravitational field. This is OK so far.
In the case of the elliptic orbit, there IS x and y linear symmetry BOTH in the orbit AND and field, but the lines of symmetry no longer align (one line of symmetry, say x, aligns, but the other, y, is translated by a distance d. d = distance from centre of gravity to centre of elliptic orbit).
To frame this observation in another way: Let us consider the centre of the ellipse as our point of interest. The orbit path is symmetric in BOTH x and y around our point of interest, but the field that is governing it is ONLY symmetric in one of the axes around our point of interest. (picture attached)
I find this quite strange. Why is it the case that a less symmetrical cause (the gravitational field) can result in a more symmetrical effect (orbit path) around our point of interest?.I can derive the trajectory using classical mechanics and the laws of gravitation, but this does not give me any obvious insight into the symmetry mismatch. I am aware that if we also consider the speed of the planet, the problem does becomes a little less strange, as the speed is not symmetrical in one of the axes.
Sorry for the slightly handwavey explanation XD Any response, however broad, would be appreciated!!
Nat :D
Not too long ago, I was thinking about planetary motion around a sun, both with circular orbits and elliptic orbits. However, when thinking a little longer about these two cases in a broader sense, I spotted a big difference which I found quite odd (assume purely classical mechanics  no relativity):
In the case of a perfectly circular orbit, there is circular symmetry both in the shape of the orbit AND the gravitational field, and the centres of symmetry align at exactly the same point. In other words, the symmetry of the orbit trajectory "aligns" perfectly with the symmetry of the field that is governing it. For this particular example, let us consider only linear symmetry in two perpendicular axes, x and y i.e. In the circular orbit, the x and y lines of symmetry for the orbit are the same as the x and y lines of symmetry for the gravitational field. This is OK so far.
In the case of the elliptic orbit, there IS x and y linear symmetry BOTH in the orbit AND and field, but the lines of symmetry no longer align (one line of symmetry, say x, aligns, but the other, y, is translated by a distance d. d = distance from centre of gravity to centre of elliptic orbit).
To frame this observation in another way: Let us consider the centre of the ellipse as our point of interest. The orbit path is symmetric in BOTH x and y around our point of interest, but the field that is governing it is ONLY symmetric in one of the axes around our point of interest. (picture attached)
I find this quite strange. Why is it the case that a less symmetrical cause (the gravitational field) can result in a more symmetrical effect (orbit path) around our point of interest?.I can derive the trajectory using classical mechanics and the laws of gravitation, but this does not give me any obvious insight into the symmetry mismatch. I am aware that if we also consider the speed of the planet, the problem does becomes a little less strange, as the speed is not symmetrical in one of the axes.
Sorry for the slightly handwavey explanation XD Any response, however broad, would be appreciated!!
Nat :D
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