, discovered by Leonhard Euler and Joseph-Louis Lagrange in the 18th century.\u00a0 Not only are they of special interest in celestial mechanics, they also have important applications for space missions and the placing of satellites.\u00a0 We even find asteroids caught around some of the Lagrange points: so called \"Trojan asteroids\".<\/span><\/p>\nWhy do these points exist?\u00a0 To investigate, we will again use the concept of a potential, as we saw in the discussion of Mercury's orbital precession.\u00a0 As a reminder, in physics a \"potential\" is an imagined landscape of hills and valleys that describes how an object reacts to forces.\u00a0 We can think of an orbit as being the motion of a particle trapped in a valley of a potential.\u00a0 This time however, we will generate the potential in a somewhat unusual way: we will adopt a frame of reference which is rotating.\u00a0 There are pros and cons to this approach.\u00a0 One one hand, motion in a rotating frame of reference is not very intuitive, and we will need to invoke \"non-inertial\" forces to understand them.\u00a0 On the other hand, doing it this way will make life a lot simpler, as then the masses and the Lagrange points themselves will appear stationary.\u00a0 Motions around the Lagrange points will also be much easier to visualize and interpret.<\/p>\n
But, seeing as how much I myself struggled when I was learning about non-inertial forces in rotating frames of reference, I think a quick digression into that subject would be a good idea.<\/p>\n
Frames of Reference and Non-Inertial Forces<\/h4>\n
Most of us are comfortable with physics in \"inertial\" frames.\u00a0 An inertial frame is one in which objects subjected to no forces will move in straight lines with constant speed due to inertia.\u00a0 In other words, an inertial frame is one in which the \"law of inertia\", or Newton's First law, is valid.\u00a0 But what if we put ourselves on a turn table, and try rolling a ball back and forth to each another?\u00a0 We will discover it is quite hard -- the ball will seem to be deflected by some mysterious force.\u00a0\u00a0<\/p>\n
Someone watching from the side will still say the ball moves in a straight line relative to the ground.\u00a0 It is us on the spinning turn-table that accounts for the weird trajectory we see.\u00a0\u00a0However, we can be just as valid in saying that the ball is deflected by something.\u00a0 We may invoke new forces, unique to our rotating frame.\u00a0 There is a\u00a0<\/span>centrifugal force <\/span><\/b>which tends to fling things away from the axis of rotation, and a C<\/b><\/span>oriolis force<\/span><\/b>\u00a0which tends to make moving objects turn and follow curling paths.\u00a0 We all have felt centrifugal force whenever riding in a car going around a turn.\u00a0 And many are probably familiar with how Coriolis forces on the spinning Earth causes air currents to deflect, and explains the rotation of hurricanes.\u00a0 These forces are sometimes said to be \"fictitious\", in the sense that there is no real physical interaction responsible for them.\u00a0 They are simply consequences of being in a strange frame of reference.\u00a0 But when we are in that reference frame, they seem totally real.<\/span><\/p>\nI have found no more wonderful a demonstration and explanation of these fictitious or non-inertial forces than in this film put together by University of Toronto physicists in 1960.\u00a0 If you have not seen this, it is an absolute must watch.\u00a0 Or if you're in a hurry:<\/p>\n
-The first 13 minutes build the ideas of different frames of reference.<\/span>
\n-13:25 to 17:00 looks at non-inertial (accelerated) frames.\u00a0\u00a0<\/span>
\n-17:04 is a beautiful example of motion in a rotating frame.\u00a0 This segment is extremely relevant to the physics we will be looking at here.<\/span><\/p>\n