4 ms·

This would need way more explanation to make sense. The part I like is that the equation pyramid looks like a Christmas tree ;)

tgb 10y ago
The principle of least action states that the path that a system takes is the one that minimizes it's 'action'. Famously, this says that light travelling through a lens will go through in the fastest path (taking into account the fact that it is slower moving through the lens than through the air). The Euler-Lagrange equations give an approach to solving 'any' classical system using this principle. In order to do so, you first parametrize the system by specifying some variables that describe the state of the system. Then you describe the energy of the system in terms of its parameters. Then you plug that into Euler-Lagrange and you can solve for the motion of the system based. (The 'action' in this sense is determined by the energy along a path that the system might take.) (This is in contrast to Physics 101 approach of a free body diagram where you add up a bunch of forces on an object. In particular the 'constraints' that the system faces (in this example, it's that the pendulum is confined to moving along a circle) are implicitly included in the parametrization of the system and so there's no need to ever consider directly the force of the bar of the pendulum that holds the weight in place.) The first example on the page parametrizes the pendulum in the 'obvious' way, by noting that you know the state of a pendulum if you know what angle the weight forms relative to the vertical (plus also the rate at which that angle is changing). This is nice, but the solution involves trigonometric functions. The second one parametrizes the same system in a different way. Here the mu parameter is, roughly speaking, the height of the weight above the minimum (but not really). It still specifies where the pendulum is (the equation M(mu) = ... tells you the (x,y) coordinates of the pendulum's weight for a particular mu), so we can still calculate the energy of the pendulum in terms of mu. Energy has two components, kinetic and potential. Potential is the just height from 0 times mass times gravitational constant g. The kinetic is mv^2/2 and the speed v can be determined by using the parametrization above M(mu)= .... The end result is a different means of expressing the same system, but interesting in a manner that does not involve trigonometric functions despite being 'so obviously' trigonometric in nature. One might hope that eliminating a trig function would speed up simulation. I think it's clear that one would need a more demanding simulation to make the difference noticeable. I think this is a fun little project and it was nice to see it introduce me to a new concept (rational trig).

tgb 10y ago

The principle of least action states that the path that a system takes is the one that minimizes it's 'action'. Famously, this says that light travelling through a lens will go through in the fastest path (taking into account the fact that it is slower moving through the lens than through the air). The Euler-Lagrange equations give an approach to solving 'any' classical system using this principle. In order to do so, you first parametrize the system by specifying some variables that describe the state of the system. Then you describe the energy of the system in terms of its parameters. Then you plug that into Euler-Lagrange and you can solve for the motion of the system based. (The 'action' in this sense is determined by the energy along a path that the system might take.) (This is in contrast to Physics 101 approach of a free body diagram where you add up a bunch of forces on an object. In particular the 'constraints' that the system faces (in this example, it's that the pendulum is confined to moving along a circle) are implicitly included in the parametrization of the system and so there's no need to ever consider directly the force of the bar of the pendulum that holds the weight in place.) The first example on the page parametrizes the pendulum in the 'obvious' way, by noting that you know the state of a pendulum if you know what angle the weight forms relative to the vertical (plus also the rate at which that angle is changing). This is nice, but the solution involves trigonometric functions. The second one parametrizes the same system in a different way. Here the mu parameter is, roughly speaking, the height of the weight above the minimum (but not really). It still specifies where the pendulum is (the equation M(mu) = ... tells you the (x,y) coordinates of the pendulum's weight for a particular mu), so we can still calculate the energy of the pendulum in terms of mu. Energy has two components, kinetic and potential. Potential is the just height from 0 times mass times gravitational constant g. The kinetic is mv^2/2 and the speed v can be determined by using the parametrization above M(mu)= .... The end result is a different means of expressing the same system, but interesting in a manner that does not involve trigonometric functions despite being 'so obviously' trigonometric in nature. One might hope that eliminating a trig function would speed up simulation. I think it's clear that one would need a more demanding simulation to make the difference noticeable. I think this is a fun little project and it was nice to see it introduce me to a new concept (rational trig).