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02ps-01 Symplectic Euler

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    Now we come to the first problem of Unit 2. This time we're still dealing with orbits.
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    We have a pendulum and we want to create expressions
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    for the position, velocity, and acceleration.
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    As always, we've given you some code to help you out.
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    Here you can see the time set, the magnitude of the acceleration due to gravity,
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    and the length of the pendulum, which is just the length of the
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    string to which the bulb of the pendulum is attached.
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    What we're asking you to do is to first fill in this definition of the acceleration of the pendulum
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    showing how it depends on the position of the weight.
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    Now if you think about the way that a pendulum swings, you can imagine
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    that if we extend the trajectory, we would get a circle.
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    So position is the length of this curve path.
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    The next thing that you need to do is to fill in this function called symplectic Euler
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    in which logically you will use symplectic Euler method to calculate both distance and velocity.
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    To help you out a little bit, here is a refresher on what the symplectic Euler method says.
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    Another important piece of information if you're not super-comfortable with physics stuff
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    is Newton's second law right here showing the relationship between force, mass, acceleration.
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    So looking back at our code, you can see that we've created
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    empty arrays for you--for position and for velocity.
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    It's up to you to fill these arrays in including the initial conditions.
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    Remember you'll need to initialize x and v where x is zero and v is zero
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    equals something that you're going to figure out.
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    I'm going to give you a few hints though--you can see that we've created
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    this constant called num_initial condition and set it to equal to 50.
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    What you're really doing overall in this problem is looking at 50 different pendulums
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    which each have different initial values for x and v, and to give you a visual of what this looks like,
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    I'm going to show you the final plot that you'll get with this program.
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    So here's the set of graph that you should get as your final result if the program is working correctly.
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    You're going to ignore these top two graphs for now. Let's focus on this bottom plot.
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    This is showing velocity on the vertical axis and position on the horizontal axis.
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    If we look at the top two graphs, we can see that this green set of points right here
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    corresponds to what's happening at time zero.
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    Since green ellipse down here is showing the set of initial conditions
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    for the 50 different pendulums that we're looking at.
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    This green ellipse is kind of like a snap shot of what's happening
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    to all these different pendulums at time zero.
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    The way that I want you to figure out how to set
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    the values for each pendulum for x and for v is just think about this ellipse.
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    Now in Unit 1 you learned about orbits and you'll actually be able to use
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    some of that knowledge in this problem knowing that this green shape is an ellipse.
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    You can see that its major axis right here lies along the line v=0
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    and its minor axis lies along this line x=2.
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    Now the half length of the major axis is 0.25
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    so I guess actually it don't looks like the major axis here.
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    It's really the minor axis if these two sets of axes had the same scales applied to them.
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    Either way, this rightmost point on ellipse corresponds to x=2.25
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    and the leftmost point corresponds to x=1.75.
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    In terms of v, we have values ranging from -2 down here to 2 up here.
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    So think about what equations for x and v you'll need to create an ellipse with these dimensions.
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    That would be how you set the initial conditions for x and v.
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    I think this is a very interesting problem so I hope you enjoy doing it. Good luck.
Title:
02ps-01 Symplectic Euler
Description:

Re-recording of ps2-1

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Team:
Udacity
Project:
CS222 - Differential Equations
Duration:
03:14
Amara Bot edited English subtitles for 02ps-01 Symplectic Euler
Gundega added a translation

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