Lagrange Maths 101

[agor95]

@ME

Thank you for the reminder. Some more grey cells are in mutiny!

Python , Latex Formula, HTML5, x3Dom and maths - overload :-)

[agor95]

This is some simple formula. However it took me some time to find them.

http://www.citycollegiate.com/workpowerenergy_Xc.htm

The Kinetic Energy formula above

http://www.citycollegiate.com/kinematicsXb.htm

The Third Equation of Motion

P.S. Just some revision

[agor95]

The is a basic example of Lagrange.

http://www.physicsinsights.org/lagrange_1.html

I am still learning what it means.

[agor95]

This a method that is used to optimize solutions.

https://en.wikipedia.org/wiki/Linear_programming

I a way we want to optimize the math solutions.

It looks like Lagrange uses the gradient of the function to find a solution.

[Grimer]

Can't see any sign of change in acceleration there.

Seems to be very two dimensional.

[agor95]

You are correct this is more of a foundation method to optimization.

However the restriction formula could be cube or quadratic.

Also maximizing formula is yours to choose so that also could have that property.

What I think I see is the combining of restriction formulas into one then get the gradient for the rate of change and again for acceleration.

[agor95]

Just some web pages to record the maths.

Basic Notes
https://steampunks.ddns.net/formula101.html

Work In Progress main formula
https://steampunks.ddns.net/formula01.html

[agor95]

In the formula01 link above references are made to components.
Those components are related to this image.

l_1_a = rod length on the right of the pivot point. The pivot point is shown by the vertical line.

m_1_a = mass 0.5 kg of the right side of the rod.

m_2_a = mass 1 kg on the end of the right rod.

These are mirrored on the left with l_1_b, m_1_b, m_2_b.

l_2_a & l_2_b are the end mass lengths

The total length is 2 meters = (l_1_a - l_2_a) + (l_1_b - l_2_b)

As you see the rod length is reduced to accept the 'End Mass' length.

[Grimer]

https://en.wikipedia.org/wiki/Lagrange% ... re_theorem

I thought that was interesting. Who'd have thunk it.

I wonder if it represents the three dimensions plus the dimension of time.

[ME]

As you see the rod length is reduced to accept the 'End Mass' length.

Here are two scenario's to think about, which may or may not apply.

1. One of these days you want to insert such mass somewhere half way.
2. You create some situation where you can attach many mass-elements with arbitrary weights and sizes to put them at arbitrary positions.
3. What if it slides...?

[agor95]

@ME

You know where I am going with this maths.

The labels should be fixed to the components.
So when it slides and rotates the label goes along for the ride.

The full unit will slide and rotate.
The slide action I am using is a change in rod length.
That being complementary l_1_a + l_1_b = 2 meters.
So as one increases the other reduces.

Formula will have set initial values.
What changes is the slide amount and rotation angle over time.

Note. After doing one then several units can be combined.

P.S. Thanks for the maths you have supplied during the years.

Sticking the label to the components is on the To-Do list.

[ME]

I only have a rough idea what you're trying.

As usual with these things: you can go left, right, bottom-up, or top-down.
So please don't mind my next opinion too much, but here is my trigger:

For example: I'm no huge fan of this Lagrange notation simply because I dot my paper both intentionally as unintentionally -- it makes such dot-notation besides some benefits, in my case, prone to significant errors depending on mere dot consistency, flecks, ink- and coffee-stains...
For the same reason I see your rod calculus, imagine the physical equivalence of sawing those rods to specific lengths, tap some thread, and screw on some weight.
While, In my opinion, it's much easier to have a solid rod and then slide'n-fix as many hollow weights onto it as you'd like.
Or in your Python case: use a list/collection of weight objects.
Now your rod-length does not depend on the attached weights which now slide by def., and with the possible issue of missing that needed recalc of rod-part-lengths with each their rod-part-masses and where each rod-part-mass have their own rod-part-leverage-effect on the pivot.

Again, never mind when things work great.

[agor95]

@ME

We are in agreement. I also have a rough idea what I'am trying :-)

However you can see the movements on my website.
That is were the image has been taken from on the resent post.

I hear you with the derivative dot notation.

What international notation would be acceptable?

Yes the method is an attractive one and has been contemplated.

The target is to model the unit described as it can be built.

Either way the result is complex.

[agor95]

Moments of Inertia

https://www.miniphysics.com/uy1-calcula ... d-rod.html

[daxwc]

agor95 Why don't you think about putting in a lever instead of bouncing it off the rim?