EOM using Lagrange and least action

[GeorgeHanna04]

With the assistance of Dr. Cooper’s Copilot Julia Assistant, I defined the system parameters, kinetic and potential energy, and constructed the Lagrangian L=T−V. I then computed the required derivatives and assembled the Euler–Lagrange equation of motion. Finally, I implemented this procedure in Julia using Symbolics.jl and Latexify.jl, obtaining a working script that symbolically derives and outputs the equation of motion in LaTeX.

What hasn't been addressed yet:

• solve for the motion for a slow rotation speed and a fast rotation speed
• visualize the solution with plots and animations

*Having Trouble saving my work using pluto's "save notebook" feature as a .jl file as opposed to a .html on my macbook. Any tips are appreciated.

In the meantime my julia code is below:

begin
using Symbolics
using Latexify

@variables t Ω L g w1 h1 m
@variables θ(t)
D = Differential(t)

θdot = D(θ)

x_prime = w1 + L*sin(θ)
z_prime = h1 - L*cos(θ)

T = 0.5*m*(L^2*θdot^2 + Ω^2*x_prime^2)
V = m*g*z_prime
Lag = T - V

dL_dθ    = expand_derivatives(Symbolics.derivative(Lag, θ))
dL_dθdot = expand_derivatives(Symbolics.derivative(Lag, θdot))
eom      = expand_derivatives(D(dL_dθdot) - dL_dθ)

println("LAGRANGIAN:")
println()
println(latexify(Lag))

println("EQUATION OF MOTION:")
println()
println(latexify(eom))


println("SIMPLIFIED FORM (divide by m·L):")
println()
println("L·θ̈ + g·sinθ - Ω²·w₁·cosθ - Ω²·L·sinθ·cosθ = 0")

end

[Hlpred]

Thanks for starting the project. I just submitted my own PR that builds off of your code. For my code, I decided to use the built in equation visualization so that I could see what I was working on.

I'm not sure what the issue is with your notebook not saving properly. If you have a local copy of the notebook that you are trying to edit, I think you just have to do ctrl+s.

[GeorgeHanna04]

The visuals look great. Thank you for keeping the flow going!

Still no luck saving the .jl file, I am running Pluto using a local web server via safari. I also just tried the CTRL + S method, but unfortunately it seemed to do nothing.