Learning MTK - Derivatives

[rain fox]

What did you try?

[mystic whale]

###

@parameters t m[1:3]

###

@variables θ(t) ϕ(t) r(t)

###

begin
    @variables dθ dϕ dr
    [dθ, dϕ, dr] .~ Differential(t).([θ, ϕ, r])
end

###

begin
    @variables ddθ ddϕ ddr
    [ddθ, ddϕ, ddr] .~ Differential(t).([dθ, dϕ, dr])
end

###

expand_derivatives(ddθ)

(here's my code)

[mystic whale]

Mostly things similar to the code I just sent, but organized in different ways -

###Particle Symbolic Position

@parameters t; @variables θ(t) ϕ(t) r(t)

@variables p[1:3]; p~[θ, ϕ, r] # Position Vector

### Particle Symbolic Velocity Components

begin
    dθ~D(θ)
    dϕ~D(ϕ)
    dr~D(r)
end

### Velocity Vector

@variables v(θ,ϕ,r,t)[1:3]; v~[r*dθ, r*sin(θ)*dϕ, dr]

### Symbolic Dipole Vector Potential

@variables A[1:3]; A ~ (μ₀ / (4π * r^3)) .* cross(m, p) # Dipole vector potential

### Symbolic Lagrangian

@variables L(θ, ϕ, r, dθ, dϕ, dr); L ~ KE-PE

KE = 0.5 * sum(collect(v .^ 2))

PE = q*dot(A, v)

### Solving Symbolic Equations of Motion

eqs=[
    D(Ddθ(L))~Dθ(L)
    D(Ddϕ(L))~Dϕ(L)
    D(Ddr(L))~Dr(L)
]

This is some code from a previous Pluto notebook where I organized things differently

(I abandoned this and started from scratch, since I was getting confused on how things related to one another)

[mystic whale]

Since I made this post I made some jumps in understanding some things- I'll post some finished stuff tomorrow

[mystic whale]

Here's where I am right now. I went through the Julia discourse on the relevant packages and the documentation before getting here, but I might've missed something in either of those places. Here's all of my code:

using ModelingToolkit, LinearAlgebra

#### Parameters (Independent): Magnetic Vacuum Permeability, Time, Magnetic Moment, Particle Mass, Charge

@parameters μ₀, t, m[1:3], mₚ, qₚ

#### Variables (dependent): Positions

@variables θ(t), ϕ(t), r(t)

p = [θ, ϕ, r]

d=sqrt(sum(collect(p) .^ 2))

#### Differential Operators

dt = Differential(t)

#### Velocity Definitions

vθ = r * dt(θ)

vϕ= r*sin(θ)*dt(ϕ)

vr=dt(r)

v=[vθ, vϕ, vr]

#### Defining the Lagrangian

KE = 1/2 * mₚ * sum(collect(v) .^ 2)

A = μ₀/(4π*d^3) * cross(m, v)

PE = qₚ*dot(A, v)

L = KE - PE

#### Euler-Lagrange Template Function

generate_euler_lagrange(L, q) = (expand_derivatives(dt(Differential(q)(L)) - Differential(q)(L)) ~ 0)

#### Defining the Euler-Lagrange Equations

equations_of_motion = generate_euler_lagrange.(L, [θ, ϕ, r])

[rain fox]

you don't have to isolate, just give the differential algebraic equations and let structural simplify handle them

[mystic whale]

Okay, I'll check that out

Thanks for the tip ^w^

[mystic whale]

This works great- thanks for the advice!!

Im still wondering if there's a better way to simplify these expressions though, they seem much more complicated then they should be

I derived these by hand to check against and they aren't supposed to be nearly as large

(Maybe there are easier to run? I dont know)