# Day_08: Filing and tracking issues in GitHub

## Contents

# Day_08: Filing and tracking issues in GitHub¶

Today I ran into some strange behavior when I tried to generalize my Lagrangian equations of motion. It boils down to something happening with Symbolic `Vectors`

Take for example

```
using Symbolics
@variables t x[1:2](t) # independent and dependent variables
@variables k # parameters
Dx1 = Differential(x[1])
Dx2 = Differential(x[2])
V = 1/2*k*(x[2]- x[1])^2
```

I define a symbolic function `V`

that is

\(\frac{1}{2}\left(x_2 - x_1\right)^2\).

Now, I take a derivative with respect to \(x_1\)

\(\frac{\partial L}{\partial x_1} = -k(x_2 - x_1)\)

but instead I get

```
Symbolics.expand_derivatives(Dx1(V))
```

*Weird!*

Expanding the polynomial there are 3 terms

\(1/2 kx_1^2\)

\(1/2 kx_2^2\)

\(-kx_1x_2\)

So, I’ll take derivatives of these terms and see what I get

```
Vterms = [1/2*k*x[1]^2,1/2*k*x[2]^2, -k*x[1]*x[2]]
```

```
Symbolics.expand_derivatives.(Dx1.(Vterms))
```

Here is the root of my problem, for some reason the partial derivative `Differential(x[1])`

is treating `x[1]`

and `x[2]`

as the same variable. The result should have been

\(kx_1\)

\(0\)

\(-kx_2\)

## Unindexed variables work¶

Now, I can compare this to an unindexed set of variables

```
@variables t y1(t) y2(t) # independent and dependent variables
@variables k # parameters
Dy1 = Differential(y1)
Dy2 = Differential(y2)
V = 1/2*k*(y2- y1)^2
```

```
Symbolics.expand_derivatives(Dy1(V))
```

This returns the correct derivative,

\(\frac{\partial V}{\partial y_1} = -k(y_2 - y_1)\)

```
@variables z_1(t)
```

```
D = Differential(t)
```

```
(::Differential) (generic function with 2 methods)
```

```
D.(x)
```

```
(broadcast(Differential(t), map(Symbolics.CallWith((t,)), x)))[1:2]
```

```
D.([y1,y2])
```

## Symbolics still adding support for arrays¶

The Symbolics.jl package is incredible. Its still working on supporting differentiation with arrays. I filed this behavior on github.com/Symbolics.jl issue #571. From what I can see, the main driver behind symbolic differentiation is in the `diff.jl`

file.

I was getting the error from line 58

```
if symtype(expr) <: AbstractArray
error("Differentiation of expressions involving arrays and array variables is not yet supported.")
end
```

but @shashi added some great exceptions that remove this error in #570 and #568. There is always more work to be done on projects like this and because Julia is open source, I can dig into the code, offer help and try to add my experiences to the solution.

## Wrapping up¶

Today, I found some strange behavior in the Symbolics.jl package. I’ll continue to update/add to the Julia projects as issues or hopefully merge some improvements. It would be great to document the process of a PR merge from start-to-finish.

For now, the work-around appears to be that `jacobian`

and `Differential`

only support scalar values. So I can create a `Vector`

of individual values as such

```
@variables x1(t) x2(t)
x = [x1, x2]
```

The development behind Julia and Symbolics.jl is active and inspiring, so I’m sure this work-around won’t need to exist much longer.