# ModelingToolkit: troubles using ode\_order\_lowering (A differentiated state's operation must be a \`Sym\`)

**URL:** <https://discourse.julialang.org/t/modelingtoolkit-troubles-using-ode-order-lowering-a-differentiated-states-operation-must-be-a-sym/63716>\
**Category:** Modelling & Simulations\
**Tags:** modelingtoolkit\
**Created:** [June 28, 2021, 9:24pm UTC](https://discourse.julialang.org/t/modelingtoolkit-troubles-using-ode-order-lowering-a-differentiated-states-operation-must-be-a-sym/63716 "2021-06-28T21:24:14Z")\
**Posts on this page:** 2\
**Page:** 1

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**Author:** ![zdenek\_hurak](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/zdenek_hurak/32/53118_2.png) [@zdenek\_hurak](https://discourse.julialang.org/u/zdenek_hurak)\
**Post date:** [June 28, 2021, 9:24pm UTC](https://discourse.julialang.org/t/modelingtoolkit-troubles-using-ode-order-lowering-a-differentiated-states-operation-must-be-a-sym/63716/1 "2021-06-28T21:24:14Z")

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I am trying to model a simple mechanical chain composed of a few masses interconnected with springs and exposed to friction. Such model comes in the form of a few coupled second-order differential equations. These I model below.

```julia
using ModelingToolkit
using OrdinaryDiffEq

@parameters a b # Some physical parameters.
@variables t z[1:10](t) # Positions of individual masses.
D = Differential(t) # Derivative with respect to time.
n = 5 # Number of masses.

F = [a*(z[i+1] - 2*z[i] + z[i-1]) for i=2:n-1]
pushfirst!(F,a*(z[2] - z[1]))
push!(F,a*(z[n-1] - z[n])) # Models stiffness of the springs to the left and right.

eqs = [D(D(z[i])) ~ -b*D(z[i]) + F[i] for i in 1:n]
                              # Acceleration given by the friction and interaction with neighbors.

sys = ODESystem(eqs)

```

So far so good. Now I want to reduce the set of second-order ODEs to a set of first-order ODEs:

```julia
sys = ode_order_lowering(sys)
ERROR: ArgumentError: A differentiated state's operation must be a `Sym`, so states like `D(u + u)` are disallowed. Got `z[1]`.
Stacktrace:
 [1] diff2term(O::Term{Real, Nothing})
   @ Symbolics ~/.julia/packages/Symbolics/AuRtL/src/utils.jl:109
 [2] lower_varname(var::Term{Real, Nothing}, idv::Sym{Real, Nothing}, order::Int64)
   @ Symbolics ~/.julia/packages/Symbolics/AuRtL/src/utils.jl:160
 [3] ode_order_lowering(eqs::Vector{Equation}, iv::Sym{Real, Nothing}, states::Vector{Term{Real, Nothing}})
   @ ModelingToolkit ~/.julia/packages/ModelingToolkit/MYBNE/src/systems/diffeqs/first_order_transform.jl:29
 [4] ode_order_lowering(sys::ODESystem)
   @ ModelingToolkit ~/.julia/packages/ModelingToolkit/MYBNE/src/systems/diffeqs/first_order_transform.jl:9
 [5] top-level scope
   @ none:1

```

I confess I am not able to decipher the error message and make the model work. Anyone can help?

---

<div class="post-metadata">

**Author:** ![dawbarton](https://sea2.discourse-cdn.com/julialang/user_avatar/discourse.julialang.org/dawbarton/32/215461_2.png) [@dawbarton](https://discourse.julialang.org/u/dawbarton)\
**Post date:** [June 30, 2021, 11:56am UTC](https://discourse.julialang.org/t/modelingtoolkit-troubles-using-ode-order-lowering-a-differentiated-states-operation-must-be-a-sym/63716/2 "2021-06-30T11:56:25Z")

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I think this is because of the recent change to MTK that implements arrays of symbolic variables as symbolic arrays (in time it will allow more straightforward vector operations in MTK). The easiest fix is to generate an array of symbolic variables to mimic the old behaviour of MTK. The way I’d do it is

```julia
@parameters a b # Some physical parameters.
@variables t
z = map(first, [@variables $sym(t) for sym in (Symbol(:z, i) for i in 1:10)]) # Positions of individual masses.
D = Differential(t) # Derivative with respect to time.
n = 5 # Number of masses.

```

which uses interpolation to create the desired symbolic variables. (There might be a cleaner way of doing this but I couldn’t work out what it is short of copying code out of the `@variables` macro, but that seems a bit brittle.) This gives you

```julia
julia> z = map(first, [@variables $sym(t) for sym in (Symbol(:z, i) for i in 1:10)])
10-element Vector{Num}:
  z1(t)
  z2(t)
  z3(t)
  z4(t)
  z5(t)
  z6(t)
  z7(t)
  z8(t)
  z9(t)
 z10(t)

```

If you want the nice subscripts like before (e.g., `z₁`), you can use the `map_subscripts` function from Symbolics (available via `ModelingToolkit.Symbolics.map_subscripts`.