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add Lagrangian explanation to DAE reduction tutorial. #3257

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Dec 4, 2024
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add Lagrangian explanation to DAE reduction tutorial. #3257

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8 changes: 8 additions & 0 deletions docs/src/examples/modelingtoolkitize_index_reduction.md
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Expand Up @@ -51,6 +51,14 @@ In this tutorial, we will look at the pendulum system:
\end{aligned}
```

These equations can be derived using the [Lagrangian equation of the first kind.](https://en.wikipedia.org/wiki/Lagrangian_mechanics#Lagrangian)
Specifically, for a pendulum with unit mass and length $L$, which thus has
kinetic energy $\frac{1}{2}(v_x^2 + v_y^2)$,
potential energy $gy$,
and holonomic constraint $x^2 + y^2 - L^2 = 0$.
The Lagrange multiplier related to this constraint is equal to half of $T$,
and represents the tension in the rope of the pendulum.

As a good DifferentialEquations.jl user, one would follow
[the mass matrix DAE tutorial](https://docs.sciml.ai/DiffEqDocs/stable/tutorials/dae_example/#Mass-Matrix-Differential-Algebraic-Equations-(DAEs))
to arrive at code for simulating the model:
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