Euler-Legrange Equations

[ChrisRackauckas]

From a Lagrangian, the Euler-Lagrange equations can be defined. It would nice to be able to just specify the Lagrangian, and have that specify the PDE that can be solved. This won't be useful until the PDE tools are created, so this may need to sit here for a bit.

[ChrisRackauckas-Claude]

Investigation Update (December 2024)

Hi @ChrisRackauckas,

I investigated the current status of this feature request. Here's what I found:

Current State

• The codebase currently only has HamiltonianProblem implemented (in src/hamiltonian.jl)
• There is no Lagrangian mechanics support in DiffEqPhysics.jl
• PR WIP: Add LagrangianProblem #24 ("WIP: Add LagrangianProblem" by @YingboMa) was closed in 2018 without being merged
• Related issues Legrangian problem types #7, Numerical solution for Euler–Lagrange equation #10, and Numerical solution for Hamilton's equations #11 remain open

Ecosystem Changes Since 2017

The original issue noted that "this won't be useful until the PDE tools are created." The SciML ecosystem has evolved significantly since then:

1. ModelingToolkit.jl now has a mature PDESystem interface
2. MethodOfLines.jl provides automated finite difference discretization
3. NeuralPDE.jl offers physics-informed neural network solvers for PDEs
4. Decapodes.jl provides discrete exterior calculus methods

Questions for Triage

Given the ecosystem evolution, I'd like to clarify the intended scope:

1. Is this feature still relevant? Should the Euler-Lagrange equation generation be part of DiffEqPhysics.jl, or has this been superseded by ModelingToolkit's symbolic capabilities?

2. What's the target interface? The original vision mentioned specifying a Lagrangian to generate PDEs. Would this mean:

   • A LagrangianProblem type (like HamiltonianProblem) that automatically derives Euler-Lagrange equations using AD?
   • Or integration with ModelingToolkit for symbolic derivation of Euler-Lagrange PDEs?

3. ODE vs PDE scope: For classical mechanics problems (particles, rigid bodies), Euler-Lagrange produces ODEs. For field theories/continuum mechanics, you get PDEs. Which is the intended scope?

If this feature is still desired, I'm happy to work on an implementation. Please let me know the preferred direction.

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This comment was generated during a proactive issue investigation.