mechanicsdsl-core 1.6.1

A Domain-Specific Language and Transpiler for Classical Mechanics

MechanicsDSL

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MechanicsDSL is a computational physics framework that lets you define physical systems in a natural, LaTeX-inspired syntax and automatically generates high-performance simulations. From pendulums to planetary orbits, from Lagrangian mechanics to fluid dynamics—describe it once, simulate it anywhere.

✨ Why MechanicsDSL?

Feature	Description
Symbolic Engine	Automatically derives equations of motion from Lagrangians or Hamiltonians
12+ Code Generators	C++, Rust, Julia, CUDA, WebAssembly, Unity, Unreal, Modelica, and more
GPU Acceleration	JAX backend with JIT compilation and automatic differentiation
Inverse Problems	Parameter estimation, sensitivity analysis, MCMC uncertainty
Jupyter Native	%%mechanicsdsl magic commands for notebooks
Real-time API	FastAPI server with WebSocket streaming
IDE Support	LSP server for VS Code with autocomplete and diagnostics
Plugin Architecture	Extensible with custom physics domains and solvers

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📦 Installation

pip install mechanicsdsl-core

With optional features:

pip install mechanicsdsl-core[jax]      # GPU acceleration + autodiff
pip install mechanicsdsl-core[server]   # FastAPI real-time server
pip install mechanicsdsl-core[jupyter]  # Notebook magic commands
pip install mechanicsdsl-core[lsp]      # VS Code language server
pip install mechanicsdsl-core[all]      # Everything

Requirements: Python 3.8+ with NumPy, SciPy, SymPy, and Matplotlib (installed automatically).

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Quick Start

The Famous Figure-8 Three-Body Orbit

Define a gravitational three-body system and watch it trace the celebrated Figure-8 periodic orbit:

from mechanics_dsl import PhysicsCompiler

# Define the system using LaTeX-inspired DSL
figure8_code = r"""
\system{figure8_orbit}
\defvar{x1}{Position}{m} \defvar{y1}{Position}{m}
\defvar{x2}{Position}{m} \defvar{y2}{Position}{m}
\defvar{x3}{Position}{m} \defvar{y3}{Position}{m}
\defvar{m}{Mass}{kg} \defvar{G}{Grav}{1}

\parameter{m}{1.0}{kg} \parameter{G}{1.0}{1}

\lagrangian{
    0.5 * m * (\dot{x1}^2 + \dot{y1}^2 + \dot{x2}^2 + \dot{y2}^2 + \dot{x3}^2 + \dot{y3}^2)
    + G*m^2/\sqrt{(x1-x2)^2 + (y1-y2)^2}
    + G*m^2/\sqrt{(x2-x3)^2 + (y2-y3)^2}
    + G*m^2/\sqrt{(x1-x3)^2 + (y1-y3)^2}
}
"""

# Compile and simulate
compiler = PhysicsCompiler()
compiler.compile_dsl(figure8_code)
compiler.simulator.set_initial_conditions({
    'x1': 0.97000436,  'y1': -0.24308753, 'x1_dot': 0.466203685, 'y1_dot': 0.43236573,
    'x2': -0.97000436, 'y2': 0.24308753,  'x2_dot': 0.466203685, 'y2_dot': 0.43236573,
    'x3': 0.0,         'y3': 0.0,         'x3_dot': -0.93240737, 'y3_dot': -0.86473146
})
solution = compiler.simulate(t_span=(0, 6.326), num_points=2000)

Dam Break Fluid Simulation

Simulate fluid dynamics with the integrated SPH solver:

from mechanics_dsl import PhysicsCompiler

fluid_code = r"""
\system{dam_break}

\parameter{h}{0.04}{m}
\parameter{g}{9.81}{m/s^2}

\fluid{water}
\region{rectangle}{x=0.0 .. 0.4, y=0.0 .. 0.8}
\particle_mass{0.02}
\equation_of_state{tait}

\boundary{walls}
\region{line}{x=-0.05, y=0.0 .. 1.5}
\region{line}{x=1.5, y=0.0 .. 1.5}
\region{line}{x=-0.05 .. 1.5, y=-0.05}
"""

compiler = PhysicsCompiler()
compiler.compile_dsl(fluid_code)
compiler.compile_to_cpp("dam_break.cpp", target="standard", compile_binary=True)

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🆕 New in v1.6.0

Jupyter Magic Commands

%load_ext mechanics_dsl.jupyter

%%mechanicsdsl --animate --t_span=0,20
\system{pendulum}
\defvar{theta}{Angle}{rad}
\parameter{m}{1.0}{kg}
\lagrangian{\frac{1}{2}*m*l^2*\dot{theta}^2 - m*g*l*(1-\cos{theta})}
\initial{theta=2.5, theta_dot=0.0}

Parameter Estimation

from mechanics_dsl.inverse import ParameterEstimator

estimator = ParameterEstimator(compiler)
result = estimator.fit(observations, t_obs, ['m', 'k'])
print(f"Fitted: m={result.parameters['m']:.3f}, k={result.parameters['k']:.3f}")

Real-time API Server

python -m mechanics_dsl.server
# -> http://localhost:8000/docs

External Integrations

Platform	Module	Purpose
OpenMDAO	integrations.openmao	Multidisciplinary optimization
ROS2	integrations.ros2	Robotics simulation
Unity	integrations.unity	Game engine (C#)
Unreal	integrations.unreal	Game engine (C++)
Modelica	integrations.modelica	Standards-based simulation

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Core Capabilities

Classical Mechanics (17 Modules)

• Lagrangian & Hamiltonian formulations with automatic EOM derivation
• Constraints: Holonomic, non-holonomic, rolling, knife-edge (Baumgarte stabilization)
• Dissipation: Rayleigh function, viscous/Coulomb/Stribeck friction
• Stability Analysis: Equilibrium points, linearization, eigenvalue analysis
• Noether's Theorem: Symmetry detection, conservation laws, cyclic coordinates
• Central Forces: Effective potential, Kepler problem, orbital mechanics
• Canonical Transformations: Generating functions, action-angle, Hamilton-Jacobi
• Normal Modes: Mass/stiffness matrices, coupled oscillators, modal decomposition
• Rigid Body: Euler angles, quaternions, gyroscopes, symmetric top
• Perturbation Theory: Lindstedt-Poincaré, averaging, multi-scale analysis
• Collisions: Elastic/inelastic, impulse, center of mass frame
• Scattering: Rutherford, cross-sections, impact parameter
• Variable Mass: Tsiolkovsky rocket equation, conveyor belts
• Continuous Systems: Vibrating strings, membranes, field equations

Quantum Mechanics

• Bound States: Infinite well, finite square well, hydrogen atom
• Scattering: Step potential, delta barriers, transmission/reflection coefficients
• Quantum Tunneling: Rectangular barriers, WKB approximation, Gamow factor
• Semiclassical: WKB wavefunctions, Bohr-Sommerfeld quantization
• Hydrogen Atom: Energy levels, Bohr radius, spectral series (Lyman, Balmer, etc.)
• Ehrenfest Theorem: Quantum-classical correspondence

Electromagnetism

• Charged Particles: Lorentz force, cyclotron motion, Larmor radius
• Waves: Plane waves, Poynting vector, radiation pressure
• Antennas: Hertzian dipole, λ/2 dipole, radiation resistance
• Waveguides: TE/TM modes, cutoff frequencies, group velocity
• Traps: Penning trap, magnetic dipole traps, gradient/curvature drift

Special Relativity

• Kinematics: Lorentz boosts, velocity addition, time dilation, length contraction
• Four-Vectors: Spacetime intervals, invariants, metric signature (+,-,-,-)
• Doppler Effect: Longitudinal, transverse, cosmological redshift
• Radiation: Synchrotron radiation, Thomas precession, twin paradox

General Relativity

• Black Holes: Schwarzschild metric, Kerr (rotating), ergosphere
• Geodesics: Light bending, ISCO, photon sphere
• Lensing: Deflection angle, Einstein radius, magnification
• Cosmology: FLRW metric, Hubble law, comoving distance

Statistical Mechanics

• Ensembles: Microcanonical, canonical, grand canonical
• Distributions: Boltzmann, Fermi-Dirac, Bose-Einstein
• Models: Ising model, ideal gas, quantum harmonic oscillator
• Thermodynamic Quantities: Partition functions, entropy, free energy

Thermodynamics

• Heat Engines: Carnot, Otto, Diesel cycles
• Equations of State: Ideal gas, van der Waals
• Phase Transitions: Clausius-Clapeyron, latent heat
• Heat Capacity: Debye, Einstein models

Fluid Dynamics

• SPH Solver: Smoothed Particle Hydrodynamics for incompressible fluids
• Kernels: Poly6, Spiky, Viscosity with Tait equation of state
• Boundaries: No-slip, periodic, reflective conditions

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📚 Examples & Tutorials

Interactive Tutorials (Jupyter)

#	Tutorial	Topics
1	Getting Started	DSL basics, simple pendulum, export
2	Double Pendulum	Chaos, sensitivity, phase space
3	Parameter Estimation	Inverse problems, Sobol analysis

Example Scripts

The examples/ directory contains 30+ progressive examples:

Level	Examples
Beginner	Harmonic oscillator, Simple pendulum, Plotting basics
Intermediate	Double pendulum, Coupled oscillators, 2D motion, Damping
Advanced	3D gyroscope, Hamiltonian formulation, Phase space, Energy analysis
Expert	C++ export, WebAssembly targets, SPH fluid dynamics

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Documentation

Full documentation with tutorials, API reference, and DSL syntax guide:

Read the Docs

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Contributing

We welcome contributions! See CONTRIBUTING.md for guidelines.

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License

MIT License — see LICENSE for details.

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Built with ❤️ for physicists, engineers, and curious minds.