mechanicsdsl-core 1.6.0

A Domain-Specific Language and Transpiler for Classical Mechanics

MechanicsDSL

---

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
Fluid Dynamics	Built-in SPH solver for dam breaks, waves, and liquid physics
High Performance	Generates optimized C++, OpenMP, and WebAssembly code
Rich Visualization	Phase space plots, energy analysis, and smooth animations
Research Ready	Validated against analytical solutions and conservation laws

---

Installation

pip install mechanicsdsl-core

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

---

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)

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

---

Examples

The examples/ directory contains 30 progressive tutorials:

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

---

Documentation

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

Read the Docs

---

Contributing

We welcome contributions! See CONTRIBUTING.md for guidelines.

---

License

MIT License — see LICENSE for details.

---

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