Noether’s Theorem & Symmetries

The symmetry module implements Noether’s theorem for automatic detection of conservation laws from symmetries.

Overview

Noether’s theorem establishes a profound connection between symmetries and conservation laws:

For every continuous symmetry of a physical system, there exists a corresponding conserved quantity.

The module provides:

Cyclic Coordinate Detection: Find coordinates absent from the Lagrangian
Conjugate Momentum Calculation: Compute conserved momenta
Symmetry Detection: Identify translation, rotation, and time symmetries
Conservation Law Derivation: Automatically derive conserved quantities

Theory

Cyclic Coordinates

A coordinate \(q_i\) is cyclic (or ignorable) if it does not appear in the Lagrangian:

\[\frac{\partial L}{\partial q_i} = 0\]

For a cyclic coordinate, the conjugate momentum is conserved:

\[p_i = \frac{\partial L}{\partial \dot{q}_i} = \text{constant}\]

Common Symmetry-Conservation Pairs

Symmetry	Conserved Quantity	Cyclic Coordinate
Time translation	Energy	(explicit time)
Space translation	Linear momentum	Position
Rotation about axis	Angular momentum	Angle

Usage Examples

Detecting Cyclic Coordinates

from mechanics_dsl.domains.classical import NoetherAnalyzer, detect_cyclic_coordinates
import sympy as sp

# Free particle in 2D
m = sp.Symbol('m', positive=True)
x_dot = sp.Symbol('x_dot', real=True)
y_dot = sp.Symbol('y_dot', real=True)

# L = (1/2)m(ẋ² + ẏ²)  - no x, y dependence!
L = sp.Rational(1, 2) * m * (x_dot**2 + y_dot**2)

cyclic = detect_cyclic_coordinates(L, ['x', 'y'])
# Returns: ['x', 'y'] - both cyclic, so p_x and p_y are conserved

Central Force Problem

from mechanics_dsl.domains.classical import NoetherAnalyzer
import sympy as sp

analyzer = NoetherAnalyzer()

# Central force in polar coordinates
m = sp.Symbol('m', positive=True)
k = sp.Symbol('k', positive=True)
r = analyzer.get_symbol('r')
phi = analyzer.get_symbol('phi')  # angle
r_dot = analyzer.get_symbol('r_dot')
phi_dot = analyzer.get_symbol('phi_dot')

# L = (1/2)m(ṙ² + r²φ̇²) - V(r)
# Note: φ does not appear explicitly!
L = sp.Rational(1, 2) * m * (r_dot**2 + r**2 * phi_dot**2) - k/r

# Detect cyclic coordinates
cyclic = analyzer.detect_cyclic_coordinates(L, ['r', 'phi'])
print(cyclic)  # ['phi'] - angle is cyclic

# Get conserved quantity (angular momentum)
p_phi = analyzer.compute_conjugate_momentum(L, 'phi')
print(p_phi)  # m * r² * φ̇ = L (angular momentum)

Energy Conservation

from mechanics_dsl.domains.classical import NoetherAnalyzer
import sympy as sp

analyzer = NoetherAnalyzer()

# For time-independent Lagrangian, energy is conserved
m = sp.Symbol('m', positive=True)
k = sp.Symbol('k', positive=True)
x = analyzer.get_symbol('x')
x_dot = analyzer.get_symbol('x_dot')

L = sp.Rational(1, 2) * m * x_dot**2 - sp.Rational(1, 2) * k * x**2

# Compute energy (Hamiltonian)
E = analyzer.compute_energy(L, ['x'])
print(E)  # (1/2)m*ẋ² + (1/2)k*x² = T + V

Finding All Symmetries

from mechanics_dsl.domains.classical import NoetherAnalyzer
import sympy as sp

analyzer = NoetherAnalyzer()

# Isotropic 2D harmonic oscillator
m, k = sp.symbols('m k', positive=True)
x = analyzer.get_symbol('x')
y = analyzer.get_symbol('y')
x_dot = analyzer.get_symbol('x_dot')
y_dot = analyzer.get_symbol('y_dot')

L = sp.Rational(1, 2) * m * (x_dot**2 + y_dot**2) - \
    sp.Rational(1, 2) * k * (x**2 + y**2)

symmetries = analyzer.find_all_symmetries(L, ['x', 'y'])

for sym in symmetries:
    print(f"{sym.symmetry_type}: {sym.conserved_quantity}")

# Output:
# TIME_TRANSLATION: Energy (T + V)
# ROTATION: Angular momentum (x*p_y - y*p_x)

API Reference

Classes

class NoetherAnalyzer

Analyzer for symmetries and conservation laws.

detect_cyclic_coordinates(lagrangian, coordinates)

Find all cyclic (ignorable) coordinates.

Returns:: List of cyclic coordinate names

compute_conjugate_momentum(lagrangian, coordinate)

Compute \(p_i = \partial L / \partial \dot{q}_i\).

Returns:: Sympy expression for momentum

compute_energy(lagrangian, coordinates)

Compute energy \(E = \sum p_i \dot{q}_i - L\).

Returns:: Energy expression

find_all_symmetries(lagrangian, coordinates)

Detect all symmetries and corresponding conserved quantities.

Returns:: List of SymmetryInfo objects

class ConservedQuantity

Represents a conserved quantity from Noether’s theorem.

name: Human-readable name (e.g., “angular_momentum”)

expression: Sympy expression for the conserved quantity

symmetry: The symmetry that gives rise to this conservation law

Enums

class SymmetryType

TIME_TRANSLATION: Energy conservation
SPACE_TRANSLATION: Linear momentum conservation
ROTATION: Angular momentum conservation
BOOST: Center of mass motion