Class LagrangianMechanics

java.lang.Object

org.episteme.natural.physics.LagrangianMechanics

public class LagrangianMechanics
extends Object

Lagrangian mechanics - analytical mechanics using generalized coordinates.

Based on Hamilton's principle of least action: δS = δ∫L dt = 0 Euler-Lagrange equation: d/dt(∂L/∂q̇) - ∂L/∂q = 0

Since:

1.0

Author:

Silvere Martin-Michiellot, Gemini AI (Google DeepMind)

Constructor Summary

Constructors

Constructor	Description
LagrangianMechanics()

Method Summary

Modifier and Type	Method	Description
static Real	action(Real lagrangian, Real timeInterval)	Action integral: S = ∫₁² L dt (Simplified as L * Δt for constant Lagrangian)
static Vector<Real>	generalizedMomentum(Real mass, Vector<Real> velocity)
static Real	generalizedMomentum(Real mass, Real generalizedVelocity)	Generalized momentum: p = ∂L/∂q̇
static Real	lagrangian(Real kineticEnergy, Real potentialEnergy)	Lagrangian for particle in potential: L = T - V = ½mv² - V(x)
static Real	lagrangianCentralForce2D(Real mass, Real radialVel, Real radius, Real angularVel, Real potential)	Lagrangian for particle in central force (2D polar): L = ½m(ṙ² + r²θ̇²) - V(r)
static Real	lagrangianFreeParticle(Real mass, Vector<Real> velocity)
static Real	lagrangianFreeParticle(Real mass, Real velocity)	Lagrangian for free particle: L = T = ½mv²
static Real	lagrangianHarmonicOscillator(Real mass, Real velocity, Real springConstant, Real position)	Lagrangian for harmonic oscillator: L = ½mẋ² - ½kx²
static Real	lagrangianPendulum(Real mass, Real length, Real angularVel, Real angle, Real g)	Lagrangian for pendulum: L = ½ml²θ̇² + mgl cos(θ)

Methods inherited from class Object

clone, equals, finalize, getClass, hashCode, notify, notifyAll, toString, wait, wait, wait

Constructor Details

LagrangianMechanics

public LagrangianMechanics()

Method Details

lagrangianFreeParticle

public static Real lagrangianFreeParticle(Real mass,
 Real velocity)

Lagrangian for free particle: L = T = ½mv²

lagrangianFreeParticle

public static Real lagrangianFreeParticle(Real mass,
 Vector<Real> velocity)

lagrangian

public static Real lagrangian(Real kineticEnergy,
 Real potentialEnergy)

Lagrangian for particle in potential: L = T - V = ½mv² - V(x)

generalizedMomentum

public static Real generalizedMomentum(Real mass,
 Real generalizedVelocity)

Generalized momentum: p = ∂L/∂q̇

generalizedMomentum

public static Vector<Real> generalizedMomentum(Real mass,
 Vector<Real> velocity)

action

public static Real action(Real lagrangian,
 Real timeInterval)

Action integral: S = ∫₁² L dt (Simplified as L * Δt for constant Lagrangian)

lagrangianHarmonicOscillator

public static Real lagrangianHarmonicOscillator(Real mass,
 Real velocity,
 Real springConstant,
 Real position)

Lagrangian for harmonic oscillator: L = ½mẋ² - ½kx²

lagrangianPendulum

public static Real lagrangianPendulum(Real mass,
 Real length,
 Real angularVel,
 Real angle,
 Real g)

Lagrangian for pendulum: L = ½ml²θ̇² + mgl cos(θ)

lagrangianCentralForce2D

public static Real lagrangianCentralForce2D(Real mass,
 Real radialVel,
 Real radius,
 Real angularVel,
 Real potential)

Lagrangian for particle in central force (2D polar): L = ½m(ṙ² + r²θ̇²) - V(r)