Principle of Least Action Visualization

Explore how the true trajectory is an extremum of the action functional S = ∫ L dt

Drag start and end points to adjust boundary conditions

Action Values

Optimal Path: 0.00
Average Action: 0.00
Worst Path: 0.00

Scenario Selection

Trajectory Parameters

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Current Formula

L = T - V
S = ∫t0t1 L dt

Mathematical Principles

What is Action?

Action S is the integral of the Lagrangian L over time, a core quantity in the evolution of mechanical systems.

S = ∫t₀t₁ L(q, q̇, t) dt
其中 L = T - V（动能减去势能）

Principle of Least Action

The true physical path is the one that makes the action an extremum (usually a minimum).

δS = 0
Variational principle: variation of action is zero

Euler-Lagrange Equation

From the principle of least action, we can derive the equations of motion:

d/dt (∂L/∂q̇) - ∂L/∂q = 0

Lagrangians for Various Scenarios

Harmonic Oscillator (Spring)

L = (1/2)mẋ² - (1/2)kx²

Simple Pendulum

L = (1/2)ml²θ̇² - (1/2)mglθ²

Free Fall

L = (1/2)mẋ² - mgx

Brachistochrone Problem

最小化时间 T = ∫ ds/v

Key Insights

🎯

Functional Extremum

Action is a function of paths (a functional), and the true path makes this functional an extremum.

⚡

Energy Perspective

The Lagrangian L = T - V combines kinetic and potential energy, reflecting energy conversion and balance.

🌐

Universal Principle

The principle of least action applies to all areas of physics, from classical mechanics to quantum field theory.

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Calculus of Variations

Through calculus of variations, we can derive all equations of motion from the principle of least action.