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damped harmonic oscillator (physical system) | particle in a box in 1D (physical system)

### 1. Damped Harmonic Oscillator

This is a system where an object oscillates (like a mass on a spring), but its motion gets smaller over time due to a damping force (e.g., friction or air resistance).

#### Simple Explanation

Imagine a weight on a spring bouncing up and down. Without damping, it would keep bouncing forever at a steady rhythm (natural frequency, ω₀). With damping (controlled by ζ), it slows down and eventually stops. The damping ratio ζ decides how fast it stops:

- ζ < 1: Underdamped (oscillates but fades out).

- ζ = 1: Critically damped (stops as fast as possible without oscillating).

- ζ > 1: Overdamped (slowly returns to rest, no oscillation).

#### Key Features

- Degrees of Freedom: 1 (motion only happens along one direction, say x).

- Lagrangian (L): A mathematical function that describes the system’s energy. Here, it includes kinetic energy (½ m x'(t)²) minus potential energy (½ m ω₀² x(t)²), adjusted by a damping factor exp(2 ζ ω₀ t).

- Equation of Motion: x''(t) + 2 ζ ω₀ x'(t) + ω₀² x(t) = 0  

  This is a second-order differential equation. It says: acceleration (x''), damping (x'), and spring force (x) balance out.

- Solution:  

  x(t) = e^(-ζ ω₀ t) [terms with sinh and cosh], where α = ω₀ √(ζ² - 1).  

  The e^(-ζ ω₀ t) part makes the motion decay, and sinh/cosh describe the shape of the oscillation (or lack thereof).

- Hamiltonian (ℋ): Represents total energy, with momentum p(t) and position x(t), adjusted for damping.

- Hamiltonian Equations:  

  - x'(t) = exp(-2 ζ ω₀ t) p(t)/m (velocity from momentum).  

  - p'(t) = -exp(2 ζ ω₀ t) m ω₀² x(t) (force changes momentum).  

- **Period**: T = 2π / (ω₀ √(1 - ζ²)), but only for ζ < 1 (underdamped case where it oscillates).

#### Physical Quantities

- m: Mass of the object.

- ω₀: Natural frequency (how fast it oscillates without damping).

- ζ: Damping ratio (how strong the damping is).

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### 2. Particle in a 1D Box

This is a quantum system where a particle is trapped between two walls (at x = 0 and x = l) and can’t escape.

#### Simple Explanation

Think of a tiny particle bouncing between two walls, like a ping-pong ball in a narrow tube. It can only move left or right (1D), and its energy comes in specific “steps” (quantized). Unlike the oscillator, there’s no spring or damping—just hard boundaries.

#### Key Features

- Degrees of Freedom: 1 (motion only along the x-axis).

- Solution:  

  x(t) = TriangleWave[{0, l}, (t x'(0) + x(0))/(2 l) - 1/4] for 0 ≤ x(0) ≤ l.  

  This describes a particle moving back and forth linearly between 0 and l, like a wave with sharp turns at the walls. (Note: This is a classical approximation; quantum mechanics uses standing waves like sin(nπx/l).)

- No Damping: Unlike the oscillator, there’s no friction or energy loss here.

- No Lagrangian or Hamiltonian Given: Typically, for a free particle in a box, the Lagrangian is just kinetic energy (½ m x'(t)²), and the Hamiltonian is p²/(2m), but boundaries enforce quantization in quantum mechanics.

#### Physical Quantities

- m: Mass of the particle.

- l: Length of the box (distance between walls).

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### Comparison

- Damped Harmonic Oscillator: Oscillates with a spring-like force, loses energy due to damping, and motion decays over time.

- Particle in a 1D Box: Moves freely between walls, no forces inside the box, just bounces back and forth (classically) or forms standing waves (quantum).

- Degrees of Freedom: Both have 1 (motion in one direction).

- Energy: Oscillator has potential and kinetic energy; particle in a box only has kinetic energy between walls.

Damped harmonic oscillator | particle in a box in 1D

degrees of freedom  1

action | S(t_1, t_2) = integral_(t_1)^(t_2) exp(2 ζ ω_0 τ) (1/2 m x'(τ)^2 - 1/2 m ω_0^2 x(τ)^2) dτ | 

Lagrangian | L = exp(2 ζ ω_0 t) (1/2 m x'(t)^2 - 1/2 m ω_0^2 x(t)^2) | 

equations of motion | x''(t) + 2 ζ ω_0 x'(t) + ω_0^2 x(t) = 0 | 

solution of the equations of motion | x(t) = e^(ζ (-t) ω_0) ((x'(0) sinh(α t))/α + x(0) ((ζ ω_0 sinh(α t))/α + cosh(α t)))

with

α = sqrt(-1 + ζ^2) ω_0 | x(t) = TriangleWave[{0, l}, (t x'(0) + x(0))/(2 l) - 1/4] for 0<=x(0)<=l

Hamiltonian | ℋ = exp(-2 ζ ω_0 t) p(t)^2/(2 m) + exp(2 ζ ω_0 t) 1/2 m ω_0^2 x(t)^2 | 

Hamiltonian equations of motion | x'(t) = exp(-2 ζ ω_0 t) p(t)/m | p'(t) = -exp(2 ζ ω_0 t) m ω_0^2 x(t) | 

solution of the Hamiltonian equations of motion | x(t) = e^(ζ (-t) ω_0) ((p(0) sinh(α t))/(α m) + x(0) ((ζ ω_0 sinh(α t))/α + cosh(α t))) 

with

α = sqrt(-1 + ζ^2) ω_0

p(t) = e^(ζ t ω_0) (p(0) (cosh(α t) - (ζ ω_0 sinh(α t))/α) - (m x(0) ω_0^2 sinh(α t))/α)

with

α = sqrt(-1 + ζ^2) ω_0 | 

canonical momenta | p(t) = m e^(2 ζ t ω_0) x'(t) | 

period | T = (2 π)/(sqrt(1 - ζ^2) ω_0) for abs(ζ)<1 | 

 | description | physical quantity | basic dimensions | D

m | mass | mass | [mass] | R^+

ω_0 | natural angular frequency | angular frequency | [angle] [time]^-1 | R^+

ζ | damping ratio | multiplicative constant | [dimensionless] | R^+

l | length of the box | length | [length] | R^+

m | mass of a test particle | mass | [mass] | R^+ ..