Hamiltonian Mechanics - BinaryVisual

What is Hamiltonian Mechanics?

A reformulation of classical mechanics that uses canonical coordinates and Hamilton's equations to describe system dynamics

Core Concept

Hamiltonian mechanics is a reformulation of classical mechanics that describes the evolution of a system in terms of canonical coordinates and momenta. Developed by William Rowan Hamilton, it provides a powerful mathematical framework that reveals deep connections between symmetries and conservation laws.

Unlike Lagrangian mechanics which uses generalized coordinates and velocities, Hamiltonian mechanics employs canonical coordinates and conjugate momenta, leading to a more symmetric and geometrically intuitive description of system dynamics in phase space.

Key Principles of Hamiltonian Mechanics

Understanding the foundational concepts that make Hamiltonian mechanics powerful

Canonical Coordinates

Pairs of generalized coordinates and their conjugate momenta that provide a complete description of a system's state.

Phase Space Conjugate Variables

Hamilton's Equations

A set of first-order differential equations that describe the time evolution of canonical coordinates.

First Order Symplectic Structure

Phase Space

A mathematical space where each point represents a possible state of a system, with coordinates and momenta as axes.

State Space Trajectories

Fundamental Equations

The mathematical foundation of Hamiltonian mechanics

Hamiltonian Function

H(q, p, t) = Σ (pᵢ q̇ᵢ) - L(q, q̇, t)

Where H is the Hamiltonian, pᵢ are conjugate momenta, q̇ᵢ are generalized velocities, L is the Lagrangian

Hamilton's Equations

dqᵢ/dt = ∂H/∂pᵢ
dpᵢ/dt = -∂H/∂qᵢ

A system of 2n first-order differential equations for n degrees of freedom

Poisson Bracket

{A, B} = Σ (∂A/∂qᵢ ∂B/∂pᵢ - ∂A/∂pᵢ ∂B/∂qᵢ)

Mathematical operation encoding the algebraic structure of classical mechanics

Key Insight

Hamiltonian mechanics provides a geometric interpretation of dynamics in phase space and naturally incorporates conservation laws through Poisson brackets.

Applications of Hamiltonian Mechanics

Where Hamiltonian mechanics shines in solving complex problems

Quantum Mechanics

Hamiltonian mechanics provides the classical foundation for quantum mechanics, where observables become operators and Poisson brackets become commutators.

Schrödinger equation formulation
Operator quantization
Heisenberg picture

Statistical Mechanics

The Hamiltonian formulation is essential for statistical mechanics, enabling the derivation of ensemble averages and partition functions.

Microcanonical ensemble
Liouville's theorem
Ergodic hypothesis

Space Mission Design

Phase space techniques and canonical transformations are crucial for designing fuel-efficient spacecraft trajectories.

Halo orbits
Interplanetary transfers
Gravity assists

Control Theory and Robotics

Hamilton-Jacobi theory and optimal control formulations benefit from Hamiltonian approaches to dynamical systems.

Optimal control
Feedback linearization
Stability analysis

Interactive Simulations

Experience Hamiltonian mechanics through hands-on visualizations

Parameters

Mass (kg) 1.0 kg

Spring Constant (N/m) 20 N/m

Initial Position (m) 0.5 m

Initial Momentum (kg⋅m/s) 0.0

Simulation Data

Time: 0.00 s

Position: 0.50 m

Momentum: 0.00 kg⋅m/s

Hamiltonian: 2.50 J

Kinetic Energy: 0.00 J

Potential Energy: 2.50 J

Simulation Progress

Parameters

Mass (kg) 1.0 kg

Gravitational Parameter (m³/s²) 1.0

Initial Distance (m) 5.0 m

Initial Angle (rad) 0.0 rad

Angular Momentum (kg⋅m²/s) 2.0

Simulation Data

Time: 0.00 s

Distance: 5.00 m

Angle: 0.00 rad

Hamiltonian: -0.10 J

Kinetic Energy: 0.10 J

Potential Energy: -0.20 J

Simulation Progress

Parameters

Moment of Inertia I₁ (kg⋅m²) 1.0

Moment of Inertia I₂ (kg⋅m²) 1.5

Moment of Inertia I₃ (kg⋅m²) 2.0

Initial Angular Velocity ω₁ (rad/s) 1.0

Initial Angular Velocity ω₂ (rad/s) 0.5

Initial Angular Velocity ω₃ (rad/s) 0.1

Simulation Data

Time: 0.00 s

ω₁: 1.00 rad/s

ω₂: 0.50 rad/s

ω₃: 0.10 rad/s

Hamiltonian: 0.70 J

Kinetic Energy: 0.70 J

Simulation Progress

Parameters

Length (m) 1.0 m

Mass (kg) 1.0 kg

Initial Angle θ (rad) 0.5 rad

Initial Angle φ (rad) 0.0 rad

Initial Momentum pθ (kg⋅m²/s) 0.0

Initial Momentum pφ (kg⋅m²/s) 0.5

Simulation Data

Time: 0.00 s

θ: 0.50 rad

φ: 0.00 rad

Hamiltonian: 4.91 J

Kinetic Energy: 0.13 J

Potential Energy: 4.78 J

Simulation Progress

Worked Examples

Step-by-step solutions to classic Hamiltonian mechanics problems

Problem Statement

Find the Hamiltonian and equations of motion for a one-dimensional harmonic oscillator with mass m and spring constant k.

Solution Steps

Step 1: Write the Lagrangian

The Lagrangian for a harmonic oscillator is:

L = T - V = ½mv² - ½kx² = ½mx̄² - ½kx²

Step 2: Find the conjugate momentum

The conjugate momentum is defined as p = ∂L/∂ẋ:

p = ∂L/∂ẋ = ∂/∂ẋ(½mx̄² - ½kx²) = mx̄

Solving for ẋ:

ẋ = p/m

Step 3: Compute the Hamiltonian

Using the Legendre transformation H = px̄ - L:

H = px̄ - L = p(p/m) - (½m(p/m)² - ½kx²) = p²/2m + ½kx²

Step 4: Derive Hamilton's equations

Applying Hamilton's equations:

dx/dt = ∂H/∂p = p/m

dp/dt = -∂H/∂x = -kx

Step 5: Solve the equations

Differentiating the first equation and substituting the second:

d²x/dt² = (1/m)dp/dt = (1/m)(-kx) = -(k/m)x

This is the familiar equation for simple harmonic motion with angular frequency ω = √(k/m).

Problem Statement

Derive the Hamiltonian for the Kepler problem (motion under gravitational force) and show how it leads to the conservation of energy and angular momentum.

Solution Steps

Step 1: Write the Lagrangian in polar coordinates

For a particle of mass m moving under gravitational force:

L = T - V = ½m(ṙ² + r²θ̄²) - (-k/r) = ½m(ṙ² + r²θ̄²) + k/r

Step 2: Find the conjugate momenta

Calculate the conjugate momenta for r and θ:

pᵣ = ∂L/∂ṙ = mṙ

pθ = ∂L/∂θ̄ = mr²θ̄

Step 3: Express velocities in terms of momenta

ṙ = pᵣ/m

θ̄ = pθ/mr²

Step 4: Compute the Hamiltonian

Using H = Σ pᵢq̄ᵢ - L:

H = pᵣṙ + pθθ̄ - L = pᵣ(pᵣ/m) + pθ(pθ/mr²) - [½m((pᵣ/m)² + r²(pθ/mr²)²) + k/r]

H = pᵣ²/2m + pθ²/2mr² - k/r

Step 5: Identify conserved quantities

Since H does not explicitly depend on time, energy is conserved:

dH/dt = 0 ⇒ Energy conservation

Since H does not explicitly depend on θ, angular momentum is conserved:

dpθ/dt = -∂H/∂θ = 0 ⇒ pθ = constant

Problem Statement

Show that the fundamental Poisson brackets for canonical coordinates are {qᵢ, pⱼ} = δᵢⱼ, {qᵢ, qⱼ} = 0, and {pᵢ, pⱼ} = 0.

Solution Steps

Step 1: Recall the definition of Poisson brackets

For any two functions A and B of canonical coordinates:

{A, B} = Σ (∂A/∂qᵢ ∂B/∂pᵢ - ∂A/∂pᵢ ∂B/∂qᵢ)

Step 2: Calculate {qᵢ, pⱼ}

{qᵢ, pⱼ} = Σₖ (∂qᵢ/∂qₖ ∂pⱼ/∂pₖ - ∂qᵢ/∂pₖ ∂pⱼ/∂qₖ)

= Σₖ (δᵢₖ δⱼₖ - 0 ⋅ 0) = δᵢⱼ

Step 3: Calculate {qᵢ, qⱼ}

{qᵢ, qⱼ} = Σₖ (∂qᵢ/∂qₖ ∂qⱼ/∂pₖ - ∂qᵢ/∂pₖ ∂qⱼ/∂qₖ)

= Σₖ (δᵢₖ ⋅ 0 - 0 ⋅ δⱼₖ) = 0

Step 4: Calculate {pᵢ, pⱼ}

{pᵢ, pⱼ} = Σₖ (∂pᵢ/∂qₖ ∂pⱼ/∂pₖ - ∂pᵢ/∂pₖ ∂pⱼ/∂qₖ)

= Σₖ (0 ⋅ δⱼₖ - δᵢₖ ⋅ 0) = 0

Step 5: Interpret the results

These fundamental Poisson brackets encode the canonical structure of Hamiltonian mechanics:

Coordinates and their conjugate momenta have unit Poisson bracket
Coordinates among themselves have zero Poisson bracket
Momenta among themselves have zero Poisson bracket

Hamiltonian Mechanics Simulations

Hamiltonian Mechanics Simulations Visually

What is Hamiltonian Mechanics?

Core Concept

Key Principles of Hamiltonian Mechanics

Canonical Coordinates

Hamilton's Equations

Phase Space

Fundamental Equations

Hamiltonian Function

H(q, p, t) = Σ (pᵢ q̇ᵢ) - L(q, q̇, t)

Hamilton's Equations

dqᵢ/dt = ∂H/∂pᵢdpᵢ/dt = -∂H/∂qᵢ

Poisson Bracket

{A, B} = Σ (∂A/∂qᵢ ∂B/∂pᵢ - ∂A/∂pᵢ ∂B/∂qᵢ)

Key Insight

Applications of Hamiltonian Mechanics

Quantum Mechanics

Statistical Mechanics

Space Mission Design

Control Theory and Robotics

Interactive Simulations

Parameters

Simulation Data

Parameters

Simulation Data

Parameters

Simulation Data

Parameters

Simulation Data

Worked Examples

Simple Harmonic Oscillator

Problem Statement

Solution Steps

Step 1: Write the Lagrangian

Step 2: Find the conjugate momentum

Step 3: Compute the Hamiltonian

Step 4: Derive Hamilton's equations

Step 5: Solve the equations

Kepler Problem

Problem Statement

Solution Steps

Step 1: Write the Lagrangian in polar coordinates

Step 2: Find the conjugate momenta

Step 3: Express velocities in terms of momenta

Step 4: Compute the Hamiltonian

Step 5: Identify conserved quantities

Poisson Brackets

Problem Statement

Solution Steps

Step 1: Recall the definition of Poisson brackets

Step 2: Calculate {qᵢ, pⱼ}

Step 3: Calculate {qᵢ, qⱼ}

Step 4: Calculate {pᵢ, pⱼ}

Step 5: Interpret the results

Related Topics

Lagrangian Mechanics

Central Force Motion

Rigid Body Dynamics

dqᵢ/dt = ∂H/∂pᵢ
dpᵢ/dt = -∂H/∂qᵢ