Learn & Review: Lagrangian and Hamiltonian Mechanics in Under

Introduction to Classical Mechanics Formulations

This summary introduces three formulations of classical mechanics: Newtonian, Lagrangian, and Hamiltonian. While Newtonian mechanics is typically taught first, Lagrangian and Hamiltonian mechanics are more widely used by modern physicists and are crucial for understanding quantum mechanics. The simple pendulum is used as an example to illustrate each approach.

1. Newtonian Mechanics (F=ma)

• Core Concept: This is the foundational approach, applying Newton's second law, F = ma, to systems.
• Pendulum Example:
  • Forces acting on the pendulum bob are gravity (mg) and tension (T).
  • The focus is on the tangential component of force and acceleration along the path of motion.
  • The tangential component of gravity is mg sin(θ), directed towards the equilibrium position.
  • Using the arc length coordinate (s) or the angle (θ) is possible. The relationship is s = lθ.
  • The equation of motion in terms of arc length is: ms'' = -mg sin(θ)
  • Rewriting in terms of the angle θ (using s'' = lθ''): θ'' = -(g/l) sin(θ).
• Limitations: The sine term makes the general solution complex. A simple solution (sinusoidal) exists only for small angles (θ).

2. Lagrangian Mechanics

• Core Concept: This approach starts with the system's energies rather than forces. It defines a function called the Lagrangian (L), which is the difference between kinetic energy (K) and potential energy (U): L = K - U.
• Pendulum Example:
  • Kinetic Energy (K): K = 1/2 * m * v^2. In terms of θ, K = 1/2 * m * (lθ')^2 = 1/2 * ml^2 * θ'^2.
  • Potential Energy (U): Assuming the pivot is ground level, U = mgy. With y = -l cos(θ), U = -mgl cos(θ).
  • Lagrangian (L): L = K - U = 1/2 * ml^2 * θ'^2 + mgl cos(θ).
• Euler-Lagrange Equation: The equation of motion is derived from the Euler-Lagrange equation: d/dt (∂L/∂θ') = ∂L/∂θ
  • ∂L/∂θ' (generalized momentum) = ml^2 * θ'
  • ∂L/∂θ (generalized force) = -mgl sin(θ)
  • Taking the time derivative of ∂L/∂θ': d/dt (ml^2 * θ') = ml^2 * θ''
  • Plugging into the Euler-Lagrange equation: ml^2 * θ'' = -mgl sin(θ)
  • This simplifies to the same equation of motion: θ'' = -(g/l) sin(θ).
• Principle of Least Action: The Euler-Lagrange equation arises from the principle that systems follow paths that minimize (or extremize) the "action," defined as the integral of the Lagrangian over time.
• Advantages:
  • Avoids dealing with vector forces and accelerations.
  • Allows the choice of convenient coordinates (generalized coordinates).
  • Simplifies handling constraints and symmetries.

3. Hamiltonian Mechanics

• Core Concept: This approach starts with the Hamiltonian (H), which is often, but not always, the total energy of the system (K + U). It uses generalized coordinates (like θ) and their corresponding generalized momenta (p).
• Pendulum Example:
  • Total Energy (E): E = K + U = 1/2 * ml^2 * θ'^2 - mgl cos(θ).
  • Generalized Momentum (p): Defined as p = ∂L/∂θ' = ml^2 * θ'.
  • Hamiltonian (H): Expressed in terms of θ and p: H = p^2 / (2ml^2) - mgl cos(θ).
  • General Definition of Hamiltonian: H = p * θ' - L. This definition ensures H represents total energy in many cases and is crucial for more complex systems.
• Hamilton's Equations: These are a pair of first-order differential equations:
  • θ' = ∂H/∂p
  • p' = -∂H/∂θ
• Pendulum Example using Hamilton's Equations:
  • ∂H/∂p = p / (ml^2)
  • ∂H/∂θ = -mgl sin(θ)
  • Hamilton's equations:
    • θ' = p / (ml^2) (This is the definition of p rearranged)
    • p' = -(-mgl sin(θ)) = mgl sin(θ)
  • Taking the time derivative of the first equation (p' = ml^2 * θ'') and substituting into the second yields ml^2 * θ'' = mgl sin(θ), which again leads to θ'' = -(g/l) sin(θ).
• Advantages:
  • Provides a pair of first-order differential equations instead of a single second-order one.
  • Offers a powerful geometric perspective through phase space.
• Phase Space:
  • A space where each point represents the state of the system (position and momentum, e.g., θ and p).
  • The evolution of the system over time traces a flow on this phase space.
  • For systems with conserved energy (like the pendulum), the flow follows lines of constant Hamiltonian (constant energy).
  • The different types of curves in phase space (closed loops vs. open wavy lines) correspond to different physical behaviors of the pendulum (e.g., oscillation vs. rotation).
• Connection to Quantum Mechanics: Functions on classical phase space become operators in quantum mechanics. The Hamiltonian operator is central to the time evolution of quantum states via the Schrödinger equation (e^(-iHt/ħ)).

In summary, while Newtonian mechanics uses forces, Lagrangian mechanics uses energy (K-U) and the Euler-Lagrange equation, and Hamiltonian mechanics uses energy (often K+U) and Hamilton's equations. All three methods, when applied correctly, yield the same equations of motion but offer different insights and mathematical tools, with Lagrangian and Hamiltonian mechanics being particularly important for advanced physics and quantum mechanics.