Hamilton’s Equations of Motion

Learning Objectives:

• Understand the formulation of Hamiltonian mechanics from Lagrangian mechanics.
• Learn how to derive Hamilton’s equations of motion.
• Apply Hamiltonian formalism to simple mechanical systems.

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Key Concepts / Definitions:

• Hamiltonian ($H$): A function usually representing the total energy of a system, obtained via Legendre transformation of the Lagrangian.
• Generalized Coordinates ($q_i$): Variables that describe the configuration of a system.
• Generalized Momenta ($p_i$): Conjugate momenta defined as $p_i = \frac{\partial L}{\partial \dot{q}_i}$.

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Theory and Explanation:

Hamiltonian mechanics is an alternative formulation of classical mechanics that uses generalized coordinates $q_i$ and conjugate momenta $p_i$ instead of just coordinates and velocities.

Given the Lagrangian $L(q_i, \dot{q}_i, t)$, the conjugate momenta are defined as:

\[p_i = \frac{\partial L}{\partial \dot{q}_i}\]

The Hamiltonian $H$ is defined as the Legendre transform of the Lagrangian:

\[H(q_i, p_i, t) = \sum_i p_i \dot{q}_i - L(q_i, \dot{q}_i, t)\]

Hamilton’s equations of motion are the following first-order differential equations:

\[\dot{q}_i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q_i}\]

These equations describe the time evolution of a system in phase space and form the basis of modern theoretical physics, including quantum mechanics and statistical mechanics.

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Mathematical Formulation:

Starting with the Lagrangian $L(q_i, \dot{q}_i, t)$, define the conjugate momenta:

\[p_i = \frac{\partial L}{\partial \dot{q}_i}\]

Perform a Legendre transformation to obtain the Hamiltonian:

\[H(q_i, p_i, t) = \sum_i p_i \dot{q}_i - L\]

Then Hamilton’s equations of motion follow as:

\[\dot{q}_i = \frac{\partial H}{\partial p_i}, \quad \dot{p}_i = -\frac{\partial H}{\partial q_i}\]

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Solved Examples:

• Example 1:
  Problem: Derive Hamilton’s equations for a free particle of mass $m$.
  Solution:
  Lagrangian:

  \[L = \frac{1}{2} m \dot{q}^2\]

  Conjugate momentum:

  \[p = \frac{\partial L}{\partial \dot{q}} = m \dot{q} \Rightarrow \dot{q} = \frac{p}{m}\]

  Hamiltonian:

  \[H = p \dot{q} - L = \frac{p^2}{m} - \frac{1}{2} m \left( \frac{p}{m} \right)^2 = \frac{p^2}{2m}\]

  Hamilton’s equations:

  \[\dot{q} = \frac{\partial H}{\partial p} = \frac{p}{m}, \quad \dot{p} = -\frac{\partial H}{\partial q} = 0\]

  Hence, the particle moves with constant momentum.

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• Example 2:
  Problem: Apply Hamilton’s equations to a simple harmonic oscillator.
  Solution:
  Lagrangian:

  \[L = \frac{1}{2} m \dot{q}^2 - \frac{1}{2} k q^2\]

  Conjugate momentum:

  \[p = \frac{\partial L}{\partial \dot{q}} = m \dot{q} \Rightarrow \dot{q} = \frac{p}{m}\]

  Hamiltonian:

  \[H = \frac{p^2}{2m} + \frac{1}{2} k q^2\]

  Hamilton’s equations:

  \[\dot{q} = \frac{\partial H}{\partial p} = \frac{p}{m}, \quad \dot{p} = -\frac{\partial H}{\partial q} = -k q\]

  These yield the standard equations of motion for a harmonic oscillator.

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Important Points / Summary:

• Hamiltonian mechanics provides a symmetrical, phase-space-based approach to classical mechanics.
• The Hamiltonian often corresponds to the total energy.
• Hamilton’s equations are first-order and are useful in analytical and quantum mechanics.
• Canonical transformations preserve the form of Hamilton’s equations.

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Practice Questions:

• Short Answer:
  1. Define the Hamiltonian and explain how it is related to the Lagrangian.
  2. Write down Hamilton’s equations for a charged particle in an electromagnetic field.
• Numerical:
  1. Derive the Hamiltonian for a particle of mass $m$ in a potential $V(q) = \lambda q^4$.
  2. Compute $\dot{q}$ and $\dot{p}$ for a particle in the potential $V(q) = \frac{1}{2}kq^2$ using Hamilton’s equations.
• MCQs:
  1. Hamilton’s equations are:
     • a) Second-order equations in time
     • b) First-order equations in time
     • c) Algebraic equations
     • d) None of the above
       Answer: b)
  2. The Hamiltonian for a free particle is:
     • a) $H = \frac{p^2}{2m}$
     • b) $H = \frac{1}{2} m q^2$
     • c) $H = p q$
     • d) $H = m p$
       Answer: a)

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