This lecture contains principle of least action, Hamilton’s equations of motion, and solved examples.

The principle of least action states that, out of all imaginable paths by which a system can move from one configuration to another, the actual path followed by nature is the one for which the action has an extremum value. In modern language, this is stated more accurately as the principle of stationary action, because the action need not always be strictly minimum; it may be minimum, maximum, or simply stationary against small variations of the path.

\[ S = \int_{t_1}^{t_2} L(q,\dot q,t)\,dt \]

where \(S\) is the action and \(L\) is the Lagrangian of the system.

Thus, the physical motion of a particle or system is obtained by requiring that a small variation in the action vanishes:

\[ \delta S = \delta \int_{t_1}^{t_2} L(q,\dot q,t)\,dt = 0 \]

This variational condition leads directly to the Euler--Lagrange equations, which govern the dynamics of the system.

Although the two expressions are often used interchangeably in elementary mechanics, there is a subtle distinction between them. The term principle of least action is the older and more popular name, suggesting that nature chooses the path for which the action is the minimum. However, this is not always strictly true.

Solved Examples:

  • Example 1:
    Problem: Derive the equation of motion for a free particle using the principle of least action.
    Solution:
    The Lagrangian is $L = \frac{1}{2} m \dot{x}^2$.
    Action:
    \(S[x(t)] = \int_{t_1}^{t_2} \frac{1}{2} m \dot{x}^2 \, dt\) Applying the Euler-Lagrange equation:
    \(\frac{d}{dt}(m \dot{x}) = 0 \Rightarrow \ddot{x} = 0\) This corresponds to uniform motion.

  • Example 2:
    Problem: Use the principle of least action to derive the motion of a harmonic oscillator.
    Solution:
    The Lagrangian is $L = \frac{1}{2} m \dot{x}^2 - \frac{1}{2} k x^2$.
    Euler-Lagrange equation:
    \(\frac{d}{dt}(m \dot{x}) + k x = 0 \Rightarrow m \ddot{x} + k x = 0\) This is the equation for simple harmonic motion.

Practice Questions:

  • Short Answer:
    1. Define the principle of least action.
    2. What is meant by stationary action?
  • Numerical:
    1. Find the equation of motion for a particle in a linear potential $V(x) = Fx$ using least action.
    2. Compute the action for a particle moving at constant speed $v$ from $x=0$ to $x=L$ in time $T$.
  • MCQs:
    1. Which of the following is minimized in the principle of least action?
      • a) Kinetic energy
      • b) Potential energy
      • c) Action
      • d) Hamiltonian
        Answer: c)
    2. The Euler-Lagrange equation is obtained from:
      • a) Newton’s laws
      • b) Hamilton’s equations
      • c) Principle of least action
      • d) Gauss’s law
        Answer: c)

Hamilton’s Equations of Motion

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.

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:

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

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.

  • 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.

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)