The Principle of Least Action
Learning Objectives:
- Understand the physical and mathematical meaning of the principle of least action.
- Learn how it leads to the Euler-Lagrange equations of motion.
- Apply the principle to solve simple problems in mechanics.
Key Concepts / Definitions:
- Action ($S$): A scalar quantity defined as the time integral of the Lagrangian, $S = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt$.
- Lagrangian ($L$): The function $L = T - V$, where $T$ is kinetic energy and $V$ is potential energy.
- Principle of Least Action: The path taken by a physical system between two states is the one for which the action is stationary (usually minimized).
Theory and Explanation:
The Principle of Least Action is a powerful and unifying concept in classical mechanics. It asserts that:
A system evolves between two configurations in such a way that the action functional $S$ is stationary.
This principle is central to Lagrangian mechanics and underlies modern formulations of physics, including quantum mechanics and field theory.
Let a mechanical system move from point $A$ at time $t_1$ to point $B$ at time $t_2$. Among all possible paths it could take, the system follows the one for which the action
\[S[q(t)] = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt\]is stationary — meaning that small variations in the path do not change $S$ to first order.
This leads directly to the Euler-Lagrange equations, which describe the system’s motion.
Mathematical Formulation:
Let $q(t)$ be the generalized coordinate of the system.
Action is given by:
\[S[q(t)] = \int_{t_1}^{t_2} L(q, \dot{q}, t) \, dt\]Consider a small variation: $q(t) \rightarrow q(t) + \varepsilon \eta(t)$, where $\eta(t_1) = \eta(t_2) = 0$.
Then,
\[\delta S = \int_{t_1}^{t_2} \left( \frac{\partial L}{\partial q} \eta + \frac{\partial L}{\partial \dot{q}} \dot{\eta} \right) dt\]Integrating by parts the second term:
\[\delta S = \int_{t_1}^{t_2} \left( \frac{\partial L}{\partial q} - \frac{d}{dt} \frac{\partial L}{\partial \dot{q}} \right) \eta(t) \, dt\]Since $\eta(t)$ is arbitrary, for $\delta S = 0$:
\[\frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}} \right) - \frac{\partial L}{\partial q} = 0\]This is the Euler-Lagrange equation.
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.
Important Points / Summary:
- The principle of least action is a variational principle for deriving the equations of motion.
- It unifies many physical laws under a single formalism.
- The action is stationary, not necessarily minimal.
- Leads to Euler-Lagrange equations which generalize Newton’s second law.
Practice Questions:
- Short Answer:
- Define the principle of least action.
- What is meant by stationary action?
- Numerical:
- Find the equation of motion for a particle in a linear potential $V(x) = Fx$ using least action.
- Compute the action for a particle moving at constant speed $v$ from $x=0$ to $x=L$ in time $T$.
- MCQs:
- Which of the following is minimized in the principle of least action?
- a) Kinetic energy
- b) Potential energy
- c) Action
- d) Hamiltonian
Answer: c)
- 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)
- Which of the following is minimized in the principle of least action?