Variational Principle

Introduction

The variational principle is a fundamental concept in physics and mathematics, particularly in the fields of classical mechanics, quantum mechanics, and the calculus of variations. It asserts that the path taken by a physical system between two states is the one for which a certain quantity, known as the action, is stationary (usually a minimum or a maximum). This principle provides a powerful method for deriving the equations of motion for a system and has profound implications in both theoretical and applied sciences.

Historical Background

The origins of the variational principle can be traced back to the work of Pierre de Fermat in the 17th century, who formulated the principle of least time in optics. This principle states that light follows the path that takes the least time to travel from one point to another. Later, in the 18th century, Joseph-Louis Lagrange and William Rowan Hamilton extended these ideas to mechanics, leading to the development of Lagrangian and Hamiltonian mechanics.

Mathematical Formulation

The variational principle can be mathematically formulated using the concept of the action, \( S \), which is defined as the integral of the Lagrangian, \( L \), over time:

\[ S = \int_{t_1}^{t_2} L \, dt \]

where \( L \) is a function of the generalized coordinates, velocities, and possibly time. The principle states that the actual path taken by the system makes the action stationary, which can be expressed as:

\[ \delta S = 0 \]

This condition leads to the Euler-Lagrange equations, which are the fundamental equations of motion in Lagrangian mechanics:

\[ \frac{d}{dt} \left( \frac{\partial L}{\partial \dot{q}_i} \right) - \frac{\partial L}{\partial q_i} = 0 \]

where \( q_i \) are the generalized coordinates and \( \dot{q}_i \) are the generalized velocities.

Applications in Classical Mechanics

In classical mechanics, the variational principle is used to derive the equations of motion for a wide range of systems. For example, in the case of a simple pendulum, the Lagrangian is given by:

\[ L = T - V = \frac{1}{2} m \ell^2 \dot{\theta}^2 - mg\ell (1 - \cos \theta) \]

where \( T \) is the kinetic energy, \( V \) is the potential energy, \( m \) is the mass, \( \ell \) is the length of the pendulum, \( \theta \) is the angular displacement, and \( g \) is the acceleration due to gravity. Applying the Euler-Lagrange equation yields the equation of motion for the pendulum.

Quantum Mechanics and the Variational Principle

In quantum mechanics, the variational principle is used in the context of the Schrödinger equation. The principle states that the ground state energy of a quantum system is the minimum value of the expectation value of the Hamiltonian, \( H \), over all possible normalized wavefunctions, \( \psi \):

\[ E_0 = \min_{\psi} \left( \frac{\langle \psi | H | \psi \rangle}{\langle \psi | \psi \rangle} \right) \]

This approach is particularly useful in the variational method, where trial wavefunctions are used to approximate the ground state energy of complex systems.

Calculus of Variations

The calculus of variations is a branch of mathematical analysis that deals with finding the extrema of functionals, which are mappings from a space of functions to the real numbers. The variational principle is a central concept in this field, and it is used to derive the Euler-Lagrange equations, which are the necessary conditions for a functional to have an extremum.

Noether's Theorem

One of the most profound results related to the variational principle is Noether's theorem, which states that every differentiable symmetry of the action of a physical system corresponds to a conservation law. For example, the invariance of the action under time translations leads to the conservation of energy, while invariance under spatial translations leads to the conservation of momentum.

Image

A pendulum swinging in a gravitational field.

References

Goldstein, H. (1980). Classical Mechanics. Addison-Wesley.
Landau, L. D., & Lifshitz, E. M. (1976). Mechanics. Pergamon Press.
Dirac, P. A. M. (1981). The Principles of Quantum Mechanics. Oxford University Press.