4.4 Applications to mechanics and field theory

Variational principles form the backbone of classical mechanics, connecting the behavior of physical systems to mathematical optimization. These principles, like the principle of least action, provide a powerful framework for deriving equations of motion and understanding system dynamics.

The Lagrangian and Hamiltonian formulations offer alternative perspectives on mechanics, each with unique advantages. These approaches extend beyond point particles to continuous systems and fields, providing a unified way to describe a wide range of physical phenomena.

Variational Principles in Mechanics

Calculus of variations in mechanics

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• Euler-Lagrange equation forms fundamental equation in calculus of variations derived from minimizing action integral
• Action integral defined as time integral of Lagrangian $S = \int_{t_1}^{t_2} L(q, \dot{q}, t) dt$ quantifies system's behavior over time
• Principle of least action states system path between two points minimizes action making it stationary
• Deriving equations of motion applies Euler-Lagrange equation to Lagrangian yielding second-order differential equations describing system motion (Newton's laws)

Variational principles for field theories

• Extends point particle mechanics to continuous systems like electromagnetic fields
• Field theory Lagrangian density $\mathcal{L}(\phi, \partial_\mu\phi)$ depends on fields and their derivatives
• Action functional for fields integrates Lagrangian density over spacetime $S[\phi] = \int \mathcal{L}(\phi, \partial_\mu\phi) d^4x$
• Euler-Lagrange equations for fields $\frac{\partial \mathcal{L}}{\partial \phi} - \partial_\mu \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} = 0$ obtained by varying action with respect to fields
• Applications span various physical theories (electromagnetic field theory, Klein-Gordon field, Dirac field)

Lagrangian and Hamiltonian Formulations

Lagrangian and Hamiltonian from variations

• Lagrangian formulation uses Lagrangian function $L = T - V$ and Euler-Lagrange equations $\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = 0$
• Hamiltonian formulation employs Legendre transformation introducing generalized momenta $p_i = \frac{\partial L}{\partial \dot{q}_i}$
• Hamiltonian function $H = \sum_i p_i \dot{q}_i - L$ leads to Hamilton's equations $\dot{q}_i = \frac{\partial H}{\partial p_i}$, $\dot{p}_i = -\frac{\partial H}{\partial q_i}$
• Variational derivation of Hamilton's principle involves action integral variation considering boundary conditions and constraints

Variations for continuous systems

• Functional derivatives generalize partial derivatives for functionals measuring change in functional due to infinitesimal function change
• Variational principles for continuous systems apply to phenomena (string vibrations, elastic deformations)
• Field theory variational principles use action principle for fields deriving field equations
• Noether's theorem connects symmetries and conservation laws in continuous systems and field theories
• Hamiltonian density for fields results from Legendre transformation of Lagrangian density yielding canonical field equations