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 = ∫ t 1 t 2 L ( q , q ˙ , t ) d t S = \int_{t_1}^{t_2} L(q, \dot{q}, t) dt S = ∫ t 1 t 2 L ( q , q ˙ , t ) d t 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 L ( ϕ , ∂ μ ϕ ) \mathcal{L}(\phi, \partial_\mu\phi) L ( ϕ , ∂ μ ϕ ) depends on fields and their derivatives
Action functional for fields integrates Lagrangian density over spacetime S [ ϕ ] = ∫ L ( ϕ , ∂ μ ϕ ) d 4 x S[\phi] = \int \mathcal{L}(\phi, \partial_\mu\phi) d^4x S [ ϕ ] = ∫ L ( ϕ , ∂ μ ϕ ) d 4 x
Euler-Lagrange equations for fields ∂ L ∂ ϕ − ∂ μ ∂ L ∂ ( ∂ μ ϕ ) = 0 \frac{\partial \mathcal{L}}{\partial \phi} - \partial_\mu \frac{\partial \mathcal{L}}{\partial (\partial_\mu \phi)} = 0 ∂ ϕ ∂ L − ∂ μ ∂ ( ∂ μ ϕ ) ∂ L = 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 from variations
Lagrangian formulation uses Lagrangian function L = T − V L = T - V L = T − V and Euler-Lagrange equations d d t ∂ L ∂ q ˙ i − ∂ L ∂ q i = 0 \frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = 0 d t d ∂ q ˙ i ∂ L − ∂ q i ∂ L = 0
Hamiltonian formulation employs Legendre transformation introducing generalized momenta p i = ∂ L ∂ q ˙ i p_i = \frac{\partial L}{\partial \dot{q}_i} p i = ∂ q ˙ i ∂ L
Hamiltonian function H = ∑ i p i q ˙ i − L H = \sum_i p_i \dot{q}_i - L H = ∑ i p i q ˙ i − L leads to Hamilton's equations q ˙ i = ∂ H ∂ p i \dot{q}_i = \frac{\partial H}{\partial p_i} q ˙ i = ∂ p i ∂ H , p ˙ i = − ∂ H ∂ q i \dot{p}_i = -\frac{\partial H}{\partial q_i} p ˙ i = − ∂ q i ∂ H
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