Glossary

4.4 Applications to mechanics and field theory

2 min read•july 25, 2024

form the backbone of classical mechanics, connecting the behavior of physical systems to mathematical optimization. These principles, like the , 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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  • forms fundamental equation in calculus of variations derived from minimizing
  • Action integral defined as time integral of Lagrangian S=∫t1t2L(q,q˙,t)dtS = \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 L(ϕ,∂μϕ)\mathcal{L}(\phi, \partial_\mu\phi) depends on fields and their derivatives
  • for fields integrates Lagrangian density over spacetime S[ϕ]=∫L(ϕ,∂μϕ)d4xS[\phi] = \int \mathcal{L}(\phi, \partial_\mu\phi) d^4x
  • for fields ∂L∂ϕ−∂μ∂L∂(∂μϕ)=0\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 (, , )

Lagrangian and Hamiltonian Formulations

Lagrangian and Hamiltonian from variations

  • Lagrangian formulation uses Lagrangian function L=T−VL = T - V and Euler-Lagrange equations ddt∂L∂q˙i−∂L∂qi=0\frac{d}{dt}\frac{\partial L}{\partial \dot{q}_i} - \frac{\partial L}{\partial q_i} = 0
  • Hamiltonian formulation employs Legendre transformation introducing pi=∂L∂q˙ip_i = \frac{\partial L}{\partial \dot{q}_i}
  • Hamiltonian function H=∑ipiq˙i−LH = \sum_i p_i \dot{q}_i - L leads to q˙i=∂H∂pi\dot{q}_i = \frac{\partial H}{\partial p_i}, p˙i=−∂H∂qi\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

  • 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
  • connects symmetries and conservation laws in continuous systems and field theories
  • Hamiltonian density for fields results from Legendre transformation of Lagrangian density yielding

Key Terms to Review (17)

Action Functional: The action functional is a mathematical expression that summarizes the dynamics of a physical system by integrating the Lagrangian over time. It plays a central role in the formulation of the principle of least action, which states that the path taken by a system between two states is the one that minimizes this action functional. This concept connects deeply with the Euler-Lagrange equations, which arise from the action functional and provide the equations of motion for systems described by Lagrangian mechanics.
Action Integral: The action integral is a fundamental quantity in physics defined as the integral of the Lagrangian function over time. It plays a crucial role in the principle of least action, where the path taken by a system is the one that minimizes the action, connecting concepts in mechanics and field theory. This concept also extends to constrained variations and Hamiltonian mechanics, making it essential for understanding the dynamics of both classical and quantum systems.
Canonical field equations: Canonical field equations are mathematical expressions that describe the dynamics of fields in both classical and quantum mechanics, serving as a bridge between the variational principle and field theory. They arise from the principle of least action, allowing for the formulation of equations of motion for fields such as electromagnetic or gravitational fields. This framework provides essential insights into the behavior and interactions of various physical systems in mechanics and field theory.
Dirac Field: A Dirac field is a quantum field that describes fermions, which are particles that obey the Pauli exclusion principle, like electrons and quarks. This field is crucial for understanding the behavior of spin-1/2 particles in quantum mechanics and serves as the foundation for quantum field theory. The Dirac equation, which governs this field, combines principles of quantum mechanics and special relativity, revealing how these particles interact with each other and with other fields.
Electromagnetic field theory: Electromagnetic field theory is the study of electric and magnetic fields, their interactions with charged particles, and the way they propagate through space. This theory forms the foundation for understanding how electromagnetic waves, such as light, are generated and transmitted, and is essential for applications in mechanics and various technologies.
Euler-Lagrange Equation: The Euler-Lagrange equation is a fundamental equation in the calculus of variations that provides a necessary condition for a function to be an extremum of a functional. This equation connects the concept of functionals, which map functions to real numbers, with variational principles by expressing how changes in a function affect the value of the functional. By deriving this equation, one can derive the paths that systems take in mechanics and field theories based on the principle of least action.
Euler-Lagrange Equations: The Euler-Lagrange equations are fundamental equations in the calculus of variations, providing a method to derive the equations of motion for a system described by a Lagrangian. These equations relate to the path taken by a system to minimize or maximize a certain quantity, usually the action, which is an integral of the Lagrangian over time. They form the backbone of classical mechanics and connect closely with principles such as conservation laws and symmetry operations.
Functional Derivatives: Functional derivatives are a generalization of ordinary derivatives that apply to functionals, which are mappings from a space of functions to the real numbers. They help us understand how a functional changes when the function it depends on is varied slightly. This concept is crucial in many areas such as variational principles, mechanics, and quantum mechanics, allowing us to derive equations of motion and optimize functionals.
Generalized momenta: Generalized momenta are a set of quantities that arise in the formulation of classical mechanics and field theory, defined as the partial derivative of the Lagrangian with respect to the generalized velocities. They play a crucial role in transforming the equations of motion and formulating the Hamiltonian framework, linking kinetic and potential energies to the motion of systems.
Hamilton's equations: Hamilton's equations are a set of first-order differential equations that describe the evolution of a physical system in classical mechanics. They provide a reformulation of Newtonian mechanics, emphasizing energy conservation and enabling a clearer understanding of systems with multiple degrees of freedom. These equations are fundamental in connecting classical mechanics to other areas, such as field theory, adiabatic processes, and advanced mathematical techniques like Poisson brackets.
Hamiltonian formalism: Hamiltonian formalism is a reformulation of classical mechanics that describes a physical system in terms of its Hamiltonian function, which represents the total energy of the system as a function of generalized coordinates and momenta. This approach emphasizes the evolution of systems over time and allows for the analysis of both classical mechanics and quantum mechanics using similar mathematical structures, making it fundamental to many areas of physics, including field theory and canonical transformations.
Klein-Gordon Field: The Klein-Gordon field is a relativistic quantum field theory model that describes scalar particles, represented mathematically by a scalar field satisfying the Klein-Gordon equation. This framework connects the principles of quantum mechanics with special relativity and lays the groundwork for understanding particle interactions and dynamics in field theory and mechanics, including concepts like symmetries and conservation laws.
Lagrangian Density: Lagrangian density is a scalar function that encapsulates the dynamics of a physical system, expressed in terms of fields and their derivatives. It serves as the foundation for the formulation of the equations of motion through the principle of least action, playing a crucial role in both classical field theories and quantum field theories. By analyzing the Lagrangian density, one can derive conservation laws and symmetries that govern the behavior of physical systems.
Lagrangian Mechanics: Lagrangian mechanics is a reformulation of classical mechanics that uses the principle of least action to derive the equations of motion for a system. Instead of focusing on forces, it emphasizes energy, specifically the difference between kinetic and potential energy, known as the Lagrangian. This approach provides a powerful framework for analyzing both mechanical systems and fields, enabling applications that range from particle dynamics to field theory.
Noether's Theorem: Noether's Theorem is a fundamental principle in theoretical physics that connects symmetries and conservation laws, stating that every continuous symmetry of a physical system corresponds to a conserved quantity. This concept bridges various areas of mechanics and field theory, revealing how invariance under transformations leads to the conservation of energy, momentum, and other quantities.
Principle of Least Action: The principle of least action states that the path taken by a system between two states is the one for which the action functional is minimized. This principle connects various aspects of physics, as it provides a unifying framework for understanding motion in both classical mechanics and quantum mechanics.
Variational Principles: Variational principles are fundamental concepts in physics and mathematics that involve finding the extrema (minimum or maximum) of a functional, often related to physical systems. These principles are widely used to derive the equations of motion and to describe the behavior of mechanical systems and fields, connecting to concepts such as energy conservation and the principle of least action.
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