Glossary

4.1 Functionals and variational principles

2 min read•july 25, 2024

Functionals and variational principles are powerful tools in physics and math. They map function spaces to real numbers and help optimize systems by finding functions that maximize or minimize certain quantities.

These concepts are crucial in classical mechanics, optics, and quantum physics. They lead to elegant formulations like Hamilton's principle and the , which describe how systems evolve to minimize or maximize specific quantities.

Functionals and Variational Principles

Functionals in calculus of variations

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  • Functionals map function spaces to real numbers denoted as J[f]J[f] where ff is a function (arc length, surface area, potential energy)

  • Properties of functionals encompass linearity, continuity, and differentiability

  • Calculus of variations optimizes functionals finding functions that maximize or minimize a given

  • of a functional represents change due to small function alterations analogous to derivatives in ordinary calculus

Variational principles for motion equations

  • Variational principles express fundamental physics laws as optimization problems where systems evolve to minimize or maximize certain quantities

  • Deriving motion equations through variational principles provides alternative to Newtonian mechanics leading to more elegant and general formulations

  • Hamilton's principle states system path minimizes action defined as time integral of Lagrangian

  • Fermat's principle dictates light travels along path minimizing travel time leading to reflection and refraction laws

Least action in physical systems

  • Principle of least action guides system evolution along path minimizing action S=∫t1t2L(q,q˙,t)dtS = \int_{t_1}^{t_2} L(q, \dot{q}, t) dt where LL is Lagrangian and qq are generalized coordinates

  • principle generalizes least action principle action stationary (not necessarily minimum) for physical path δS=0\delta S = 0 for small variations around true path

  • Applications span classical mechanics (particle motion, rigid body dynamics), optics (light propagation in media), and quantum mechanics ()

Derivation of Euler-Lagrange equations

  • Euler-Lagrange equations describe stationary points of functionals fundamental to calculus of variations
  • Derivation steps:
  1. Start with action functional S[q(t)]=∫t1t2L(q,q˙,t)dtS[q(t)] = \int_{t_1}^{t_2} L(q, \dot{q}, t) dt
  2. Consider variation q(t)→q(t)+δq(t)q(t) \rightarrow q(t) + \delta q(t)
  3. Expand variation of action to first order
  4. Apply integration by parts
  5. Use stationary action principle δS=0\delta S = 0
  • Resulting equations ddt(∂L∂q˙i)−∂L∂qi=0\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0 for each generalized coordinate qiq_i
  • Significance equivalent to Newton's second law in generalized coordinates forms basis for Lagrangian and Hamiltonian mechanics

Key Terms to Review (16)

Boundary Conditions: Boundary conditions are specific constraints applied to a physical system at its boundaries, which help define the behavior and solutions of differential equations that govern the system. They play a crucial role in both classical and quantum mechanics, influencing how functions and states behave at limits, ensuring that solutions are physically meaningful and consistent with the realities of the system being studied.
Convex Functional: A convex functional is a type of functional that satisfies the property of convexity, meaning that for any two points in its domain and any number between 0 and 1, the functional evaluated at a weighted average of those two points is less than or equal to the weighted average of the functional's values at those points. This property is crucial in the context of variational principles, as it often guarantees the existence and uniqueness of solutions to optimization problems in physics and mechanics.
Differentiation of Functionals: Differentiation of functionals refers to the process of finding the variation of a functional, which is a mapping from a space of functions to the real numbers, with respect to small changes in the functions themselves. This concept is essential in understanding variational principles, as it helps identify how the functional behaves when the input function is slightly altered, thereby revealing critical information about the system's dynamics and stability.
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.
Extremal: An extremal refers to a function or functional that achieves a maximum or minimum value under certain constraints. In the context of variational principles, extremals are significant because they represent the solutions to problems where one seeks to optimize a quantity, often subject to specific conditions or boundaries.
Feynman Path Integral: The Feynman Path Integral is a formulation of quantum mechanics that sums over all possible paths a particle can take between two points, assigning a probability amplitude to each path based on its action. This approach connects quantum mechanics to classical mechanics through the principle of least action and provides a framework for understanding various phenomena in physics, including propagators and statistical mechanics.
Functional: A functional is a mathematical entity that maps functions into real numbers, acting as a higher-level abstraction than regular functions. It plays a crucial role in variational principles, where the goal is often to find the function that minimizes or maximizes a given functional, ultimately leading to insights about physical systems and their behaviors. In mechanics, functionals are key when determining the paths taken by systems based on Lagrangian formulations and principles of least action.
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.
Hamiltonian Functional: The Hamiltonian functional is a mathematical construct used in classical and quantum mechanics that describes the total energy of a system in terms of its generalized coordinates and momenta. It plays a crucial role in variational principles, as it allows one to derive the equations of motion by finding the stationary points of action, making it central to the formulation of mechanics.
Lagrangian Functional: The Lagrangian functional is a mathematical construct used in physics and calculus of variations that describes the dynamics of a system by encapsulating its kinetic and potential energies. It serves as the basis for deriving the equations of motion for a system through the principle of least action, which states that the path taken by the system will minimize the action, defined as the integral of the Lagrangian functional over time. This concept connects deeply with variational principles, allowing for the analysis of complex mechanical systems in both classical and quantum contexts.
Linear Functional: A linear functional is a specific type of linear map from a vector space to its underlying field, often the real or complex numbers. It satisfies the properties of additivity and homogeneity, meaning that the functional evaluated at a sum of vectors is the sum of the functionals evaluated at each vector, and scaling a vector by a scalar scales the output of the functional by that same scalar. This concept is crucial for understanding variational principles, as linear functionals often represent quantities like energy or action in physical systems.
Path Integral Formulation: The path integral formulation is a method in quantum mechanics that represents the evolution of a quantum system as a sum over all possible paths that the system can take between two points. This approach connects classical mechanics, specifically through variational principles and the principle of least action, to quantum mechanics by considering each path's contribution to the probability amplitude, allowing for a new perspective on dynamics and interactions.
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.
Principle of stationary action: The principle of stationary action is a fundamental concept in physics stating that the path taken by a system between two states is the one for which the action is stationary (usually minimized or maximized). This principle connects the dynamics of a system to variational methods, where the action is defined as a functional that depends on the trajectory of the system over time.
Stationary Action: Stationary action refers to the principle that the path taken by a physical system between two states is the one for which the action integral is stationary, meaning it can be a minimum, maximum, or saddle point. This principle is fundamental in deriving the equations of motion in classical mechanics and connects deeply with the concept of functionals and variational principles, where action is considered a functional that maps paths to real numbers, allowing for optimization of physical trajectories.
Variation: Variation refers to the process of changing or altering a quantity or function in order to understand its effects or to find an optimal solution. In the context of functionals and variational principles, variation is crucial because it allows for the exploration of how small changes in functions can lead to changes in corresponding functionals, which are mappings from functions to real numbers.
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