4.1 Functionals and variational principles

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 principle of least action, 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 $J[f]$ where $f$ 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 functional

• Variation 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 $S = \int_{t_1}^{t_2} L(q, \dot{q}, t) dt$ where $L$ is Lagrangian and $q$ are generalized coordinates

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

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

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 $S[q(t)] = \int_{t_1}^{t_2} L(q, \dot{q}, t) dt$
2. Consider variation $q(t) \rightarrow q(t) + \delta q(t)$
3. Expand variation of action to first order
4. Apply integration by parts
5. Use stationary action principle $\delta S = 0$

• Resulting equations $\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}_i}\right) - \frac{\partial L}{\partial q_i} = 0$ for each generalized coordinate $q_i$
• Significance equivalent to Newton's second law in generalized coordinates forms basis for Lagrangian and Hamiltonian mechanics