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11: Calculus of Variations

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    The previous chapters surveyed various energy conversion devices. The purpose of Chapters 11 and 12 is to establish a general framework to describe any energy conversion process. By placing energy conversion processes in a larger framework, we may be able to see relationships between processes or identify additional energy conversion processes to study. Establishing this framework requires some abstraction and hence some mathematics. In the next section, we define the Principle of Least Action and the idea of calculus of variations. In the following sections, we apply these ideas to two example energy conversion systems: a mass spring system and a capacitor inductor system.

    An advantage of using calculus of variations over other techniques is that the analysis is based on energy, which is a scalar, instead of the potential, which may be a scalar or vector. Working with a scalar quantity like energy instead of a vector can make the mathematics quite a bit more manageable.

    • 11.1: Lagrangian and Hamiltonian
      This page covers the conversion of energy between different forms, emphasizing the generalized path \(y(t)\) that evolves over time or variables. It introduces the Lagrangian \(\mathcal{L}\) as the energy difference and the Hamiltonian \(H\) as the total energy. It distinguishes generalized potential from potential energy, defines generalized momentum, and presents key concepts like conservation of energy and generalized capacity, underscoring their importance in energy conversion processes.
    • 11.2: Principle of Least Action
      This page defines the action \(\mathbb{S}\) as the integral of the Lagrangian \(\mathcal{L}\) and explains the Principle of Least Action, which asserts nature chooses paths that minimize action. It discusses the calculus of variations used to derive the Euler-Lagrange equation, which relates to equations of motion and connects energy conversion and conservation laws.
    • 11.3: Derivation of the Euler-Lagrange Equation
      This page covers the derivation and significance of the Euler-Lagrange equation from the Principle of Least Action, emphasizing its connection to Hamilton's equations. It also explains the importance of the second-order Euler-Lagrange equation in solving variational problems, underlining its utility in classical mechanics and dynamic system analysis.
    • 11.4: Mass Spring Example
      This page covers the application of the Euler-Lagrange equation to derive motion equations for a mass-spring system, explaining energy transformations and introducing Hamiltonian and Lagrangian concepts. It showcases the system's oscillatory behavior via a second-order linear differential equation. Additionally, it derives Hamilton's equations from the Euler-Lagrange equation, linking generalized paths to energy storage, represented through first-order differential equations.
    • 11.5: Capacitor Inductor Example
      This page explores the calculus of variations in an electrical circuit featuring a capacitor and inductor, focusing on energy conversion between electrical and magnetic energy. It discusses the relationships between current, voltage, charge, and energy storage for both components, introducing Hamiltonian and Lagrangian methods to derive motion equations that exemplify oscillatory behavior. The page emphasizes energy conservation throughout the system's dynamics.
    • 11.6: Schrödinger's Equation
      This page covers the fundamentals of quantum mechanics, emphasizing microscopic systems such as electrons and atoms. It discusses the Hamiltonian's role and calculus of variations, explaining how the wave function (\(\psi\)) predicts particle probabilities in energy states. The Hamiltonian combines kinetic and potential energy to derive the time-independent Schrödinger equation, which is central to the subject.
    • 11.7: Problems
      This page explores functions and functionals in physics, focusing on mechanics and action principles. It includes examples like parabolas and Lagrangians, with applications such as deriving equations of motion, energy conservation in systems like torsion springs and optics.

    Thumbnail: Minimizing function and trial functions. (CC BY-SA 2.5; Banerjee).

    This page titled 11: Calculus of Variations is shared under a CC BY-NC 4.0 license and was authored, remixed, and/or curated by Andrea M. Mitofsky via source content that was edited to the style and standards of the LibreTexts platform.