• Bookmarks

    Bookmarks

  • Concepts

    Concepts

  • Activity

    Activity

  • Courses

    Courses


The Action Principle is a fundamental concept in physics that states the path taken by a system is the one for which the action is stationary, typically a minimum. This principle underlies the derivation of equations of motion in classical mechanics, quantum mechanics, and field theory, unifying various physical laws under a common framework.
Lagrangian Mechanics is a reformulation of classical mechanics that provides a powerful framework for analyzing the dynamics of systems by focusing on energy rather than forces. It uses the principle of least action to derive equations of motion, making it particularly useful for complex systems and systems with constraints.
Hamiltonian Mechanics is a reformulation of classical mechanics that provides a powerful framework for analyzing the dynamics of systems, particularly in contexts where energy conservation is more natural to describe than forces. It uses the Hamiltonian function, which typically represents the total energy of the system, to derive equations of motion that are often simpler to solve than their Lagrangian counterparts.
The Euler-Lagrange equation is a fundamental equation of calculus of variations, used to find the path that minimizes or extremizes a functional. It is essential in deriving the equations of motion in classical mechanics and plays a crucial role in fields like physics, engineering, and optimization problems.
The variational principle is a fundamental concept in physics and mathematics that determines the state or path of a system by finding the extrema of a functional, often representing energy. It is widely used in fields like quantum mechanics, classical mechanics, and general relativity to derive equations of motion or field equations by minimizing or maximizing an action integral.
The path integral formulation is a quantum mechanics framework that generalizes the action principle of classical mechanics, allowing for the calculation of quantum amplitudes by summing over all possible paths a particle can take. Developed by Richard Feynman, it provides a powerful tool for understanding quantum field theory and has applications in statistical mechanics and quantum gravity.
Stationary action, also known as the principle of least action, is a fundamental principle in physics that states the path taken by a system between two states is the one for which the action is stationary (usually a minimum). This principle underlies the equations of motion in classical mechanics, quantum mechanics, and general relativity, providing a unifying framework for understanding physical laws.
Classical mechanics is a branch of physics that deals with the motion of bodies under the influence of force, providing a framework for understanding the physical world from the macroscopic to the astronomical scale. It is based on principles such as Newton's laws of motion and the conservation of energy, serving as the foundation for more advanced theories like quantum mechanics and relativity.
Quantum mechanics is a fundamental theory in physics that describes the physical properties of nature at the smallest scales, such as atoms and subatomic particles. It introduces concepts like wave-particle duality, uncertainty principle, and quantum entanglement, which challenge classical intuitions about the behavior of matter and energy.
Field theory is a framework used in physics and mathematics to describe how physical quantities are distributed in space and time, often represented by fields such as electromagnetic or gravitational fields. It provides a systematic approach to understanding interactions and dynamics in systems, crucial for theories like electromagnetism, quantum Field theory, and general relativity.
Variation technique is a method used in mathematics and physics to find functions that optimize or extremize a given functional, often applied in calculus of variations and quantum mechanics. It provides a framework for solving problems where the solution is not a single value but a function, enabling the determination of paths, surfaces, or other quantities that satisfy specific criteria.
Hamilton's Equations provide a reformulation of classical mechanics that is particularly powerful in theoretical physics, offering a bridge between classical and quantum mechanics. They describe the evolution of a system in terms of generalized coordinates and momenta, making them highly useful for systems with constraints and in the study of dynamical systems.
The Einstein-Hilbert Action is a fundamental principle in general relativity that provides the simplest action from which the Einstein field equations can be derived, describing how matter and energy influence spacetime curvature. It is expressed as an integral of the Ricci scalar curvature over a manifold, weighted by the determinant of the metric tensor, and is central to understanding the dynamics of gravitational fields in the framework of modern physics.
Lagrangian density is a function that encapsulates the dynamics of a field in classical and quantum field theories, providing the equations of motion through the Euler-Lagrange equations. It is the field-theoretic generalization of the Lagrangian function used in classical mechanics, and its integral over space yields the action, a central quantity in determining the evolution of physical systems.
Variational calculus is a field of mathematical analysis that deals with finding extrema of functionals, which are mappings from a set of functions to the real numbers. It is fundamental in deriving equations of motion and optimizing systems in physics, economics, and engineering through principles like the Euler-Lagrange equation.
Path integrals, introduced by Richard Feynman, provide a formulation of quantum mechanics where probabilities are computed as sums over all possible paths a particle can take between two points. This approach is particularly powerful in quantum field theory and statistical mechanics, offering intuitive insights and computational techniques for complex systems.
The Euler-Lagrange equations are fundamental in the calculus of variations and provide necessary conditions for a functional to have an extremum, often used to derive equations of motion in classical mechanics. They are derived from the principle of stationary action and are pivotal in fields like physics and engineering for optimizing systems and processes.
The Feynman Path Integral is a formulation of quantum mechanics that expresses the probability amplitude for a particle's state as a sum over all possible paths the particle could take, weighted by an exponential of the classical action. This approach provides deep insights into quantum field theory and has applications in various areas of physics, including statistical mechanics and quantum gravity.
Functional differentiation is a mathematical process used to determine the derivative of a functional, which is a mapping from a space of functions to the real numbers. This concept is crucial in fields like calculus of variations and quantum mechanics, where it helps in finding functions that optimize certain criteria or describe physical systems.
Path Integral Quantization is a formulation of quantum mechanics that generalizes the action principle of classical mechanics, allowing for the calculation of quantum amplitudes by summing over all possible paths a particle can take. It provides a powerful framework for understanding quantum field theory and the transition from quantum to classical behavior through the principle of least action.
Classical Field Theory is a framework in physics that describes how fields, such as the electromagnetic field, propagate and interact in space and time without invoking quantum mechanics. It serves as the foundation for understanding various physical phenomena through differential equations that govern the dynamics of these fields.
3