Glossary

5.2 Lagrangian formulation and principle of least action

2 min read•july 25, 2024

The formulation is a powerful tool in classical mechanics. It simplifies complex systems by using and the difference between kinetic and potential energies. This approach often makes solving problems easier than traditional Newtonian methods.

The heart of Lagrangian mechanics is the , derived from the . By applying this equation to various systems, we can analyze everything from simple pendulums to planetary motion, uncovering conservation laws and simplifying problem-solving.

Lagrangian Formulation Fundamentals

Definition of Lagrangian function

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  • Lagrangian function L=T−VL = T - V represents difference between kinetic (T) and potential (V) energies
  • Describes system dynamics and simplifies analysis of complex systems (coupled oscillators, planetary motion)
  • Utilizes generalized coordinates to describe system configuration reducing variables needed (polar coordinates for circular motion)

Derivation of Euler-Lagrange equations

  • Principle of least action states system follows path minimizing of Lagrangian over time
  • finds extremum of action
  • Derivation steps:
    1. Write action integral S=∫t1t2L(q,q˙,t)dtS = \int_{t_1}^{t_2} L(q, \dot{q}, t) dt
    2. Consider small variations in path q(t)→q(t)+δq(t)q(t) \rightarrow q(t) + \delta q(t)
    3. Apply integration by parts
    4. Set variation of action to zero δS=0\delta S = 0
  • Resulting Euler-Lagrange equation 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

Application of Lagrangian Mechanics

Application of Euler-Lagrange equations

  • Steps to apply Euler-Lagrange equations:
    1. Identify generalized coordinates (angle for pendulum, Cartesian coordinates for particle)
    2. Express kinetic and potential energies in terms of these coordinates
    3. Form Lagrangian L=T−VL = T - V
    4. Calculate partial derivatives ∂L∂qi\frac{\partial L}{\partial q_i} and ∂L∂q˙i\frac{\partial L}{\partial \dot{q}_i}
    5. Substitute into Euler-Lagrange equation
  • Common systems: simple pendulum, double pendulum, particle in central force field (planetary motion)
  • Constraints: holonomic incorporated directly into generalized coordinates (rigid rod), non-holonomic require additional treatment (rolling without slipping)

Solutions using Lagrangian formulation

  • Analytical solutions for simple harmonic oscillator mx¨+kx=0m\ddot{x} + kx = 0 and particle in uniform gravitational field y¨=−g\ddot{y} = -g
  • Numerical methods for complex systems: Runge-Kutta, symplectic integrators preserve geometric structure
  • Conservation laws: energy conservation for time-independent Lagrangian, momentum conservation for translational symmetry
  • Cyclic coordinates not appearing explicitly in Lagrangian lead to conserved quantities (angular momentum in central force)
  • Lagrangian approach often simplifies problem-solving compared to Newtonian mechanics especially for constrained systems (bead on wire)

Key Terms to Review (16)

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.
Calculus of Variations: Calculus of variations is a field of mathematical analysis that deals with finding functions that optimize or extremize functionals, which are mappings from a space of functions to the real numbers. It is particularly important in physics and engineering as it provides a framework for deriving equations of motion and principles governing physical systems, connecting directly to the concepts of the Lagrangian formulation and the principle of least action.
Constrained Motion: Constrained motion refers to the movement of a system where the motion is restricted by specific conditions or constraints, either holonomic or non-holonomic. These constraints can limit the degrees of freedom available to the system, impacting how it evolves over time. In the context of Lagrangian mechanics, understanding constrained motion is essential because it allows for the formulation of equations of motion that respect these restrictions, leading to accurate predictions of system behavior.
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.
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.
Generalized coordinates: Generalized coordinates are a set of variables used in physics to describe the configuration of a mechanical system in terms of its degrees of freedom. They allow for a more flexible approach to modeling systems with constraints and complex movements, making it easier to apply principles like the Euler-Lagrange equations, the Lagrangian formulation, and to analyze particle dynamics and rigid body motion.
Hamiltonian dynamics: Hamiltonian dynamics is a reformulation of classical mechanics that uses the Hamiltonian function to describe the evolution of a physical system over time. This framework highlights the role of energy and transforms the equations of motion into a set of first-order differential equations, making it particularly useful in both classical and quantum contexts. The connection between Hamiltonian dynamics and other formulations like the Lagrangian approach and path integral formulation reveals deep insights into the nature of motion and quantum behavior.
Joseph-Louis Lagrange: Joseph-Louis Lagrange was an influential mathematician and physicist known for his contributions to classical mechanics, particularly in the formulation of the Lagrangian framework. His work provided powerful mathematical tools, such as the principle of least action and the method of Lagrange multipliers, which help to analyze systems with constraints and derive equations of motion in a systematic way.
Lagrangian: The Lagrangian is a mathematical function that summarizes the dynamics of a system in classical mechanics, defined as the difference between the kinetic and potential energy of the system. This function plays a crucial role in formulating equations of motion through the principle of least action, where it helps derive equations that describe how systems evolve over time.
Mechanical Systems: Mechanical systems refer to a collection of interconnected components that work together to perform a specific function, typically involving the principles of motion, forces, and energy. In the context of physics, these systems can be analyzed using various formulations, especially the Lagrangian formulation, which allows for the derivation of equations of motion based on energy rather than forces. This approach highlights the importance of constraints and the principle of least action, which states that the path taken by a system is the one that minimizes action, effectively unifying mechanics through an energy-focused lens.
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.
Pendulum Dynamics: Pendulum dynamics refers to the study of the motion of a pendulum, which is a mass suspended from a fixed point that swings back and forth under the influence of gravity. This concept is essential in understanding how forces and energy interact in a system, allowing us to apply principles such as energy conservation and oscillatory motion in various mechanical systems.
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.
Quantum Lagrangian: The Quantum Lagrangian is a mathematical formulation used to describe the dynamics of quantum systems, extending the classical Lagrangian framework into the quantum realm. It incorporates principles of quantum mechanics, enabling the derivation of equations of motion and predictions about physical phenomena through the path integral formulation. This connects deeply with the principle of least action, which asserts that the path taken by a system is the one that minimizes action, providing a bridge between classical and quantum descriptions.
Richard Feynman: Richard Feynman was a renowned American theoretical physicist known for his contributions to quantum mechanics and quantum electrodynamics. His innovative ideas, such as the path integral formulation, revolutionized our understanding of particle physics and made complex concepts more accessible through intuitive visuals like Feynman diagrams.
Variational Principle: The variational principle is a fundamental concept in physics that states the path taken by a system between two states is the one for which a certain quantity, typically the action, is minimized or made stationary. This principle underlies several formulations of mechanics and quantum mechanics, emphasizing that systems tend to evolve in a way that optimizes certain quantities. It connects to various concepts such as the dynamics of systems, energy conservation, and perturbation effects in quantum systems.
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