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In physics, Lagrangian mechanics is a formulation of classical mechanics founded on the stationary-action principle (also known as the principle of least action). It was introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in his presentation to the Turin Academy of Science in 1760 [ 1 ] culminating in his 1788 ...
File:Lagrangian vs Eulerian [further explanation needed] Eulerian perspective of fluid velocity versus Lagrangian depiction of strain. In classical field theories , the Lagrangian specification of the flow field is a way of looking at fluid motion where the observer follows an individual fluid parcel as it moves through space and time.
Part I: Newtonian Mechanics Chapter 1: Experimental Facts; Chapter 2: Investigation of the Equations of Motion; Part II: Lagrangian Mechanics. Chapter 3: Variational Principles; Chapter 4: Lagrangian Mechanics on Manifolds; Chapter 5: Oscillations; Chapter 6: Rigid Bodies; Part III: Hamiltonian Mechanics. Chapter 7: Differential forms
In Lagrangian mechanics, according to Hamilton's principle of stationary action, the evolution of a physical system is described by the solutions to the Euler equation for the action of the system. In this context Euler equations are usually called Lagrange equations .
In field theory, the independent variable is replaced by an event in spacetime (x, y, z, t), or more generally still by a point s on a Riemannian manifold.The dependent variables are replaced by the value of a field at that point in spacetime (,,,) so that the equations of motion are obtained by means of an action principle, written as: =, where the action, , is a functional of the dependent ...
The dynamics of a rigid body system is described by the laws of kinematics and by the application of Newton's second law or their derivative form, Lagrangian mechanics. The solution of these equations of motion provides a description of the position, the motion and the acceleration of the individual components of the system, and overall the ...
William Rowan Hamilton re-formulated Lagrangian mechanics in 1833, resulting in Hamiltonian mechanics. In addition to the solutions of important problems in classical physics, these techniques form the basis for quantum mechanics: Lagrangian methods evolved in to the path integral formulation and the Schrödinger equation builds Hamiltonian ...
Perfect fluids admit a Lagrangian formulation, which allows the techniques used in field theory, in particular, quantization, to be applied to fluids. Perfect fluids are used in general relativity to model idealized distributions of matter, such as the interior of a star or an isotropic universe.