Lagrangian mechanics | Wikipedia audio article
Channel: wikipedia tts
Date: 2018-12-08

This is an audio version of the Wikipedia Article:
Lagrangian mechanics


00:02:44 1 Introduction
00:04:08 2 From Newtonian to Lagrangian mechanics
00:06:40 2.1 Newton's laws
00:07:27 2.2 D'Alembert's principle
00:08:35 2.3 Equations of motion from D'Alembert's principle
00:08:45 2.4 Euler–Lagrange equations and Hamilton's principle
00:12:52 2.5 Lagrange multipliers and constraints
00:16:46 3 Properties of the Euler–Lagrange equation
00:16:56 3.1 Non-uniqueness
00:23:12 3.2 Invariance under point transformations
00:26:56 3.3 Cyclic coordinates and conserved momenta
00:27:12 3.4 Energy conservation
00:34:11 3.5 Mechanical similarity
00:40:01 3.6 Interacting particles
00:43:11 4 Examples
00:44:43 4.1 Conservative force
00:45:11 4.1.1 Cartesian coordinates
00:53:28 4.1.2 Polar coordinates in 2d and 3d
00:53:50 4.2 Pendulum on a movable support
00:54:51 4.3 Two-body central force problem
00:56:25 4.4 Electromagnetism
00:59:43 5 Extensions to include non-conservative forces
01:03:19 6 Other contexts and formulations
01:05:52 6.1 Alternative formulations of classical mechanics
01:07:33 6.2 Momentum space formulation
01:07:49 6.3 Higher derivatives of generalized coordinates
01:08:29 6.4 Optics
01:11:29 6.5 Relativistic formulation
01:14:58 6.6 Quantum mechanics
01:22:42 6.7 Classical field theory
01:23:15 6.8 Noether's theorem
01:24:22 7 See also
01:31:06 8 Footnotes
01:31:47 9 Notes
01:32:03 10 References
01:34:38 11 Further reading
01:37:14 12 External links
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SUMMARY
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Lagrangian mechanics is a reformulation of classical mechanics, introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange in 1788.
In Lagrangian mechanics, the trajectory of a system of particles is derived by solving the Lagrange equations in one of two forms, either the Lagrange equations of the first kind, which treat constraints explicitly as extra equations, often using Lagrange multipliers; or the Lagrange equations of the second kind, which incorporate the constraints directly by judicious choice of generalized coordinates. In each case, a mathematical function called the Lagrangian is a function of the generalized coordinates, their time derivatives, and time, and contains the information about the dynamics of the system.
No new physics are necessarily introduced in applying Lagrangian mechanics compared to Newtonian mechanics. It is, however, more mathematically sophisticated and systematic. Newton's laws can include non-conservative forces like friction; however, they must include constraint forces explicitly and are best suited to Cartesian coordinates. Lagrangian mechanics is ideal for systems with conservative forces and for bypassing constraint forces in any coordinate system. Dissipative and driven forces can be accounted for by splitting the external forces into a sum of potential and non-potential forces, leading to a set of modified Euler–Lagrange (EL) equations. Generalized coordinates can be chosen by convenience, to exploit symmetries in the system or the geometry of the constraints, which may simplify solving for the motion of the system. Lagrangian mechanics also reveals conserved quantities and their symmetries in a direct way, as a special case of Noether's theorem.
Lagrangian mechanics is important not just for its broad applications, but also for its role in advancing deep understanding of physics. Although Lagrange only sought to describe classical mechanics in his treatise Mécanique analytique, William Rowan Hamilton later developed Hamilton's principle that can be used to derive the Lagrange equation and was later recognized to be applicable to much of fundamental theoretical physics as well, particularly quantum mechanics and the theory of relativity. It can also be applied to other systems by analogy, for instance to coupled ...