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# p09fl10 - Lecture 10 Lagrange examples(25 Sep 09 A...

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Lecture 10: Lagrange examples (25 Sep 09) A. D’Alembert’s principle, Lagrange 1. Transforming Newton’s equations to eliminate forces of constraint and to reduce number of variables to n - k where k = number of constraints led to the Lagrange equations of motion ( T = kinetic energy and Φ = potential energy): L T - Φ d dt ∂L ˙ q σ - ∂L ∂q σ = 0 , σ = 1 , ..., n - k 2. This transform started from F i - ˙ p i = 0, which included all forces, then calculate the work in a virtual displacement and eliminate the forces of constraint (which do no net work when summed over all the particles). 3. This was set up for a system with constraints, but could have been sim- ply a transformation to another coordinate system, such as Cartesians spherical polar coordinates. 4. The construction assumed conservative potentials F i = -∇ i Φ. How- ever, the important special case of the Lorentz force for charged particle in an electromagnetic field also fits into the formalism even though the magnetic term has a velocity dependent force. (G. Sec 1.5, FW Sec.

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