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pccrp

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pccrp

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Another method is to keep the Cartesian coordinates [itex]\vec{r}_i[/itex] and implement the constraints with help of Lagrange multipliers. Then you can vary the [itex]\delta \vec{r}_i[/itex] independently. This leads to equations of motion, where additional forces from the constraints are taken into account.

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UltrafastPED

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The Principle of Virtual Work tells us that constraints do no work.

D'Alembert's Principle permits the impressed forces of dynamical systems to be handled in the same way as constraints - the application of this is to find the unknown trajectories caused by the impressed forces via the calculus of variations - thanks to Euler and Lagrange.

For a system in equilibrium the independence of the coordinates tells us that each of the terms must be zero; for dynamics it is only the sum which must be zero.

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pccrp

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But why in dynamic situations only the sum must be zero, and not each member of the sum?

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UltrafastPED

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In the dynamical case you have a difference which must be zero ... but now the expression depends upon the trajectories via Newton's second law of motion: so yes, each term must go to zero, and this is the variational principle which is used to find those trajectories.

Sorry if my earlier statement was misleading and unclear.

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