# The Calculus of Variations

# The Calculus of Variations

This chapter introduces the calculus of variations in the context of the finite-dimensional configuration space discussed previously. The calculus of variations is concerned with the comparison of line integrals along different paths. The difference between the integral along some chosen path and the integral of the same quantity along other paths is called the variation of that integral. This chapter discusses paths in an *N*-dimensional space, variations of coordinates, variations of functions, variation of a line integral, finding extremum paths, example of an extremum path calculation, invariance and homogeneity, the Brachistochrone problem, calculus of variations with constraints, example with constraints, reduction of degrees of freedom, example of a reduction, example of a better reduction, and the coordinate parametric method.

*Keywords:*
calculus of variations, configuration space, line integrals, paths, extremum paths, invariance, constraints, degrees of freedom, coordinate parametric method, Brachistochrone problem

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