The prior chapters have focussed on the intuitive Newtonian approach to classical mechanics, which is based on vector quantities like force, momentum, and acceleration. Newtonian mechanics leads to se...The prior chapters have focussed on the intuitive Newtonian approach to classical mechanics, which is based on vector quantities like force, momentum, and acceleration. Newtonian mechanics leads to second-order differential equations of motion. The calculus of variations underlies a powerful alternative approach to classical mechanics that is based on identifying the path that minimizes an integral quantity.
In general, and unless \(f\) is a function of \(x\) and \(y\) alone, and not of \(y'\), the value of this integral will depend on the route (i.e.\(y = y(x)\) ) over which this line integral is calcula...In general, and unless \(f\) is a function of \(x\) and \(y\) alone, and not of \(y'\), the value of this integral will depend on the route (i.e.\(y = y(x)\) ) over which this line integral is calculated. An element \(ds\) of the curve can be written as \( \sqrt{1 + y'^2 dx} \) and the distance moved by the element \(ds\) (which is at a distance \(x\) from the \(y\)-axis) during the rotation is \(\phi x\).
