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This scenario is plagued by the choice of convenient coordinates, the coupling of the two pendulums, and the constraint that keeps the two pendulums from breaking apart.Įach of these problems would be extremely difficult to resolve by the usual approach of identifying forces, writing down constraints, transforming to a convenient system of coordinates, and possibly uncoupling equations by clever substitutions. Finding a representation that is at once easy to think about, and straightforward to calculate in, is no easy task.Īs a final example, consider the coupled pendulum, where one pendulum hangs from the end of another. This situation does not easily yield to a vector description because gravity beckons us to use Cartesian coordinates, while the constraint of the hoop begs that we use polar coordinates.

R G MECHANICS FREE

We might also consider a bead that is free to slide on a rotating hoop. The second is that the normal force between M M M and m m m depends upon the motion of M M M and m m m. The first is that the mass M M M moves along the floor while m m m moves at an angle, which creates a tension in the choice of coordinates (the Cartesian grid in the frame of the ground, and another in the frame of the tilted surface). Setting this problem up is difficult for two reasons. Both the block and the wedge are free to move without friction under the force of gravity.

Without making an attempt at solution, we point out some challenging problems for Newton's approach.Ĭonsider the block-shaped mass on a sliding wedge below. Let's begin by reviewing some hard problems of Newtonian mechanics and pointing out what makes them so difficult to resolve. Such is the aim of the Lagrangian formulation of mechanics. That is, to write down numbers like mass, energy, or momentum squared which are invariant under a change in coordinates. What we imagine is the ability to describe our systems in terms of scalars, instead of vectors. The scheme would be more useful if our important quantities could be easily rewritten and solved in the most convenient coordinate system. Moreover, choosing the wrong reference frame can give rise to confusing artifacts. We also need an insightful choice of coordinate system to simplify all our calculations. One can of course rewrite a vector in any system of coordinates, but it is not the most transparent operation. This all stems from the fact that Newton's laws are written in terms of vector quantities that are easiest to use in Cartesian coordinates. The main difficulty in applying the Newtonian algorithm is in identifying all the forces between objects, which requires some ingenuity. Everything from celestial mechanics to rotational motion, to the ideal gas law, can be explained by the powerful principles that Newton wrote down. Newton's laws of motion are the foundation on which all of classical mechanics is built.

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