Computational vs Analytical Mechanics

  • The reader will find no figures in this work. The methods which I set forth do not require either constructions or geometrical or mechanical reasonings: but only algebraic operations, subject to a regular and uniform rule of procedure. (Lagrange)
  • For since the fabric of the universe is most perfect and the work of a most wise Creator, nothing at all takes place in the universe in which some rule of maximum or minimum does not appear. (Euler)
  • Madam, I have come from a country where people are hanged if they talk. (Euler in Berlin to the Queen Mother of Prussia on his return from Russia)

Classical Analytical Mechanics

Newton’s mechanics initiated the scientific and industrial revolution in the late 17th century. Newton’s masterpiece was to solve the 2-body problem for a small planet orbiting around a fixed big Sun, showing the orbit to be an ellipse with the Sun in one of the focal points of the ellipse. Newton appeared like a Master of the Universe in charge of all motion according to his immutable marvellous analytical mathematics! Now man was in charge of his fate with the possibility to control the World according to his wishes, if only the analytical mathematics would work out.

But it did not, really: Newton could not even solve the 3-body problem with two planets (and nobody else either), not to speak of the N-body problem N\ge 3 bodies, and to keep the illusion of a World governed by analytical mathematics, intense efforts were made by mathematicians including in particular the top stars Euler and Lagrange to reformulate Newton’s Laws mathematically as Euler-Lagrange equations characterizing system states as minimizing Lagrangians being combinations of potential and kinetic energies. The objective of the reformulation was to describe system states with as few degrees of freedom (variables) as posssible, so that analytical solution could become possible, like for the 2-body problem but with more complicated formulas.

This developed into the discipline of analytical mechanics which has been taught as a core subject in engineering and science education with a long tradition, essentially unchanged during the last 100 (or 200) years.

Analytical mechanics is focussed on rigid-body mechanics, because the motion of a rigid (non-deformable) body can be described with few degrees of freedom, like the position of its moment of inertia and rotation around some axis, if the total mass and moment of interia is known.

High points of analytical mechanics are the 2-body and a spinning top. But problems quickly become very difficult to solve, and various tricks have been developed, which over the centuries have caused head-ache to engineering students.

Analytical mechanics is difficult: special formulation is tricky and the solution work is done by symbolic computation by pen and paper.

A standard classical course in (analytical) mechanics includes:

Contact between rigid-body mechanics is expressed as constraints on motion, which can be tricky to express mathematically.

Contact forces between rigid bodies are determined implicitely by global force balance, and thus can be tricky to compute.

Newton playing Master of the Universe

 

Computational Mechanics

The computer now opens entirely new possibilities to use Newtonian mechanics to model and simulate the World, e.g as a large N-body problem as in the Millennium Run with N=10^{10}, simply by solving Newton’s equations by a computer instead of analytical mathematics. This brings fundamental changes to the teaching, science and engineering practice of mechanics, by changing both the scope and the tools: The analytically impossible N-body problem becomes a simple computational problem, and so it goes:

Basically any thinkable problem of mechanics becomes possible to model and simulate computationally, the only limit being computer power, which is already impressive and continues to increase according to Moore’s law with doubling every 18 months.

Computational mechanics is useful: general formulation is not tricky and the solution work is done by computer.

BodyandSoul includes a lot of computational mechanics:

Contact between deformable bodies can be expressed through local elastic spring forces easily implemented in computational models.

We see that there is little overlap between a classical analytical mechanics course (special rigid-body) and modern computational mechanics (general deformable-body fluid-solid). Of course the general essentially covers also the special: Classical building is collapsing…

Perspectives

Take a look at:

And recall the Circus Cow:

showing the essence of the argument.

High point of classical analytical mechanics: Spinning top!

 

Looking Forward

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\item \hyperref[sessionanalmech]{Session: Analytical Mechanics}

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