Why mechanics is a fundamental science: does Statics logically precede Dynamics?

Traditionally, Mechanics is divided into three parts: the Statics (a study of forces without regard of motion), Kinematics (a study of motion regardless forces), and Dynamics (the study of forces and motions in combination). Well, this is a continental classification, while in the UK and US sometimes the term Kinetics is used, combining both Statics and Dynamics.

It sounds logical that the fundamental concepts of mechanics related to forces and stresses should be defined within Statics and then used in Dynamics. However, it may not work this way logically. The reason is that it is quite difficult to define the mechanical force in statics, i.e., in the world without motion.

Both in mechanics and in thermodynamics we often deal with pairs of conjugate parameters. Typically, in each pair one parameter can be observed and measured directly whereas the definition of the other parameter is quite complex, as it cannot be observed directly and should be calculated in an indirect manner (often using the concept of energy). An example is the coordinate and the force. Both are parts of the expression for the energy: dU = F dx. Other examples will be the rotation angle and torque, or the stress and strain tensors.

In thermodynamics we have pairwise quantities in the expressions for the thermodynamic potentials, such as the Gibbs free energy

dG = V dP – S dT + μ dN + (other terms in the form Y dX)

where V is volume, P is pressure, T is the temperature, S is the entropy, μ is the chemical potential, and N is the number of molecules. One can measure directly the temperature (with a thermometer), the volume, and the number of molecules. However, to determine the pressure, entropy and chemical potential, one would need certain indirect tools. The way to measure pressure is to measure a force it creates upon a surface of a known area. To measure a force one will need a calibrated device, which converts the force into a displacement. For example, a calibrated spring with a known spring constant k would work. One can attempt using a different instrument instead of the spring, for example, to measure an electrostatic force. However, the experimentalist will end up with calibrating force vs. displacement or another observable quantity, which can change with time.

In a static world where nothing changes, it may be impossible to define what is the force!

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Keeping all this in mind, it may be interesting to revisit the argument between the physicist Frank Wilczek and mechanician Walter Noll, which I have already mentioned recently in my blog.

Wilczek brings an interesting “psychological” (= Machist) argument, why what he calls “the culture of Force” prevails in mechanics, while being considered irrelevant in physics, which instead operates with energy and momentum:

Here I conclude with some remarks on the psychological
question, why force was— and usually still is— introduced in
the foundations of mechanics, when from a logical point of
view energy would serve at least equally well, and arguably
better. The fact that changes in momentum— which
correspond, by definition, to forces— are visible, whereas
changes in energy often are not, is certainly a major factor.
Another is that, as active participants in statics— for example,
when we hold up a weight— we definitely feel we are doing
something, even though no mechanical work is performed.
Force is an abstraction of this sensory experience of exertion.
D’Alembert’s substitute, the virtual work done in response to
small displacements, is harder to relate to. (Though ironically
it is a sort of virtual work, continually made real, that explains
our exertions. When we hold a weight steady, individual
muscle fibers contract in response to feedback signals they get
from spindles; the spindles sense small displacements, which
must get compensated before they grow.4) Similar reasons
may explain why Newton used force. A big part of the
explanation for its continued use is no doubt (intellectual)

The reason, according to Wilczek, is that we can feel a static force with our muscles when we hold a weight! In order to rigorously define the concept of force within statics, one needs the conjugate parameter, the displacement. Or at least a virtual displacement, which results in a virtual work. However, the justification for this virtual displacement is that we can feel it?!

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By the way, in a much more technical sense, this applies to other areas of physics. Most people know that the electromagnetic field consists of photons. What about the electrostatic Coulomb force between two charged spheres? The quantum electrodynamics (QED) tells us that they exchange virtual photons. If we greatly simplify the quantum concept of a virtual particle and apply it to mechanics, we should say the following. In elastodynamics, a solution of a dynamic problem can be presented as a superposition of waves with various wavelengths spreading with a particular speed, which is a material constant. In statics, a solution can be presented as a Fourier-superposition of various wavelengths with zero velocity and zero frequency. These are in a sense the “virtual waves” of the QED.

To put it differently, the dispersion relation states that the wavelength, L, is equal to the phase speed of the wave, V, divided by the frequency f, or L = V / f. V is a material constant (the speed of sound). However, in the limit of low frequency, f → 0, which corresponds to the statics, a new solution emerges: V = 0 while L > 0. This zero-velocity solution is a “virtual wave”. This is similar to how a linear problem [A]x=F has two different solutions: a non-homogeneous solution x=[A]-1F when F is not zero and a non-trivial homogeneous solution when F=0 defined by eigenvalues of the matrix [A]. The former corresponds to “dynamics” while the latter corresponds to “statics.”

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To conclude: statics is not a logically self-sufficient field. This is because in statics, one cannot define energy. Although it is possible to introduce formally potentials and the potential energy, they are pointless unless you can transform potential energy into the kinetic energy. This is impossible without motion and, therefore impossible in statics. Instead, one should first introduce dynamic properties and then study statics as a special case of dynamics without motion.

Other blog entries in this series:
1. A point-mass and a rigid body
2. Are Newton’s laws laws of nature?
3. Does Statics logically precede Dynamics?
4. Can rotational dynamics be deduced from Newton’s laws?