Lagrangian mechanics for nanomechanical circuits
When thinking about nanotechnology one usually thinks about quantum mechanics and Hamiltonian mechanics.
But because nanomechanics is barely mechanical quantummechanics (it rather behaves quite classical)
the Lagrangian mechanics formalism is the one fitting to the problem.
Basically a classical network of axles, gears, springs, masses, and differentials taking the role of branching points.
See: The mechanoelectrical correspondence
(wiki-TODO: add graphic with example system, requivalent circuit diagram, and some numbers)
Contents
Springs are dominant over flywheels
Operation frequencies are mainly determined by going slow to keeping friction losses
rather than being determined by deflections and vibrations from accelerations.
(See relatedpage: Same relative deflections across scales)
Proposed typical speeds are around ~1Mhz & ~5mm/s (see related page atomplacement frequency]]).
In free swinging unhampered spring-flywheel resonators without giant flywheels
the natural frequencies would be ways too high due to a lack of inertial mass of small flywheels.
You want a soft spring and a high flywheel inertia for a low frequency.
Other strategies like mechanical pulse width modulation maybe employable.
Advantages Lagrangian
- Ability to model non-conservative systems with dissipative systems and or non-potential forces (non gradient force-fields with nonzero rotor)
- Singular systems or systems with degeneracies in their configuration space systems with constraints
- may offer a more robust and versatile approach in pathological or chaotic systems
Why not Lagrangian for quantum mechanics
- Loss of (differential) operator formalism and commutation relations:
Commutation relations between operators determine the uncertainty relations.
What is measurable simultaneously and what is not. Complete sets of eigenvalues that are the observables and the result of wave collapse in a measurement process. - Non-conservative systems can be described but there are none. Everything is reversible at the smallest scales.
Related
- Mechanical-electrical analogies
- Nanomechanic circuits
- Mechanical circuit element
- Mechanical pulse width modulation
- Nanomechanics is barely mechanical quantummechanics
