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 showing: equivalent circuit diagram, and some numbers.)
Very complex mechanical systems can be easily mathematically modeled for simulation. Like e.g. The Drive system of a gem-gum factory linking
- many chemomechanical converters
- across temporary elastic buffer storage springs
- to the many mechanosynthetic cores
via mechanical differentials that act (in analogy to electrical wire-junctions) as distribution nodes to and from common transmission lines
- reducing losses and
- increasing system stiffness
- allowing for exothermy offloading and eventually even for dissipation sharing
Systems descriptions (typically coupled second order differential equations)
can be split into a number of decoupled first order differential equations.
So long they are not non-holonomic.
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.
Analysis of high frequency shocks
This is especially for analysis of propagation of high frequency shocks through the system.
For these the very small inertia of axles an cogs and gears may eventually become relevant.
These should probably be avoided in most cases to save energy.
See: Snapback
Normal proposed operation speeds are low (~1MHz ~5mm/s).
These are likely simulatable/calculatable with simpler methods.
As simple as Kirchhoffs law and such.
Instructions for usage
Used ChatGPT to jog my memory here. Seems plausible.
Using Lagrangian Formalism for Simulating Complex Mechanical Systems
The Lagrangian formalism is a powerful approach for simulating complex mechanical systems. Here's a step-by-step guide:
- Define the Lagrangian of the system as the difference between the kinetic energy and potential energy.
- Identify the generalized coordinates, denoted as q = (q1, q2, ..., qn), and their time derivatives, q̇.
- Apply the Euler-Lagrange equations to derive the equations of motion.
- Introduce new variables u = (u1, u2, ..., un) representing the generalized velocities.
- Express the time derivatives of the generalized coordinates q̇ in terms of the variables u.
- Substitute the expressions for q̇ into the original equations of motion.
- Rewrite the resulting coupled second-order differential equations as a set of decoupled first-order differential equations in terms of q and u.
- Use the state-space representation of the system for numerical simulation and analysis.
Note: Nonholonomic systems may require additional considerations and may not be able to be decoupled into first-order differential equations.
Systems of interest
Tips on a molecular mill wheel doing mechanochemistry provide only intermittent torque and sometimes even the wrong polarity.
Connecting wheels directly via mechanical differential would drive back other wheels in the longer frewheeling pauses
or even cancel out opposite with drive in the opposite direction.
The obvious thing to do is to couple a number of wheels as stiffly together as possible with slight phase shifts such that one overall gets a continuous positive torque in the desired direction. Only then one couples various grouped sources together to larger transmission line(s) via mechanical differentials.
The design needs to be such that if one group mill-wheels attempts to back-drive an other group,
then this other group needs to increase resistance to that.
What is desired here is a negative feedback self-regulation.
This is related to the reason why LEDs need resistors in series before putting them in parallel.
Otherwise their slight differences in threshold voltage leads to huge differences in current, leaving some dark and some burning through.
Though we want to avoid resistive dissipation elements as much as possible to stay energy efficient.
Especially at the nanoscale transmitting powers of single chemical reactions with small margins.
So other solutions than friction are preferable
Related: Drive subsystem of a gem-gum factory
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
