Are there real-world applications of Hamilton's Principle in mechanics?

In summary, Hamilton's Principle, which states that the actual path taken by a system is the one that minimizes the action, has various real-world applications in mechanics. It is utilized in fields such as classical mechanics, quantum mechanics, and engineering to derive equations of motion and optimize systems. Applications include the design of mechanical systems, understanding dynamic systems in robotics, and analyzing complex physical phenomena in theoretical physics. Its versatility allows for the formulation of problems in different contexts, making it a fundamental principle in both theoretical and applied mechanics.
  • #1
Trying2Learn
377
57
TL;DR Summary
From the schematic to the real.
Goooood Morning all!

I am going through a few problems in advanced dynamics with Hamilton's Principle.

Skjermbilde.JPG


One of them is shown above (this is NOT a question about the solution)

The spring constants, damping, mass, force, are all given. So, too, is the constraint: large disk rolls without slipping.

My question is this: Does such a mechanical device exist?

I suppose one could replace the "rolling without slipping" constraint with "rack and pinion" gear set.

  1. But is this specific schematic representative of some real device?
  2. And, more generally, but just as important to me: as I go through such examples, how can I find out if the schematics for the varied problems exists?
  3. Or should I content myself with realizing that these are indeed simple schematics to learn the language of how to apply Hamilton's Principle? (much like found in electrical circuitry examples)
 
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  • #2
Trying2Learn said:
TL;DR Summary: From the schematic to the real.

My question is this: Does such a mechanical device exist?
I am not sure about semantics of "exist" you use. The system seems reasonable and realizable at least theoretically to me.
 
  • #3
anuttarasammyak said:
I am not sure about semantics of "exist" you use. The system seems reasonable and realizable at least theoretically to me.
No, what I mean is: is it a schematic from a useful mechanism
 
  • #4
Trying2Learn said:
TL;DR Summary: From the schematic to the real.

Goooood Morning all!

I am going through a few problems in advanced dynamics with Hamilton's Principle.

View attachment 332084

One of them is shown above (this is NOT a question about the solution)

The spring constants, damping, mass, force, are all given. So, too, is the constraint: large disk rolls without slipping.

My question is this: Does such a mechanical device exist?

I suppose one could replace the "rolling without slipping" constraint with "rack and pinion" gear set.

  1. But is this specific schematic representative of some real device?
  2. And, more generally, but just as important to me: as I go through such examples, how can I find out if the schematics for the varied problems exists?
  3. Or should I content myself with realizing that these are indeed simple schematics to learn the language of how to apply Hamilton's Principle? (much like found in electrical circuitry examples)
Looks pretty much like that you could build it. Why shouldn't it be possible?
 
  • #5
vanhees71 said:
Looks pretty much like that you could build it. Why shouldn't it be possible?
Yes, I know I can build it, but that is not the question.

I see all these examples in a textbook on Hamilton's Principle.

With some of them, I want to motivate the students by saying

"Yes, we could build it, but it would not be purposeful/useful."

I am trying to find out if this mechanism actually does serve a real world purpose.

In fact, I am hoping to do that for MANY problems such as these.

Or, must I content myself, that if I add more springs or dashpots, all am doing is giving varied problems that have no real-world counterpart, but are ONLY useful to learn the language of Hamilton's Principle (a worthy endeavor, nonetheless).
 
  • #6
Hm, it's just to train the students to solve non-trivial mechanical problems. I don't know, what you mean by "real-world purpose".
 
  • #7
By "real world purpose" I only meant: "can I say to the students: yes, this is an ideal problem to learn the workings of HP, but in THIS particular case, THIS mechanism is also used for this purpose."

But I am beginning to think most of the examples given in textbooks that teach HP, are indeed didactic constructions and not related to actual useful mechanisms.
 
  • #8
A nice example is the gyrocompass at the very end of the mechanics lecture, i.e., the motion of a (restricted) top on the rotating Earth ;-).
 
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FAQ: Are there real-world applications of Hamilton's Principle in mechanics?

What is Hamilton's Principle in mechanics?

Hamilton's Principle, also known as the Principle of Stationary Action, states that the actual path taken by a mechanical system between two states is the one for which the action integral is stationary (usually a minimum). The action integral is defined as the integral of the Lagrangian (difference between kinetic and potential energy) over time.

How is Hamilton's Principle applied in classical mechanics?

In classical mechanics, Hamilton's Principle is used to derive the equations of motion for a system. By applying the principle, one can obtain the Euler-Lagrange equations, which describe the dynamics of the system. This approach is particularly useful for complex systems where direct application of Newton's laws would be cumbersome.

Are there applications of Hamilton's Principle in modern physics?

Yes, Hamilton's Principle extends beyond classical mechanics and is fundamental in modern physics, including quantum mechanics and general relativity. In quantum mechanics, it underpins the path integral formulation, while in general relativity, it is used to derive Einstein's field equations from the action principle.

Can Hamilton's Principle be used in engineering?

Absolutely. In engineering, Hamilton's Principle is applied in fields like structural mechanics, aerospace engineering, and robotics. It is used to optimize the design and control of mechanical systems, such as determining the optimal shape of a structure to minimize stress or the best trajectory for a robotic arm.

What are some real-world examples of Hamilton's Principle in action?

Real-world examples of Hamilton's Principle include the design of stable satellite orbits, optimizing the performance of mechanical linkages in machinery, and even in computer graphics for simulating realistic motion. It is also used in the development of algorithms for autonomous vehicles to determine efficient and safe paths.

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