Understand Hamilton's Principle Intuitively

In summary, Hamilton's principle is a mathematical observation that can be derived from a variational principle in classical mechanics. In terms of quantum theory, it can be understood as a path integral over all paths with the action of the analogous classical system. The Lagrangian does not represent a physical quantity in classical mechanics, but rather serves as a mathematical tool. The principle of least action means finding the one path where the change in kinetic energy matches the change in potential energy at all times.
  • #1
tut_einstein
31
0
While I understand how the Euler-Lagrange equations are derived by minimizing the integral of the Lagrangian, I don't intuitively understand why Hamilton's principle is true. Specifically, what physical quantity does the Lagrangian represent and what does minimizing it mean? I'd just like to get an intuitive feel for it, I understand it mathematically.

Thanks!
 
Physics news on Phys.org
  • #2
The deeper reason for why Hamilton's principle is true is mysterious within classical mechanics. It's just a mathematical observation that one can derive many equations of motion from a variational principle.

From the physical point of view you have to use quantum theory to understand why it holds true. If you formulate quantum theory in terms of Feynman's path integral in many cases it boils down to a path integral for the propagator of the particle over all paths with the action of the analogous classical system. In the case that the system is close to classical behavior, the action divided by [itex]\hbar[/itex] is very steeply rising around the classical path, along which it is stationary, and this means that the path integral is dominated by paths very close to the classical one, which explains why the equations of motion follow from the Hamilton's principle, which says nothing else than that the classical path is given by the stationary point of the action functional.
 
  • #3
tut_einstein said:
[...] I don't intuitively understand why Hamilton's principle is true. Specifically, what physical quantity does the Lagrangian represent and what does minimizing it mean?

The way I understand it the Lagrangian does not in itself represent a physical quantity. In classical mechanics the Lagrangian is a mathematical tool, with no direct physical counterpart.

As a starting point I recommend the following page (on Edwin Taylor's website), the page was created by Edwin Taylor and Slavomir Tuleja: Principle of least action interactive

The why of Hamilton's principle of least action is a recurrent question.
What does it mean to minimize the action?
I posted about this subject in november 2010, in a thread titled 'Lagrangian'

The mathematical discussion that I posted here on physicsforums is a shortened one, a more extensive discussion is on my own website.



Finding the one path for which the action is minimal is mathematically the same as finding the one path with the property that at all times the amount of change in kinetic energy is a match for the amount of change in potential energy.

The change in kinetic energy and the change in potential energy have to match; when an object traverses a potential field any gain/loss of kinetic energy must match the decrease/increase of potential energy.
 

1. What is Hamilton's Principle?

Hamilton's Principle is a fundamental principle in mechanics that states that the true path a system will take between two points in time is the one that minimizes the action function.

2. How does Hamilton's Principle relate to classical mechanics?

Hamilton's Principle is a generalization of Newton's laws of motion and can be used to derive the equations of motion for a system in classical mechanics.

3. Can Hamilton's Principle be understood intuitively?

Yes, Hamilton's Principle can be understood intuitively by thinking of it as a principle of least action. In other words, a system will always take the path that requires the least amount of effort or energy to reach its final state.

4. What is the significance of Hamilton's Principle in physics?

Hamilton's Principle is significant in physics because it provides a powerful tool for analyzing and predicting the behavior of physical systems. It has been applied in various fields, including mechanics, electromagnetics, and quantum mechanics.

5. How can Hamilton's Principle be applied in real-world situations?

Hamilton's Principle can be applied in real-world situations by using it to derive the equations of motion for a given system and then using those equations to make predictions and solve problems related to the system's behavior. It has been successfully applied in fields such as aerospace engineering, robotics, and thermodynamics.

Similar threads

Replies
25
Views
1K
Replies
1
Views
562
Replies
7
Views
737
  • Classical Physics
Replies
18
Views
1K
Replies
13
Views
872
Replies
2
Views
605
Replies
17
Views
2K
  • Classical Physics
Replies
20
Views
2K
Replies
3
Views
463
Back
Top