Principle of Least Action OR Hamilton's Principle

In summary, the principle of least action and the Hamilton principle are essentially the same, with the latter being a more general form of the former. The potential energy in the Hamilton principle can be written as a constant, making it equivalent to the Lagrangian in the case of orbits in celestial mechanics. However, there is also a freedom of a total time derivative in the Lagrangian, which can affect the potential energy in certain cases. The concept of reduced action and Maupertuis' principle can also provide further clarification on these principles.
  • #1
shehry1
44
0
Are the principle of least action(http://astro.berkeley.edu/~converse/Lagrange/Kepler%27sFirstLaw.htm) and the hamilton principle 'exactly' the same? As far as I know, yes. How do I go from one to the other
 
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  • #2
i believe 'Hamilton Principle' generalizes the 'Least Action principle'
 
  • #3
It looks like the first equation that you refer to is a special case of Hamilton's principle, where the potential energy is taken to be constant. In the case of [tex] U = c [/tex], the Lagrangian is simply [tex] (1/2)mv^2 + c [/tex], and hamilton's principle becomes

[tex] \delta \int \frac{1}{2} mv^2 {\mathrm d}t = 0 \Longrightarrow \delta \int mv^2 {\mathrm d}t = 0 [/tex].

Since [tex] dt = {dx}/{v} [/tex], this is equivalent to

[tex] \delta \int mv {\mathrm d}x = 0 [/tex].

This is usually written in the form

[tex] \delta \int p {\mathrm d}q = 0 [/tex]

to emphasize that q is a generalized coordinate.
 
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  • #4
dx said:
It looks like the first equation that you refer to is a special case of Hamilton's principle, where the potential energy is taken to be constant. In the case of [tex] U = c [/tex], the Lagrangian is simply [tex] (1/2)mv^2 + c [/tex], and hamilton's principle becomes

[tex] \delta \int \frac{1}{2} mv^2 {\mathrm d}t = 0 \Longrightarrow \delta \int mv^2 {\mathrm d}t = 0 [/tex].

Since [tex] dt = {dx}/{v} [/tex], this is equivalent to

[tex] \delta \int mv {\mathrm d}x = 0 [/tex].

This is usually written in the form

[tex] \delta \int p {\mathrm d}q = 0 [/tex]

to emphasize that q is a generalized coordinate.

I wondered whether that was the case. However, this being celestial mechanics, it is obvious that the potential is not constant. In fact, only one or two lines down, the energy is written as a sum of KE and Potential (as you would expect)
 
  • #5
There's also a freedom of a total time derivative of a function of (q,t) in the Lagrangian, so maybe in the case orbits the potential can be written in this way? I'm not sure.
 
  • #6
dx said:
There's also a freedom of a total time derivative of a function of (q,t) in the Lagrangian, so maybe in the case orbits the potential can be written in this way? I'm not sure.

http://en.wikipedia.org/wiki/Reduced_action
Check out the disambiguation
 
  • #8

1. What is the Principle of Least Action?

The Principle of Least Action, also known as Hamilton's Principle, is a fundamental principle in classical mechanics and physics that states that a physical system follows the path that minimizes the action. This means that out of all the possible paths that a system can take, it will take the one that requires the least amount of energy.

2. How is the Principle of Least Action used in physics?

The Principle of Least Action is used to describe the behavior of a wide range of physical systems, including particles, fluids, and fields. It is often used in classical mechanics, electromagnetism, and quantum mechanics to determine the equations of motion for a system.

3. What is the significance of the Principle of Least Action?

The Principle of Least Action is significant because it provides a powerful tool for predicting the behavior of physical systems. It allows us to determine the path that a system will take without having to solve complex equations of motion. It also has implications in other areas of physics, such as optics and thermodynamics.

4. How does the Principle of Least Action relate to the laws of motion?

The Principle of Least Action is closely related to Newton's laws of motion. In fact, the principle can be derived from Newton's second law, which states that the force acting on an object is equal to its mass times its acceleration. The principle extends this concept to systems with multiple particles and degrees of freedom.

5. Can the Principle of Least Action be applied to all physical systems?

The Principle of Least Action can be applied to a wide range of physical systems, from simple mechanical systems to complex quantum systems. However, it is important to note that there are some cases where the principle may not be applicable, such as in systems with strong quantum effects or non-conservative forces.

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