Hamiltons Principle proving Newtons laws?

In summary, my teacher asked us to prove that Hamiltons principle proves that Newtons equations of motion hold for N particles. I'm not sure that I fully grasp the concept but this is my understanding so far. Using the lagrangian, we can prove Newtons law for specific situations, however Hamiltons principle allows us to make the specific situation into generalized coordinates. This is my understanding so far, but I feel that I don't completely understand. Why do we need it to be in generalized coordinates? Why does hamiltons principle allow it to change to generalized coordinates? One of the strengths of the Lagrangian and Hamiltonian formulations is the ability to use generalized coordinates, which can often be chosen to take advantage of symm
  • #1
frenchyc
4
0
My teacher asked us too prove that Hamiltons principle proves that Newtons equations of motion hold for N particles. I'm not sure that i fully grasp the concept but this is my understanding so far:

Using the lagrangian we can prove Newtons law for specific situations, however Hamiltons principle allows us to make the specific situation into generalized coordintes. This is my understanding so far, but i feel that i don't completely understand. Why do we need it to be in generalized coordinates? Why does hamiltons principle allow it to change to generalized coordinates?
 
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  • #2
One of the strengths of the Lagrangian and Hamiltonian formulations is the ability to use generalized coordinates, which can often be chosen to take advantage of symmetries in the system. To recover Newton's equations, you should choose the generalized coordinates to be the rectangular coordinates associated with Euclidean space.

The above formulations can be given a geometrical interpretation in which the "space" associated with the Lagrangian formulation is a certain manifold called the tangent-bundle and the Hamiltonian formulation with the cotangent-bundle (more generally, a symplectic manifold).
 
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  • #3
I'm sorry but i was not referring to the Hamiltonian, but to Hamilton's principle. The principle that states that the path that a particle follows is such that the action integral of the Lagrangian from point 1 to point 2 is stationary. I am not sure if you were referring to that and i simply didn't understand it or if there was a miscommunication
 
  • #4
the configuration of the system is described using a "configuration space" Q (which could be described in any convenient set of coordinates for it)
the tangent-bundle is essentially the "space of configurations and velocities" (TQ)
the cotangent-bundle is essentially the "space of configurations and momenta" (T*Q)

possibly useful:
http://books.google.com/books?id=I2...P79&sig=UrhQn9eZ_avFCBudNiPHYVjRjIM#PPA226,M1
 
  • #5
thank you. I think my problem stems from me having difficulty wrapping my mind around this concept of changing the way we look at space.
 

1. How does Hamilton's principle prove Newton's laws?

Hamilton's principle is a mathematical formulation that states that the path taken by a system between two points in time is the one that minimizes the action of the system. This can be used to derive the equations of motion for a system, which are equivalent to Newton's laws of motion. Therefore, by using Hamilton's principle, we can prove Newton's laws.

2. Why is Hamilton's principle considered more fundamental than Newton's laws?

Hamilton's principle is considered more fundamental because it is a more general formulation that can be applied to a wider range of physical systems, including those with non-conservative forces. It also allows for the use of generalized coordinates and is applicable to both classical and quantum mechanics.

3. Can Hamilton's principle be used to prove all of Newton's laws?

Yes, Hamilton's principle can be used to prove all three of Newton's laws of motion. The first law, also known as the law of inertia, can be derived from the principle of least action. The second law, which relates force to acceleration, can be derived from the equation of motion derived from Hamilton's principle. And the third law, which states that every action has an equal and opposite reaction, can be proved by considering the symmetry of the action in Hamilton's principle.

4. Are there any limitations to using Hamilton's principle to prove Newton's laws?

While Hamilton's principle is a powerful tool for deriving the equations of motion, it does have some limitations. It is based on the principle of least action, which may not always hold true for all physical systems. Additionally, it may be difficult to apply in certain cases where the system has complex or non-conservative forces.

5. How does Hamilton's principle relate to other principles in physics?

Hamilton's principle is closely related to other fundamental principles in physics such as the principle of least action, the principle of least potential energy, and the principle of virtual work. These principles all share the common idea of minimizing a certain quantity, whether it be action, potential energy, or virtual work, to determine the behavior of a physical system.

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