Deriving the canonical equations of Hamilton

[kesgab]

As i know there are several diffrent way to derive the canonical equations.

Some of them starts from a physical principle like Hamilton's principle or the Lagrange equations.
But it can be derived also by simply make a Legendre transormation on the Lagrange function and then make derivatives on it. I don't understand how can we end up having equations that have physical meaning when we don't start from a physical law or principle and don't use any during the derivation. We actually don't state anything during this derivation.

(On Wikipedia - Hamiltonian mechanics, you can see this two kind of derivation, one uses Lagrange equations and the other is just mathematical.)

If somebody has any thought about this, i would be glad to hear, because i has been thinking on this for a couple of days now and haven't been able to come up with a solution.

Thanks,
kesgab

 

[Lojzek]

In my opinion the Hamiltonian equations don't have any physical meaning until you find the Hamiltonian which corresponds to your system. That derivation only proves that a mathematical system which is defined by least action principle with a certain Lagrangian L can also be defined with corresponding Hamiltonian H and Hamiltonian equations. L and H(L) is not specified yet.
The equations get physical meaning only when we prove experimentally that a certain physical system respects Lagrange/Hamiltonian equations and we define L/H which correctly describes the system. Experiment is not necessary for classical mechanics, since we already know that it respects Newton's laws: it is enough to find L for which the Lagrange equations are equivalen to second Newton's law.

 

[charng]

I would like to provide a response to this content by first acknowledging that there are indeed multiple ways to derive the canonical equations of Hamilton. Some of these methods involve starting from a physical principle, such as Hamilton's principle or the Lagrange equations, while others involve a more mathematical approach by using Legendre transformations.

The fact that there are different ways to derive these equations does not necessarily mean that one is more valid or correct than the other. In fact, the beauty of science is that there can be multiple approaches to solving a problem, and they can all lead to the same result. In this case, the canonical equations of Hamilton are a set of equations that describe the dynamics of a physical system, and they can be derived using different methods.

While starting from a physical principle may provide a more intuitive understanding of the equations, the mathematical approach using Legendre transformations is also valid. Just because we do not explicitly state a physical law or principle during this derivation does not mean that the resulting equations do not have physical meaning. In fact, the mathematical approach is based on the idea of transforming the Lagrangian, which is a function that already has physical significance in describing the dynamics of a system.

In summary, the different ways to derive the canonical equations of Hamilton do not diminish their physical meaning or validity. Both the physical and mathematical approaches can lead to the same result, and they both have their own merits. As scientists, it is important to have an open mind and consider different perspectives and methods in order to gain a deeper understanding of a concept.