Hamiltonian in Classical mechanics?

In summary, a Hamiltonian can be a useful tool in classical mechanics and its eigenvalues provide valuable information. However, it may be difficult to find information on the subject matter that requires linear algebra. The linear algebra course may not have been helpful, but studying normal modes and coupled oscillators can also provide insight. Additionally, the eigenvalue/eigenvector problem is important when studying the linear stability of fixed points in phase space and is often used to simplify problems in physics courses.
  • #1
BiGyElLoWhAt
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I've read a couple of places that a hamiltonian can be a tool used in classical mechanics and that it's eigenvalues are useful pieces of information. I've tried finding info on the subject matter, as I want to see something that actually requires linear algebra, or at least makes good use of it. My linear algebra course kind of sucked to be blunt, and I never really saw much use in it other than some organization.

Can someone hook me up with some links please? I'm probably just not looking up the right things. (I want some sort of instruction, unlike what's on the wikipedia page)
 
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  • #2
Google: "normal modes" "coupled oscillators"
 
  • #3
In addition to what robphy suggested, the eigenvalue/eigenvector problem comes up when you study the linear stability of fixed points in phase space. You may have noticed a trend in physics courses to "linearize" systems (wave equations, the linear stability I mentioned, etc.). The reason to do this is so that the problem we have left to solve is a linear algebra problem which is easy (at least relative to the original problem). Don't worry if you can't find linear algebra, it will always find you.
 

1. What is a Hamiltonian in classical mechanics?

A Hamiltonian in classical mechanics is a mathematical function that describes the total energy of a system. It is named after physicist and mathematician William Rowan Hamilton, who developed the concept in the 19th century.

2. How is the Hamiltonian different from the Lagrangian?

The Hamiltonian and the Lagrangian are both mathematical functions used to describe the behavior of a physical system. However, the Hamiltonian takes into account both the kinetic and potential energies of the system, while the Lagrangian only considers the kinetic energy.

3. What is the importance of the Hamiltonian in classical mechanics?

The Hamiltonian is a fundamental concept in classical mechanics and is used to describe the behavior of a wide range of physical systems, from simple pendulums to complex celestial systems. It allows for the prediction of the future state of a system, making it a powerful tool for understanding and analyzing physical phenomena.

4. How is the Hamiltonian related to the equations of motion?

The Hamiltonian is related to the equations of motion through Hamilton's equations, which are a set of differential equations that describe how a system evolves over time. The Hamiltonian is used to derive these equations, making it a crucial component of classical mechanics.

5. Can the Hamiltonian be used in quantum mechanics?

Yes, the concept of the Hamiltonian is also used in quantum mechanics. However, in this context, it is referred to as the quantum Hamiltonian and takes into account the principles of quantum mechanics. It is used to describe the state and behavior of quantum systems, such as atoms and subatomic particles.

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