Equation of motion in harmonic oscillator hamiltonian

In summary, the equations of motion in Hamiltonian mechanics are derived from the definition of the Poisson bracket and the evolution of a system in canonical coordinates is given by the partial derivatives of the Hamiltonian with respect to position and momentum. This is a fundamental concept in Hamiltonian mechanics and can be found in any textbook on the subject.
  • #1
oristo42
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nasLoSv.jpg

See attached photo please.

So, I don't get how equations of motion derived. Why is it that x dot is partial derivative of H in term of p but p dot is negative partial derivative of H in term of x.
 

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  • #2
oristo42 said:
So, I don't get how equations of motion derived. Why is it that x dot is partial derivative of H in term of p but p dot is negative partial derivative of H in term of x.
This follows from the definition of the Poisson bracket and is the basic definition of equations of motion in Hamiltonian mechanics. Any textbook covering Hamiltonian mechanics should tell you this at most 10 pages after starting the discussion on the subject.

The phase space evolution of a system in canonical coordinates is given by
$$
\dot x^i =\{x^i,H\},\quad \dot p_j = \{p_j, H\}.
$$
Indeed, for any time-independent function ##f## on phase space
$$
\dot f = \{f,H\}.
$$
 

What is the harmonic oscillator Hamiltonian equation?

The harmonic oscillator Hamiltonian equation is a mathematical expression that describes the motion of a particle in an oscillating system. It is given by H = p2/2m + 1/2mω2x2, where p is the momentum of the particle, m is its mass, and ω is the frequency of the oscillation.

What is the significance of the harmonic oscillator Hamiltonian equation?

The harmonic oscillator Hamiltonian equation is a fundamental equation in quantum mechanics that is used to describe the behavior of many physical systems, such as atoms, molecules, and solid materials. It allows us to understand and predict the energy states and properties of these systems.

How is the harmonic oscillator Hamiltonian equation derived?

The harmonic oscillator Hamiltonian equation is derived using principles from classical mechanics and quantum mechanics. It can be obtained by applying the Schrödinger equation to a particle in an oscillating potential energy field.

What are the assumptions made in the harmonic oscillator Hamiltonian equation?

The harmonic oscillator Hamiltonian equation assumes that the potential energy of the system is quadratic, the system is in a stable equilibrium, and there is no external force acting on the system. It also assumes that the system is in a vacuum and that there is no energy dissipation.

How is the harmonic oscillator Hamiltonian equation used in practice?

The harmonic oscillator Hamiltonian equation is used in many areas of physics, including quantum mechanics, statistical mechanics, and solid-state physics. It is used to calculate the energy levels and wave functions of a particle in an oscillating system, and to study the behavior of physical systems such as atoms and molecules.

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