Energy in the Hamiltonian formalism from phase space evolution

In summary, the Hamiltonian for a free falling body can be expressed as H = (p^2)/(2m) + f(y,t) where f(y,t) = mgy + g(t). However, the form of g(t) cannot be determined without explicit information about the potentials involved. Additionally, the Hamiltonian may not necessarily be a physically relevant observable.
  • #1
Jaime_mc2
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The hamiltonian ´for a free falling body is $$H = \dfrac{p^2}{2m} + mgy$$ and since we are using cartesian coordinates that do not depend on time and the potential only depends on the position, we know that ##H=E##. For this hamiltonian, using the Hamilton's equations and initial conditions ##y(0)=0## and ##p(0)=0##, we get the evolution in the phase space: $$y(t) = -\dfrac{1}{2}gt^2\quad p(t)=-mgt$$

Now, imagine the opposite problem: we don't know anything about the system and the potentials involved, but someone gives us the phase space evolution, ##x(t)## and ##p(t)##, for the same initial conditions. Can we get the energy using the hamiltonian formalism?.

From the phase space evolution, we know that ##\dot{y}=-gt = p/m## and ##\dot{p} = -mg##. Then $$ \dot{y}=\dfrac{\partial H}{\partial p} \ \Rightarrow\ H = \dfrac{p^2}{2m} + f(y,t) $$ $$ \dot{p} = -\dfrac{\partial H}{\partial y} = -\dfrac{\partial f}{\partial y} \ \Rightarrow\ f(y,t) = mgy + g(t) $$ Concluding that $$H = \dfrac{p^2}{2m} + mgy + g(t) $$

Apparently, we don't have enough information to determine the form of ##g(t)##. Two questions came to my mind:
  1. Were the Hamilton's equations integrated correctly? This seems to work when I put ##\dot{y}## as a function of ##p##, but woud it work expressing ##\dot{y}## in terms of other combinations of ##y##, ##p## or ##t##?. When is it mathematically correct to get rid of the time variable to integrate the equations?
  2. How can we know the expression for ##g(t)##, and how can we know the relation of the found hamiltonian with the energy if we don't have any explicit information about the potentials?.
 
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  • #2
I think everything is correct. The question you should answer is, whether ##g(t)## is physically relevant or not! Note that a priori the Hamiltonian is just a function to get the equations of motion (via Hamilton's least-action principle in Hamiltonian formulation) but not necessarily a physical observable!
 
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Related to Energy in the Hamiltonian formalism from phase space evolution

1. What is the Hamiltonian formalism?

The Hamiltonian formalism is a mathematical framework used to describe the dynamics of a physical system. It is based on the concept of energy conservation and uses the Hamiltonian function, which represents the total energy of the system, to describe the evolution of the system over time.

2. How is energy represented in the Hamiltonian formalism?

In the Hamiltonian formalism, energy is represented by the Hamiltonian function, which is a mathematical function that takes into account the kinetic and potential energy of a system. The Hamiltonian function is used to calculate the equations of motion for the system and determine how the system evolves over time.

3. What is phase space evolution?

Phase space evolution refers to the movement of a system through its phase space, which is a mathematical space that represents all possible states of the system. In the Hamiltonian formalism, the equations of motion describe how the system evolves through phase space over time.

4. How does the Hamiltonian formalism relate to energy conservation?

The Hamiltonian formalism is based on the principle of energy conservation, which states that the total energy of a closed system remains constant over time. In this formalism, the Hamiltonian function represents the total energy of the system, and the equations of motion describe how this energy is conserved as the system evolves.

5. What are the applications of the Hamiltonian formalism?

The Hamiltonian formalism has many applications in physics, particularly in the study of classical mechanics and quantum mechanics. It is used to describe the dynamics of a wide range of physical systems, including particles, fluids, and fields. It is also used in the development of numerical methods for solving complex physical problems.

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