Hamiltons equations of motion in terms of poisson bracket

In summary, the Hamiltonian formulation involves an expression of the form df/dt = {f, H} + ∂f/∂t, where f is a function of the phase-space variables (q, p) and time (t). In this formulation, the phase-space variables are not explicitly time dependent, and thus if the function f is only a function of q and p, its partial time derivative (∂f/∂t) will be zero. This is because q, p, and t are independent variables in this formulation.
  • #1
rahulor
1
0
In Hamiltonian formulation there is an expression
df / dt = { f , H } + ∂f / ∂t
where f is function of q, p and t.
While expressing Hamiltons equations of motion in terms of Poisson Bracket,
i.e if the function f = q of p then its partial time derivative ∂f / ∂t becomes zero..
Please explain why?
 
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  • #2
In the Hamilton formalism, by definition, the phase-space variables [itex](q,p)[/itex] are not explicitly time dependent. Of course, solving the Hamilton equations of motion leads to the trajectory of the system in phase space as function of time, but that's not what's meant in the phase-space formulation before the equations of motion are solved.
 
  • #3
p, q, and t are independent variables so [itex]\frac{\partial p}{\partial p}=1[/itex], [itex]\frac{\partial p}{\partial q}=0[/itex], [itex]\frac{\partial p}{\partial t}=0[/itex], [itex]\frac{\partial q}{\partial p}=0[/itex], [itex]\frac{\partial q}{\partial q}=1[/itex], [itex]\frac{\partial q}{\partial t}=0[/itex], [itex]\frac{\partial t}{\partial p}=0[/itex], [itex]\frac{\partial t}{\partial q}=0[/itex], [itex]\frac{\partial t}{\partial t}=1[/itex], by definition.
 

1. What are Hamilton's equations of motion in terms of Poisson bracket?

Hamilton's equations of motion in terms of Poisson bracket are a set of mathematical equations used to describe the dynamics of a physical system in classical mechanics. They are derived from Hamilton's principle, which states that the path a physical system takes between two points in time is the one that minimizes the action integral. The equations relate the time derivatives of the system's position and momentum to the Hamiltonian, a function that encapsulates the system's total energy.

2. How do Hamilton's equations differ from Newton's laws of motion?

Hamilton's equations differ from Newton's laws of motion in that they are a more generalized form of describing the dynamics of a physical system. While Newton's laws only apply to systems with a single point mass, Hamilton's equations can be applied to systems with multiple particles and can account for complex interactions and constraints within the system.

3. What is the significance of the Poisson bracket in Hamilton's equations?

The Poisson bracket in Hamilton's equations represents the fundamental relationship between the position and momentum variables of a physical system. It is a mathematical operation that allows us to calculate the time evolution of a system by taking into account the system's total energy and any external forces acting on it.

4. Can Hamilton's equations be applied to non-conservative systems?

Yes, Hamilton's equations can be applied to non-conservative systems, such as those with external forces, dissipative forces, or systems undergoing non-conservative transformations. In these cases, the Hamiltonian may include terms that account for the effects of these non-conservative forces on the system.

5. How are Hamilton's equations related to phase space?

Hamilton's equations can be used to determine the trajectory of a system in phase space, which is a multidimensional space where each point represents a unique set of position and momentum variables. The equations describe how the system's position and momentum change over time, allowing us to map out the system's path in phase space. This is useful for understanding the overall behavior of a system and predicting future states.

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