Proving Hamiltonian ≠ Energy for Rotating Ball

[M. next]

Consider a ball of mass m rotating around an axis Oz (vertical). This ball is on a circle whose center is the same O.
Given: Angular velocity of ring is d∅/dt = ω.
Mind explaining it so we can prove that Hamiltonian here is different from Energy?!

 

[vanhees71]

The question is, is your assertion true? Let's start from the Lagrangian, using $\phi$ as the generalized coordinate. The motion on a circle is then described by

$$\vec{x}=\begin{pmatrix} r \cos \phi \\ r \sin \phi \end{pmatrix}$$

with $r=\text{const}.$ The Lagrangian is

$$L=T=\frac{m}{2} r^2 \dot{\phi}^2.$$

The Hamiltonian is then defined as

$$H(q,p)=\dot{q} p-L$$

with the canonical momentum

$$p=\frac{\partial L}{\partial \dot{\phi}}=m r^2 \dot{\phi}.$$

The Hamiltonian is thus

$$H(q,p)=\frac{p^2}{2 m r^2}.$$

Written in terms of $\dot{q}=\partial_p H=p/(m r^2)$ one sees that $H=T$, and thus $H$ is the energy of the system.

 

[M. next]

Thank you for your reply, very organized, and this is so true! But why did he ask us to prove that H different from E?

 

[M. next]

And why did he mention "Angular velocity of ring is d∅/dt = ω"?

 

[PJC01]

The Hamiltonian and energy are two different concepts in physics, and it is important to understand the distinction between them. The Hamiltonian is a mathematical function that describes the total energy of a system, while energy is a physical quantity that represents the ability of a system to do work.

In the case of a rotating ball, the Hamiltonian would take into account both the kinetic energy of the ball due to its rotation and its potential energy due to its position in the gravitational field. On the other hand, energy in this system would only refer to the kinetic energy of the ball due to its rotation.

To prove that the Hamiltonian is not equal to energy in this system, we can look at the mathematical expressions for both quantities. The Hamiltonian for a rotating ball would be given by H = K + U, where K represents the kinetic energy and U represents the potential energy. On the other hand, the energy of the rotating ball would only be represented by K.

If we consider the specific case of a ball rotating at a constant angular velocity ω, the Hamiltonian would be H = 1/2 Iω^2 + mgh, where I is the moment of inertia, m is the mass, g is the acceleration due to gravity, and h is the height of the ball above the ground. On the other hand, the energy of the ball would simply be E = 1/2 Iω^2.

As we can see, the expressions for the Hamiltonian and energy are different, with the Hamiltonian taking into account both the kinetic and potential energy of the ball, while energy only considers the kinetic energy. Therefore, in this case, the Hamiltonian is not equal to energy.

In conclusion, the Hamiltonian and energy are two distinct quantities and in the case of a rotating ball, the Hamiltonian takes into account both the kinetic and potential energy while energy only represents the kinetic energy. Therefore, it is clear that the Hamiltonian and energy are not equal in this system.