The Hamiltonian vs. the energy function

[Loxias]

Homework Statement

The mechanics of a system are described by the Lagrangian:
L = \frac{1}{2}\dot{x}^2 + \dot{x}t

Homework Equations

(a) Write the Energy (Jacobi function) for the system.
(b) Show that \frac{dh}{dt} \neq \frac{\partial h}{\partial t}
(c) Write an expression for the Hamiltonian of the system.
(d) Recall that \frac{dH}{dt} = \frac{\partial H}{\partial t} allways.
explain why\frac{dH}{dt} = \frac{\partial H}{\partial t}, \frac{dh}{dt} \neq \frac{\partial h}{\partial t} , even though H and h are equal in value.

The Attempt at a Solution

a. \frac{\partial L}{\partial \dot{x}} = \dot{x} + t
and we get
h = \frac{\partial L}{\partial \dot{x}}\dot{x} - L = \frac{1}{2}\dot{x}^2

b. \frac{\partial h}{\partial t} = 0, \frac{dh}{dt} \dot{x}\ddot{x}

c. This is what I don't understand..
They both have the same expression... what is the difference between the two sections..

 

[jdwood983]

Ah, but they don't have the same expression. The Hamiltonian (H) is written in terms of the q's and p's, while the pre-Hamiltonian (h) is written in terms of q's and \dot{q}'s. Because of that little change, there is a difference.

In this case,

<br /> h(x,\dot{x},t)=\frac{1}{2}\dot{x}^2<br />

while

<br /> H(x,p_x,t)=\frac{1}{2}\left(p_x-t\right)^2<br />

where we used the Legendre transform (part a) to get the momentum (recall p_q=\partial L/\partial\dot{q}).

 

[Loxias]

Thank you for clarifying this.
Could you elaborate more about what you meant with the legendere transform?

 

[jdwood983]

Sure. The Legendre transform is a mathematical operation that transforms one set of coordinates into another set. In the case of Hamiltonian mechanics, you are turning velocity coordinates (\dot{q}) into momentum coordinates (p):

<br /> p_q=\frac{\partial L}{\partial\dot{q}}<br /> [/itex]

 

[Loxias]

Thanks