- #1
Loxias
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Homework Statement
The mechanics of a system are described by the Lagrangian:
[tex] L = \frac{1}{2}\dot{x}^2 + \dot{x}t [/tex]
Homework Equations
(a) Write the Energy (Jacobi function) for the system.
(b) Show that [tex] \frac{dh}{dt} \neq \frac{\partial h}{\partial t} [/tex]
(c) Write an expression for the Hamiltonian of the system.
(d) Recall that [tex] \frac{dH}{dt} = \frac{\partial H}{\partial t} [/tex] allways.
explain why[tex] \frac{dH}{dt} = \frac{\partial H}{\partial t}, \frac{dh}{dt} \neq \frac{\partial h}{\partial t} [/tex] , even though H and h are equal in value.
The Attempt at a Solution
a. [tex] \frac{\partial L}{\partial \dot{x}} = \dot{x} + t [/tex]
and we get
[tex] h = \frac{\partial L}{\partial \dot{x}}\dot{x} - L = \frac{1}{2}\dot{x}^2 [/tex]
b. [tex] \frac{\partial h}{\partial t} = 0, \frac{dh}{dt} \dot{x}\ddot{x} [/tex]
c. This is what I don't understand..
They both have the same expression... what is the difference between the two sections..