Hamilton and Lagrange functions

In summary, the Hamiltonian equations of motion for a particle in two dimensions are \dot{p_x}=-\frac{\partial H}{\partial x} and \dot{p_y}=-\frac{\partial H}{\partial y}, where H is the Hamiltonian given by H = (p_x^2)/2m + (p_y^2)/2m + ap_xp_y + U(x,y). To find the corresponding Lagrange function, we can use the Legendre transformation, which gives L = p_x\dot{x} + p_y\dot{y} - H. Then, the Lagrange equations are \frac{d}{dt}\left(\frac{\partial L}{\partial \
  • #1
Shafikae
39
0
The hamilton function of a particle in two dimensions is given by

H = (p[tex]\stackrel{2}{x}[/tex])/2m + (p[tex]\stackrel{2}{y}[/tex])/2m + apxpy + U(x,y)
Obtain the Hamiltonian equations of motion. Find the corresponding Lagrange function and Lagrange equations.

Would it be px = dH/dpy (of course it would be partial)
and py = - dH/dpx ?
and how do we take into account the potential?
 
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  • #2
Shafikae said:
Would it be px = dH/dpy (of course it would be partial)
and py = - dH/dpx ?
and how do we take into account the potential?

No, Hamilton's equations of motion are [tex]\dot{p_i}=-\frac{\partial H}{\partial q_i}[/tex] qnd [tex]\dot{q_i}=\frac{\partial H}{\partial p_i}[/tex], where [itex]q_i[/itex] are the generalized coordinates and [itex]p_i[/itex] are there corresponding momenta.

In this case, your generalized coordinates are [itex]x[/itex] and [itex]y[/itex] (i.e. [itex]q_1=x[/itex] and [itex]q_2=y[/itex]) )and there corresponding momenta are [itex]p_x[/itex] and [itex]p_y[/itex] (i.e. [itex]p_1=p_x[/itex] and [itex]p_2=p_y[/itex])...
 

Related to Hamilton and Lagrange functions

1. What is the difference between Hamilton and Lagrange functions?

The Hamilton function, also known as the Hamiltonian, is a function that represents the total energy of a system. It is defined as the sum of the kinetic and potential energies of a system. On the other hand, the Lagrange function, also known as the Lagrangian, is a function that represents the difference between the kinetic and potential energies of a system. It is defined as the kinetic energy minus the potential energy.

2. How are Hamilton and Lagrange functions used in physics?

Hamilton and Lagrange functions are used in classical mechanics to describe the motion of a system. They can be used to derive the equations of motion for a system, which can then be solved to determine the position, velocity, and acceleration of the system at any given time.

3. What is the purpose of using Hamilton and Lagrange functions?

The main purpose of using Hamilton and Lagrange functions is to simplify the equations of motion for a system. These functions allow for a more elegant and concise representation of the dynamics of a system, making it easier to analyze and solve complex problems in classical mechanics.

4. Can Hamilton and Lagrange functions be used for any type of system?

Yes, Hamilton and Lagrange functions can be used for any system that can be described using classical mechanics. This includes systems such as particles, rigid bodies, and fluids.

5. How do Hamilton and Lagrange functions relate to each other?

Lagrange functions are a special case of Hamilton functions, where the potential energy is equal to zero. This means that the Lagrangian is equal to the kinetic energy, which is a subset of the Hamiltonian. In other words, the Lagrange function is a simplified version of the Hamilton function, making it easier to solve for the equations of motion.

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