Find the hamiltonian and lagrangian of a chained system

[martyf]

Homework Statement

N points at distance a each other -> chain length L=Na. q$$_{}n$$
is the n-point shift.

Homework Equations

q$$\stackrel{}{}..$$$$_{}n$$=$$\Omega$$$$^{}2$$(q$$_{}n+1$$+q$$_{}n-1$$-2q$$_{}n$$

I must find the hamiltonian and lagrangian of the system.

The Attempt at a Solution

 

[Jhero]

To find the Hamiltonian and Lagrangian of this system, we first need to define our variables and parameters. Let N be the number of points in the chain, a be the distance between each point, and L be the total length of the chain.

The Hamiltonian of a system is defined as the total energy of the system, which is equal to the sum of the kinetic and potential energies. In this case, the kinetic energy can be written as the sum of the kinetic energies of each individual point in the chain:

K = 1/2 * m * v^2 = 1/2 * m * (dq/dt)^2

where m is the mass of each point and v is its velocity. Since each point in the chain is connected to the next one by a spring with a spring constant k, the potential energy can be written as:

U = 1/2 * k * (q_n+1 - q_n - a)^2

where q_n is the displacement of the n-th point from its equilibrium position.

Therefore, the Hamiltonian of the system can be written as:

H = K + U = 1/2 * m * (dq/dt)^2 + 1/2 * k * (q_n+1 - q_n - a)^2

Now, to find the Lagrangian, we need to define the generalized coordinates and their time derivatives. In this case, we can choose the displacement of each point from its equilibrium position as the generalized coordinates, and their time derivatives as the generalized velocities:

q_n = q_n
dq_n/dt = v_n

Using these generalized coordinates and velocities, the Lagrangian can be written as:

L = K - U = 1/2 * m * v_n^2 - 1/2 * k * (q_n+1 - q_n - a)^2

Finally, we can express the Lagrangian in terms of the generalized coordinates and their time derivatives:

L = 1/2 * m * (dq/dt)^2 - 1/2 * k * (q_n+1 - q_n - a)^2

This is the Hamiltonian and Lagrangian of the given system. From here, we can use the Euler-Lagrange equations to derive the equations of motion for the system.