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Involving Hamiltonian Dynamics

Involving Hamiltonian Dynamics
problem 5.6

A mass m is suspended by a massless spring of spring constant k and unstretched length b. The suspension point has a constant upward acceleration a(0 subscript of a). Gravity is acting vertically downward. Find the Lagrangian and Hamiltonian functions and obtain Hamilton's equations of motion. What is the period of motion?

Problem involving the Poisson Brackets
5.16
Determine the Poisson brackets formed from the Cartesian components of the momentum p and the angular momentum
L = r * p. HINT( * IS THE MULTIPLICATION SYMBOL)

Problem 4.5 comes from The Lagrangian Formulation of Mechanics section
4.5
Three uniform rods AB,BC,CD, each of mass m and length 2b, are smoothly hinged at B and C. This system is suspended from end A. A horizontal force F is applied at the other end, D. Find the equilibrium configuration.

Solution Preview

5.6
We assume that the system is 1-dimensional and address it in the frame moving together with the suspension point, so that we replace the free fall acceleration g by (g+a) and write the quasi-gravitational potential energy as

-m(g+a)y

where the vertical coordinate is counted from the suspension point DOWNWARDS.

The potential energy of the spring is k(y-b)^2/2 and the kinetic energy of the mass is m(dy/dt)^2/2, so that the Lagrangian
is

L = m(dy/dt)^2/2 - k(y-b)^2/2 + m(g+a)y =

= m(dy/dt)^2/2 - k(y-q)^2/2 + const,

where

q = b + m(g+a)/k

and the constant is unimportant so that we shall drop the constant hereafter.

The momemtum is

p = mdy/dt,

the Hamiltonian is

H = p^2/(2m) ...

Solution Summary

A mass m is suspended by a massless spring of spring constant k and unstretched length b. The suspension point has a constant upward acceleration a(0 subscript of a). Gravity is acting vertically downward. Find the Lagrangian and Hamiltonian functions and obtain Hamilton's equations of motion. What is the period of motion?

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