Analytical Mechanics 1st. Edition by Lemos, Nivaldo A. | 9781108416580 | 1108416586 | Holooly

Mechanical Engineering

Analytical Mechanics

64 SOLVED PROBLEMS

Question: 5.5

Investigate the stability of the circular orbit of a celestial body about the Sun. ...

Verified Answer:

According to Eq. (5.66),

E = T + V = \frac{...

Question: 5.6

Study the stability of the stationary precession (θ = θ0) of a spherical pendulum. ...

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From (5.67) one identifies

E = T + V =\frac...

Question: 5.7

Two equal pendulums perform small oscillations coupled by a spring of negligible mass. Determine the normal modes of vibration and the solution to the equations of motion if, when t =0, only one of the pendulums is slightly displaced from its equilibrium position and the motion starts with both ...

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Let us assume that the equilibrium configuration i...

Question: 5.8

Determine the normal coordinates for the coupled pendulums of Example 5.7. ...

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The two linearly independent solutions to Eqs. (5....

Question: 6.3

Interpreting the scattering of X-rays by a graphite sheet as caused by the collision of a photon with an electron, A. H. Compton showed in 1923 that λ′ = λ+ 2h/mc sin² θ/2 .(6.134) The energy and momentum transfer from the X-ray to the electron is known as the Compton effect. Derive Compton’s ...

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In Fig. 6.6 a photon of frequency ν collides with ...

Question: 7.4

In a problem in quantum cosmology (Lemos, 1996) the action has the form S = ∫^ t2 t1 dt {a pa + φpφ-N(p²a/24a-p²φ/2a³+ 6ka)},(7.111) where k is a constant related to the curvature of space and t designates the evolution parameter. Given that N and a are strictly positive quantities, obtain the ...

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The action (7.111) has the form (7.86) with N acti...

Question: 8.6

A projectile moving on the vertical xy-plane in a uniform gravitational field is described by the Hamiltonian H = (p²x +p²y )/2m + mgy. Show that F = y −tpy/m−gt²/2 and G = x−tpx/m are constants of the motion and obtain new constants of the motion by means of Poisson’s theorem. ...

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We have

\frac{dF}{dt}=\left\{F,H\right\}+ \...

Question: 8.10

Given the Lagrangian L = 1/2 (x − y)² , (8.178) find the primary constraints. ...

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The Hessian matrix

W =\begin{pmatrix} \fra...

Question: 8.11

Discuss the consistency conditions and obtain the general solution to Hamilton’s equations for the system described by the Lagrangian of Example 8.10. ...

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The Hamiltonian associated with the Lagrangian (8....

Question: 9.1

Solve the equation of motion for the free particle in one dimension by the Hamilton-Jacobi method. ...

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Since H = p²/2m, the Hamilton-Jacobi equation take...

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