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7 Lagrangian and Hamiltonian Mechanics
7.17 A simple pendulum of length l and mass m is pivoted to the block of mass M
which slides on a smooth horizontal plane, Fig. 7.3. Obtain the equations of
motion of the system using Lagrange’s equations.
Fig. 7.3
7.18 Determine the equations of motion of an insect of mass m crawling at a uniform speed v on a uniform heavy rod of mass M and length 2a which is
turning about a fixed end. Assume that at t = 0 the insect is at the middle
point of the rod and it is crawling downwards.
7.19 A uniform rod of mass M and length 2a is attached at one end by a cord of
length l to a fixed point. Calculate the inclination of the string and the rod
when the string plus rod system revolves about the vertical through the pivot
with constant angular velocity ω.
7.20 A particle moves in a horizontal plane in a central force potential U (r ). Derive
the Lagrangian in terms of the polar coordinates (r, θ). Find the corresponding
momenta p r and p θ and the Hamiltonian. Hence show that the energy and
angular momentum of the particle are conserved.
[University of Manchester 2007]
7.21 Consider the system consisting of two identical masses that can move horizontally, joined with springs as shown in Fig. 7.4. Let x, y be the horizontal
displacements of the two masses from their equilibrium positions.
(a) Find the kinetic and potential energies of the system and deduce the
Lagrangian.
(b) Show that Lagrange’s equation gives the coupled linear differential equations
m ¨
x = −4kx + 3ky
m ¨
y = 3kx − 4ky
Fig. 7.4
7 Lagrangian and Hamiltonian Mechanics
7.17 A simple pendulum of length l and mass m is pivoted to the block of mass M
which slides on a smooth horizontal plane, Fig. 7.3. Obtain the equations of
motion of the system using Lagrange’s equations.
Fig. 7.3
7.18 Determine the equations of motion of an insect of mass m crawling at a uniform speed v on a uniform heavy rod of mass M and length 2a which is
turning about a fixed end. Assume that at t = 0 the insect is at the middle
point of the rod and it is crawling downwards.
7.19 A uniform rod of mass M and length 2a is attached at one end by a cord of
length l to a fixed point. Calculate the inclination of the string and the rod
when the string plus rod system revolves about the vertical through the pivot
with constant angular velocity ω.
7.20 A particle moves in a horizontal plane in a central force potential U (r ). Derive
the Lagrangian in terms of the polar coordinates (r, θ). Find the corresponding
momenta p r and p θ and the Hamiltonian. Hence show that the energy and
angular momentum of the particle are conserved.
[University of Manchester 2007]
7.21 Consider the system consisting of two identical masses that can move horizontally, joined with springs as shown in Fig. 7.4. Let x, y be the horizontal
displacements of the two masses from their equilibrium positions.
(a) Find the kinetic and potential energies of the system and deduce the
Lagrangian.
(b) Show that Lagrange’s equation gives the coupled linear differential equations
m ¨
x = −4kx + 3ky
m ¨
y = 3kx − 4ky
Fig. 7.4