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7 Lagrangian and Hamiltonian Mechanics
Hamiltonian’s Canonical Equations
∂ H
∂ p r
= ˙
q r ,
∂ H
∂q r
= − ˙
p r
(7.4)
7.2 Problems
7.1 Consider a particle of mass m moving in a plane under the attractive force
μm/r 2 directed to the origin of polar coordinates r , θ . Determine the equations
of motion.
7.2 (a) Write down the Lagrangian for a simple pendulum constrained to move in
a single vertical plane. Find from it the equation of motion and show that
for small displacements from equilibrium the pendulum performs simple
harmonic motion.
(b) Consider a particle of mass m moving in one dimension under a force with
the potential U (x) = k(2x 3 − 5x 2 + 4x), where the constant k > 0. Show
that the point x = 1 corresponds to a stable equilibrium position of the
particle. Find the frequency of a small amplitude oscillation of the particle
about this equilibrium position.
[University of Manchester 2007]
7.3 Determine the equations of motion of the masses of Atwood machine by the
Lagrangian method.
7.4 Determine the equations of motion of Double Atwood machine which consists
of one of the pulleys replaced by an Atwood machine. Neglect the masses of
pulleys.
7.5 A particular mechanical system depending on two coordinates u and v has
kinetic energy T = v 2 ˙
u 2 + 2 ˙
v 2 , and potential energy V = u 2 − v 2 . Write
down the Lagrangian for the system and deduce its equations of motion (do not
attempt to solve them).
[University of Manchester 2008]
7.6 Write down the Lagrangian for a simple harmonic oscillator and obtain the
expression for the time period.
7.7 A particle of mass m slides on a smooth incline at an angle α. The incline is not
permitted to move. Determine the acceleration of the block.
7.8 A block of mass m and negligible size slides on a frictionless inclined plane of
mass M at an angle θ with the horizontal. The plane itself rests on a smooth
horizontal table. Determine the acceleration of the block and the inclined plane.
7.9 A bead of mass m is free to slide on a smooth straight wire of negligible mass
which is constrained to rotate in a vertical plane with constant angular speed ω
about a fixed point. Determine the equation of motion and find the distance x
from the fixed point at time t. Assume that at t = 0 the wire is horizontal.
7 Lagrangian and Hamiltonian Mechanics
Hamiltonian’s Canonical Equations
∂ H
∂ p r
= ˙
q r ,
∂ H
∂q r
= − ˙
p r
(7.4)
7.2 Problems
7.1 Consider a particle of mass m moving in a plane under the attractive force
μm/r 2 directed to the origin of polar coordinates r , θ . Determine the equations
of motion.
7.2 (a) Write down the Lagrangian for a simple pendulum constrained to move in
a single vertical plane. Find from it the equation of motion and show that
for small displacements from equilibrium the pendulum performs simple
harmonic motion.
(b) Consider a particle of mass m moving in one dimension under a force with
the potential U (x) = k(2x 3 − 5x 2 + 4x), where the constant k > 0. Show
that the point x = 1 corresponds to a stable equilibrium position of the
particle. Find the frequency of a small amplitude oscillation of the particle
about this equilibrium position.
[University of Manchester 2007]
7.3 Determine the equations of motion of the masses of Atwood machine by the
Lagrangian method.
7.4 Determine the equations of motion of Double Atwood machine which consists
of one of the pulleys replaced by an Atwood machine. Neglect the masses of
pulleys.
7.5 A particular mechanical system depending on two coordinates u and v has
kinetic energy T = v 2 ˙
u 2 + 2 ˙
v 2 , and potential energy V = u 2 − v 2 . Write
down the Lagrangian for the system and deduce its equations of motion (do not
attempt to solve them).
[University of Manchester 2008]
7.6 Write down the Lagrangian for a simple harmonic oscillator and obtain the
expression for the time period.
7.7 A particle of mass m slides on a smooth incline at an angle α. The incline is not
permitted to move. Determine the acceleration of the block.
7.8 A block of mass m and negligible size slides on a frictionless inclined plane of
mass M at an angle θ with the horizontal. The plane itself rests on a smooth
horizontal table. Determine the acceleration of the block and the inclined plane.
7.9 A bead of mass m is free to slide on a smooth straight wire of negligible mass
which is constrained to rotate in a vertical plane with constant angular speed ω
about a fixed point. Determine the equation of motion and find the distance x
from the fixed point at time t. Assume that at t = 0 the wire is horizontal.