Question

A
particle moves in a potential field V(r,z)=az/r, a is constant. Use
the cylindrical coordinates as the general coordinates.

1)Determine the Lagrangian of this particle.

2)Calculate the generalized impulse.

3)Determine the Hamiltonian of this particle and the
Hamiltonian’s equations of motion.

4)Determine the conserved quatities of this system.

Answer #1

A particle moves in a
potential field,let V(z)be the potential energy
function,V(z)=kz, use the cylindrical coordinates as general
coordinates.
(1)Determine the Lagrangian
for this particle.
(2)Calculate the generalized
impulse for this particle.
(3)Determine the Hamiltonian
and the equation of motions for this particle.
(4)Determine the conserved quantity of this
system.

Consider a vertical plane in a constant gravitational field.
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A particle of mass, m, in an isolated environment moves along a
line with speed v whilst experiencing a force proportional to its
distance from the origin.
a) Determine the Langrangian of the system
b) Determine the Hamiltonian of the system
c) Write down Hamilton’s equations of motion for the particle d)
Show that the particle executes simple harmonic motion

a particle of mass m moves in three dimension under the action
of central conservative force with potential energy v(r).find the
Hamiltonian function in term of spherical polar cordinates ,and
show φ,but not θ ,is ignorable .Express the quantity
J2=((dθ/dt)2 +sin2 θ(dφ
/dt)2) in terms of generalized momenta ,and show that it
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Let us consider a particle of mass M moving in one dimension q
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- Show that the equation of motion can be obtained from the
Lagrangian:
?=?^2?? [ (1/2) ??̇² − ?(?) ]
- show that the Hamiltonian is
?= (?² ?^−2??) / 2? +?(?)?^2??
Where ? = ??̇?^−2?? is the momentum conjugate to q.
Because of the explicit dependence of...

A particle of mass m moves in a circle of radius R at a constant
speed v as shown in the figure. The motion begins at point Q at
time t = 0. Determine the angular momentum of the particle about
the axis perpendicular to the page through point P as a function of
time.

A particle moves in a circle of radius r=1 m, with a constant
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1. What is the period?
2. What is the frequency?
3. What is the angular frequency
4. How does θ change in time
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The electric potential in an electric field is given by
V(x, y, z)= (-9.40 V/m5)x3y2 + (3.85 V/m4)y4 - (9.8 V/m2)zy.
Determine the unit vector form E = [ Ex V/m)i + (Ey V/m)j + (Ez
V/m)k] of the electric field at the point whose coordinates are
(-1.3 m, 2.3 m, 3.1 m). Give the x, y, z components of electric
field in the form "+/-abc" V/m, or, "ab.c" V/m as is appropriate.
For example, if you calculate the electric...

Please provide a brief overview on:
chapter 21
1. Can I obtain the net electric field due to two or
more point charges?
2. Can I obtain the net Coulomb force on a point charge
due to two or more point charges?
3. Can I draw physical quantities (for example,
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2 Equipartition The laws of statistical mechanics lead to a
surprising, simple, and useful result — the Equipartition Theorem.
In thermal equilibrium, the average energy of every degree of
freedom is the same: hEi = 1 /2 kBT. A degree of freedom is a way
in which the system can move or store energy. (In this expression
and what follows, h· · ·i means the average of the quantity in
brackets.) One consequence of this is the physicists’ form of...

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