Solution

The position as a function of time

Problem

Given the position vector as a function of the time t, rewrite it in cylindrical and spherical coordinates while making the curvilinear unit vectors' time dependency explicit.

The velocity

Problem

Rewrite the velocity  in cylindrical and spherical coordinates while making the curvilinear unit vectors' time dependency explicit .

The acceleration

Problem

Rewrite the acceleration in cylindrical and spherical components while making the curvilinear unit vectors' time dependency explicit.

Deriving these formulas

All these results for the position , velocity  and acceleration  are based on the differentiation rules for cylindrical and spherical unit vectors. It is thus instructive to also be able to derive any of these formulas; for that, we need the differentiation rule for the unit vectors. For example, for the spherical ones

The above result contains, in the last equation, the cylindrical radial unit vector ; rewrite it in the spherical basis

So the differentiation rules for spherical unit vectors, with the result expressed in the spherical system, are

Problem

With this information at hand, derive, in steps, the expressions for the velocity and acceleration in cylindrical and spherical coordinates

Summary

Velocity and acceleration in the case of 2-dimensional motion on the x, y plane

Problem

Derive formulas for velocity and acceleration in the case of 2-dimensional motion on the  plane, starting from the general 3-dimensional formulas above, e.g. (44) and (51) in spherical coordinates. Specialize the resulting formulas for the case of circular motion.

A single particle

The equation of motion of a single particle is Newton's  law

where  is the acceleration and  is the linear momentum, so in terms of  

We define the angular momentum  of a particle, and the torque  acting upon it, as

Differentiating the definition of

Since  is parallel to , the first term in the above cancels, and in the second term, from (64),

from which

that is, the linear and angular momentum are conserved quantities. Note that  does not require that , only that .

Many-particle systems

Lagrange equations

In the case of a closed system, or a system in a constant external field, the equations of motions can also be derived from the knowledge of the kinetic energy T and the potential energy U . For this purpose, construct the Lagrange function  and derive the equations of motion as the Lagrange equations for L.

For closed systems, the potential energy is related to the force acting on each particle by the equation . Formally,  is the gradient operator in the basis onto which  is projected, and with respect to its coordinates - say in Cartesian basis and coordinates .

The kinetic energy - say T - of a single particle is given by

Since the kinetic energy T is additive, a system of n particles has

where  is the velocity of the i^th particle. Generally speaking, the potential energy  of the system is a function of the coordinates  of the n particles, and the Lagrangian is defined as

The potential energy is related to the force acting on each particle by the equation . Formally,  is the gradient operator in the basis onto which  is projected. Knowing the Lagrangian, we can derive the (Lagrange) equations of motion  as

Work

By definition, the work performed by a force  to move a particle an infinitesimal amount  is . Thus, the work to move it from  to  along some path C is

Problem

A particle is submitted to a force whose Cartesian components are given by. Calculate the work done by this force when moving the particle along a straight line from the origin to a point ().

Conservation of the total energy of a closed system or system in a constant external field

Problem

Consider a closed system, or a system in a constant external field, for which the total force acting on the i^th particle of the system can be derived from a potential, . Show that the total energy of the system is conserved.

Problem

 Consider a system of n particles in two reference systems K and K'  that move relative to each other with constant velocity . Show that the relation between the energies of the system, E and E', is given by

Conservation of the total momentum of a closed system

The conservation of the total momentum of a closed system of one particle is clear: if the particle does not interact with anything external, the force acting on it is zero, and from Newton's 2^nd law  follows .

For a closed system of many particles, while the total force acting on the system is equal to 0, there can be internal forces different from zero acting on each particle due to its interaction with the other particles. These internal forces, however, produce no acceleration of the system; in general, they cancel each other out due to Newton's 3^rd law.

Problem

Consider a system of n particles measured in two frames of reference K and K'  that move relative to each other with velocity . Show that the system's momenta  and  are related by

.

Problem

A particle of mass m moving with velocity  leaves a half-space in which its potential energy is a constant  and enters another in which its potential energy is a different constant . Determine the change in direction of motion of the particle; that is,  where  and  are the angles between an axis perpendicular to the separating plane and the momentum  in the regions 1 and 2.

Conservation of angular momentum

Like the conservation of linear momentum, the conservation of the total angular momentum of a closed system of one particle is natural: if the particle does not interact with anything external, the force acting on it is zero and therefore its torque . From  it follows that , that is, the angular momentum  is conserved.

Problem

a) Express the Cartesian components of the angular momentum , as well as its norm, in cylindrical and spherical coordinates.

b) Rewrite  in cylindrical coordinates and using the cylindrical orthonormal basis of unit vectors, then do the same using spherical coordinates and the spherical basis.

Problem

Consider a system of n particles measured in two frames of reference K and K' whose origins have distance  from each other. Show that the momenta  and  of the system are related by

where  is the total momentum of the system as seen from K.

Problem

a) Consider a closed system of n particles, and two frames of reference K and K'  that move relative to each other with a constant velocity . Show that the momenta  and respectively measured in K and K' are related by

where  is the distance from the origin of K to the origin of K',   is the position of the center of mass as seen from K, and  and  are the total linear and angular momenta measured in K'.
b) Show that when the origin of K' is the center of mass , this formula reduces to

,

where  is the total linear momentum in K.

Cyclic coordinates

Any generalized coordinate  which does not appear explicitly in the Lagrangian is called cyclic. To any cyclic coordinate corresponds a conserved quantity. From

when  is cyclic, the right-hand side is 0 and so the quantity  is conserved.

Problem

The Lagrangian describing the movement of a particle in a central field has  as a cyclic coordinate. Using cylindrical coordinates, show that the corresponding conserved quantity is the z component of the angular momentum

Motion in one dimension

Problem
For a closed system, or any system where the total energy  is conserved, show the following:

a) The trajectory in implicit form can always be computed directly from E.

b) The turning points, if any, can be computed directly from U.

Problem

Determine the period of oscillations of a pendulum of mass m and length l in a gravitational field as a function of the amplitude of the oscillations

Problem

Integrate the equations of motion for a particle of mass m moving in a field whose potential energy is .

Reduced mass

Motion in a central field

A one-body problem in a central field, is about the motion of a single particle of mass m in a field where the potential energy, and so the magnitude of the force, depends only on the distance between the particle and a fixed point, frequently chosen as the origin of the reference system. As seen above, the two-body problem , is reducible to a one-body problem in a central field.

Problem

The angular momentum  of a particle that moves in a central field is conserved, so  evolves in time on a fixed plane perpendicular to the constant . Show that the surface swept per second by the position vector  is constant (Kepler's second law).

Problem
Starting from the constancy of the energy E and the angular momentum , compute the equations of movement and integrate them according to:

a) using differential elimination techniques to uncouple the system of equations of movement that involve both of  and  

b) interactively, one step at a time, uncouple the equations of movement eliminating  from the problem, resulting in an implicit solution .

c) eliminate t from the problem to obtain an equation for , whose solution is the trajectory as

Kepler's problem

Problem

Show that, when , with the solution (442) for the motion in a central field, part c) of the previous problem,

becomes the equation of a conic section

where .

Free oscillations in one dimension

Problem

Consider the case of 1-dimensional motion where the acting force opposes the motion as a function of the position, . This is the case, for example, of a spring, the more you stretch it (the bigger x), the more it opposes the stretching in the opposite direction . Write the equation of motion as Newton's 2^nd law, then the Lagrangian and Lagrange equation for it, and integrate the equation for generic initial conditions

Forced oscillations

Problem

Consider oscillations in 1 dimension of a system on which an external force  acts. For the oscillations to be small,  must produce only small displacements. The total force is  .

a) Write the equation of motion as Newton's 2^nd law, then write the Lagrangian and Lagrange equations.

b) Integrate the equation of motion for generic initial conditions.

c) Specialize the solution computed in b) for  to obtain

for some constants .

d) Show that a solution for the case considered in c), that is, , can be computed manually to get

for some other constants .

e) Specialize the solution of item d) in the case of resonance, when , by taking limits, thus obtaining

f) Show that the solution computed taking limits in e) can be compute directly by using dsolve and specializing the integration constants  and  that appear when solving the underlying differential equation.

Oscillations of systems with many degrees of freedom

Problem
Formulate the equations of motion for the free oscillations of a system with n degrees of freedom as

where  represents the displacement of the  generalized coordinate , the index a runs from 1 to n and there is an implicit sum over repeated indices (Einstein's convention).