Advanced Classical Mechanics/Linear Motion

We will examine the dynamics of a particle that is free to travel along only one dimension. It could be restricted to travel along a line or in a circle for example. The results for one-dimensional motion of a single particle provide an foundation to understand motion in more than one dimension and the dynamics of many-particle systems.

General ResultsEdit

Before looking at some specific examples let's look at a rather general case. A particle moves along one dimension subject to a force that only depends on position along that dimension. We have

 

Let's define the kinetic energy of the particle to be

 

The rate of change of the kinetic energy is

 

and let's calculate the total change in the kinetic energy from one time to another

 

where we have defined the potential energy

 

If we rearrange things we get

 

so   is a constant over the motion. This is the conservation of energy. We can invert the integral to calculate the force in terms of the potential energy

 

or we can calculate the velocity of the particle as a function of position using the fact that energy is conserved   so

 

In particular the motion is restricted to the region where  .

The PendulumEdit

Let's consider a small ball of mass   suspended from a pivot by a light rod of length  . The ball is restricted to move in a vertical circle. It is useful to write the position of the ball in terms of the angle   that the rod makes with the vertical direction.

The position of the bob is

 

using Cartesian coordinates. The force is  . Let's construct the potential energy

 

and the kinetic energy

 

The PeriodEdit

 
A plot of the ratio between the actual period of a pendulum and the approximate value obtained for small angles, as a function of the amplitude.

Let's say the bob is released at an angle  , how long does in take to reach the bottom,  . This is one quarter of the period of the pendulum. The total energy of the bob is

 

We can solve this equation for   to get

 

We can solve this differential equation by the separation of variables

 

where   is the period of the pendulum. The integral is not elementary but we can do it numerically or using special functions. However, if   is small we can use   to get another integral

 

     .

We can also calculate the period as a function of amplitude numerically or using special functions

Bounded and Unbounded MotionEdit

The motion of the pendulum has two different characters depending on the initial energy of the bob. In the previous example we considered a situation where the bob starts without any motion, so the initial energy is purely potential. On the other hand, the energy of the bob could be more arbitrary. Let's write

 

Remember that the total energy of the particle is defined up to an additive constant. The cosine ranges from -1 to 1, so if

 

the motion of the bob is periodic because at some value of   the velocity of the bob vanishes and then changes sign. Specifically,

 

On the other hand if   the equation has no real solution. In this case the velocity of the bob doesn't change sign and the bob goes round and round, slowing as it goes over the top.

Motion near EquilibriumEdit

The motion of a particle can only be in equilibrium if the total forces on it vanish. If the force only depends on the position of the particle (as it does for a conservative force), then the slope of potential energy must vanish at the equilibrium point because  

For simplicity (and without loss of generality) we can take the point of equilibrium to be   and take the potential energy to be zero at this point. We can expand the potential energy as a Taylor series about any point. About the point of equilibrium we have

 

For the equilibrium point the first two terms vanish so we have

 

The motion of a particle near almost any point of equilibrium is described by this potential energy, so it appears often in all branches of physics and it is useful to analyze in some detail. If the constant   is positive, the potential energy curve is concave up. The motion of a particle of any energy is periodic between the points where  ,

 

On the other hand, if   is negative, the particle approaches the origin and slows down as it gets closer. If  , the particle turns around at

 

If the energy of the particle is positive it surmounts the barrier and continues travelling on the other side.

The force corresponding to the potential energy function is  , yielding an equation of motion

 

This is a linear differential equation for   with constant coefficients. Such equations can generally be solved by linear combinations of exponential functions. Let's try the substitution

 

to get

 

which we can solve for any values of   and   as long as

  and  

If   is negative, it is handy to use the related hyperbolic functions to look at the motion

 

where  . From the discussion earlier if the energy of the particle is less than zero, then the motion is restricted to one side of the barrier and the variable   always has the same sign. Looking at the solution, this means that   On the other hand, if the energy is positive we have  .

If   is positive, the constants   and   are imaginary. In this case we can use

 

where  .

Alternative SolutionEdit

We can also solve for the motion using the same techniques as we used for the pendulum. For conservative linear motion this is always possible, but for motion in several dimensions, one can use this technique only if there are additional conserved quantities. Using the conservation of energy has the advantage that it doesn't appear that we are guessing for the solution. The energy of the particle is conserved

 

We can solve this equation for the velocity of the particle as a function of position

 

We can solve this differential equation by the separation of variables

 

We have four possibilities depending on the signs of   and  . If   and   are both positive the integral is

 

and if we invert this formula we get

 

where we have dropped the   sign for clarity.

If   is negative and   is positive, there is no solution. If   is negative, the solutions are unbounded. For positive   we have

 

and for negative   we have

 

Complex RepresentationEdit

We found earlier that the general solution for small oscillations was

 

When   is positive, the constant   is purely imaginary. It is customary to define  , yielding the periodic solution

 

This solution is of course a complex number. However, both the real and imaginary parts of this solution are also solutions of the differential equation, or one could let   yielding the real solution,

 

that we obtained earlier. In general we considering periodic phenomena, we will use solutions of the form   with the proviso that since the physical quantity   that we will take the real part of the solution.

Damped Linear OscillatorEdit

We found that the motion of a particle near a position of stable equilibrium under a conservative force approximates that of a simple harmonic oscillator. There may be some energy loss as well, so there may be a force that depends on the velocity. If we take   and   as small quantities, a good approximation to the total force is

 

yielding an equation of motion

 

Using the earlier results we will try a solution of the form   which yields an equation

 

The roots of this equation are

 

There are three regimes of the damping.

Large DampingEdit

If

 

then both roots for   are real and negative. Specifically we have

 

and the solution

 

At late times, the second term dominates and the damping time is approximately  .

Small DampingEdit

If

 

then both roots for   are complex and the general solution is

 

where  ,

 

and  .

We can define the quality factor or 'Q' of the oscillator to be

 

Critical DampingEdit

In the limiting case we have   and the two roots for   coincide. The resulting solution   cannot be general because a second-order differential equation but have two constants of integration. Here we have

 

Driven OscillatorsEdit

We can imagine some force driving the oscillator. Perhaps someone is pushing on a child on a swing. We get the following equation of motion,

 

This is an inhomogeneous equation of motion. The general solution to such an equation is the solution to the homogeneous equation (which has two constants of integration) plus a particular solution to the equation. To find the particular let's consider a periodic driving force

 

Since the equation of motion is linear, the particular solution for the sum of several driving forces is the sum of the solutions for each driving force. Let's look for a solution that varies at the same frequency as the driving force

 

Substituting this into the equation of motion yields an equation for the amplitude of the oscillation

 

and solving for the amplitude yields

 

The amplitude is complex and we can divide it into a magnitude and a phase  . Using the fact that   we have

 

This yields the real general solution to our equation of the form

 

The second term that depends on the initial conditions decreases exponentially. After a long time this transcient can be neglected and the oscillations are solely governed by the driving force.