Hooke's law

If we Taylor expand the force about a point, we find:

A restoring force linearly proportional to the displacement is named after Robert Hooke (1635 – 1703). Newton and Hooke disputed the credit for discovering the inverse-square law of gravity and despised each other.

In 1D:

F = -kx

setting the equilibrium position to be x0. Inserting this into Newton's 2nd law yields:

m d2x/dt2 = - kx ⟹ d2x/dt2 + ω2 x = 0

This is a 2nd order differential equation, which implies the general solution has 2 constant determined by 2 ICs. Several equivalent choices:

x(t) = A sin (ωt + α) = A cos(ωt + β) = B sin ωt + C cos ωt

Choose a form of the solution suited to the ICs of the problem.

Show that E = ½ k A2 = const.

Period: 𝜏 = 2π/ω Frequency: ν = 1/𝜏 = ω/2π

In 2D and the special case:

we have

F = -kr = ma

This is 2 uncoupled ODEs in Cartesian coordinates:

d2x/dt2 + ω2 x = 0d2y/dt2 + ω2 y = 0

The solutions are thus independent:

x(t) = Ax cos(ωt - βx)y(t) = Ay cos(ωt - βy)

Defining δ = βx – βy , we can show these equations imply:

y(t) = Ay cos(ωt – βx + δ) = Ay [cos(ωt – βx) cos δ – sin(ωt – βx) sin δ]= Ay [x/Ax cos δ - √[1 – (x/Ax)2] sin δ]

Isolating square-root term and squaring, we find:

δ = π/2 (left circular polarized) or 3π/2 (right circular polarized)δ = 0 (linearly polarized y/Ay = x/Ax ) or π (linearly polarized y/Ay = -x/Ax)

Generally we have elliptical motion.

When tensor of 1st derivatives is not proportional to the identity matrix, we have:

x(t) = Ax cos(ωx t – βx)y(t) = Ay cos(ωy t – βy)

Such a curve will be closed only if ωx/ωy is a rational number (frequencies are commensurable). If frequencies are incommensurable, curve will eventually fill the entire rectangle. Important for stellar orbits in galactic potentials.

Phase diagrams:

Newton's 2nd law is a 2nd order differential equation, requiring 2 ICs per degree of freedom to fully specify the solution. It is sometimes useful to consider solutions not in the N dimensional configuration space, but the 2N dimensional phase space. Trajectories in phase space cannot cross. Simple harmonic oscillators trace out ellipses in phase space:

Damped oscillators:

If the SHO has a damping force that linear in the velocity, the EOM becomes:

which implies:

where β = b/2m is the damping parameter. The solutions to linear, homogenous ODEs such as this one are x(t) = Cert. Inserting into the above, we find:

implying the general solution:

This solution has qualitatively different behavior in 3 cases:1. underdamping: ω > β2. critical damping: ω = β3. overdamping: ω < β

Underdamping:

In case 1 above, the general solution becomes:

where

In general, A1 and A2 are complex, giving 4 unknowns, but the requirement that x and dx/dt be real provides 2 constraints, leaving the desired 2 constants for our 2nd order ODE. We can either recognize that A2 is the complex conjugate of A1, or choose alternative linear combinations that we know to be real, sin ωt and cos ωt:

Important points:1. Envelope of trajectory e-βt.2. Frequency is lower than undamped oscillator ω1 < ω.3. dE/dt = Fdamp x dx/dt = -b (dx/dt)2 other energy is going into P.E. of oscillator.

Critical damping:

The solution is:

Check this.

Key feature of critical damping is that the solution approaches equilibrium faster than either of the above cases.

Overdamping:

In this case, the general solution is:

where:

First term dominates at late times.

Sinusoidal driving forces:

The solution to this equation can be expressed as x(t) = xc(t) + xp(t) where the complementary solution vanishes when inserted into the RHS as in the absence of driving. We guess that the particular solution is:

and find that:

At late times t >> 1/β, the complementary solution dies away and the system approaches the particular solution. The phase lag δ increases from 0 at ω = 0 to π/2 at ω = ω0 to π at ω >> ω0 as seen below:

The amplitude D is maximized where the denominator is minimized:

Resonance is only possible for ω0 > β√2, in which case the amplitude will be:

This amplification is often described in terms of the quality factor:

In the limit ω0 >> β, the ratio D/ω02A goes to Q. Amplification for different values of Q is shown

below:

Notice that as Q increases (β decreases), the peak of D increases and ωR approaches ω0. The FWHM of the resonance is:

The range of Q in physical systems varies greatly. Standard mechanical systems can have Q of a few, the best mechanical oscillators can have Q ~ 104, and atomic transitions can have Q > 107 .

Electrical oscillations:

An electrical circuit with a capacitor and inductor is also a simple harmonic oscillator. Apply Kirchoff's law to this circuit yields (I = -dq/dt):

This oscillator has a frequency:

2 ICs correspond to charge and current at time t = 0. Inductance corresponds to mass in the mechanical system; inverse of capacitance to spring constant. Electric potential energy of capacitor corresponds to mechanical potential energy of spring; energy in magnetic field of inductor corresponds to kinetic energy.

A constant force (such as gravity) displaces the equilibrium position of SHO, but otherwise leaves solution unchanged:

The same is true of a constant emf in an electrical circuit:

yielding a solution:

Adding a resistor to the circuit corresponds to damping:

We see that β = R/2L and the solution to this equation are as for the damped SHO:

Complex Impedance

The complex impedance Z of the element of a circuit is the ratio of the voltage across that element to the current.

V(t) = V0 eiωt

Resistor: V = IR ⟹ Z = V/I = RCapacitor: q = CV ⟹ I = C dV/dt = iωCV ⟹ Z = -i/ωCInductor: V = LdI/dt ⟹ I = ∫ V/L dt = V/iωL ⟹ Z = iωL

We add the impedance of series elements and the inverse impedances of parallel elements. For the circuit as a whole I(t) = V(t)/Z implying that for a sinusoidal driving voltage, the magnitude of the current will be V0/Z0 and the phase lag of the current will be Arg[Z].

Series and Parallel

If we had springs in series:

implying that the effective spring constant is:

If we add capacitors in series, the charge on the capacitors must be the same implying:

It appears at first that there is a direct analogy between springs and capacitors in series. However, C enters into ω0 as the inverse of the spring constant k. If we want to create an electrical circuit with the same ω0 as a mechanical circuit, we need instead

This is provided by capacitors in parallel:

Conversely, to create the electrical analogue of springs in parallel:

requires

i.e. capacitors in series. Applying a time-varying emf to an electrical oscillator is entirely analogous to

applying a time-varying driving force to a mechanical oscillator.

Principle of Superposition

The SHO operator:

is linear, implying that we can add the particular solutions for different driving functions to find the solution for the total driving function.

Fourier's theorem tells us that any periodic function (with period 𝜏 and frequency ωn = 2π/𝜏) can be expressed as a series of sin and cos terms:

where

We can therefore express the particular solution to any periodic driving function as:

Green's function

How do we deal with an arbitrary (non-periodic) driving function? An acceleration a applied at a time t0 over a short time interval 𝜏 << 2π/ω0 will lead to the particular solution:

Any acceleration a(t) can be considered as a succession of these impulses; by the principle of superposition any particular solution can therefore be expressed as a summation of these impulse responses:

where G(t, t') is an example of a Green's function. Green's function are incredibly useful for solving linear, inhomogeneous differential equations.