Motion in three dimensions and vector calculus tools

In one dimension every position-dependent force has a potential, and energy conservation solves the problem. In three dimensions neither statement is automatic. Forces can depend on all three coordinates and on velocity in ways that do no work, the right choice of coordinates can turn a hard problem into an easy one, and a force must pass a test before a potential exists. This lesson collects the tools: velocity and acceleration in polar and spherical coordinates, the gradient and curl, the criterion for a force to be conservative, and the motion of a charged particle in magnetic and electric fields, which is the standard example of a velocity-dependent force that does no work.

Polar coordinates in a plane

Many problems, including all the central-force problems of lessons 7 to 9, are naturally described by the distance \(r\) from a centre and the angle \(\phi\). Introduce unit vectors \(\hat{\mathbf r} = (\cos\phi, \sin\phi)\) pointing outward and \(\hat{\boldsymbol\phi} = (-\sin\phi, \cos\phi)\) pointing in the direction of increasing angle. Unlike \(\hat{\mathbf x}\) and \(\hat{\mathbf y}\), these unit vectors change as the particle moves: differentiating,

With \(\mathbf r = r\hat{\mathbf r}\), the velocity is \(\dot{\mathbf r} = \dot r\hat{\mathbf r} + r\dot\phi\,\hat{\boldsymbol\phi}\), and differentiating once more,

Each term has a meaning. \(\ddot r\) is the obvious radial acceleration. \(-r\dot\phi^2\) is the centripetal acceleration, present even at constant \(r\). \(r\ddot\phi\) is the tangential acceleration from changing angular speed. The last term, \(2\dot r\dot\phi\), is the Coriolis term: a particle moving outward while turning must speed up sideways, because points further out move faster at the same angular speed. The transverse component can be written \(\frac1r\frac{d}{dt}(r^2\dot\phi)\). If the force has no transverse component, \(r^2\dot\phi\) is constant, which is the conservation of angular momentum per unit mass and the key to lesson 7.

Cylindrical and spherical coordinates

Adding a height \(z\) to polar coordinates gives cylindrical coordinates, useful for anything with an axis of symmetry: a bead on a rotating rod, a charge spiralling along a magnetic field line, water in a draining tub. Spherical coordinates \((r, \theta, \phi)\), with \(\theta\) measured from the \(z\) axis, suit problems with a centre: gravity, the Coulomb force, a spherical pendulum. The velocity in spherical coordinates is

so the kinetic energy is \(\tfrac12m\big(\dot r^2 + r^2\dot\theta^2 + r^2\sin^2\theta\,\dot\phi^2\big)\). The accelerations are longer and are usually best obtained through the Lagrangian method of PHYS-344, but this kinetic energy is often all you need: combined with a potential energy and the conserved quantities of the problem, it gives the motion directly.

The gradient and conservative forces

In three dimensions a force field \(\mathbf F(\mathbf r)\) is conservative if the work \(\int\mathbf F\cdot d\mathbf r\) between any two points is the same along every path. Equivalently, the work around every closed loop is zero. When that holds, the potential energy \(U(\mathbf r) = -\int_{\mathbf r_0}^{\mathbf r}\mathbf F\cdot d\mathbf r\) is well defined, and the force is recovered as minus its gradient,

The gradient points in the direction in which \(U\) increases fastest, and its magnitude is that fastest rate. The force therefore points straight downhill on the potential energy surface. Surfaces of constant \(U\) are perpendicular to the force everywhere, which is why a ball released on a smooth hillside starts moving perpendicular to the contour lines.

How can we tell whether a given force is conservative without checking every loop? Stokes' theorem converts the work around a small loop into a local quantity. The work around a closed curve \(C\) bounding a surface \(S\) is

where the curl \(\nabla\times\mathbf F\) has components \(\big(\partial_yF_z - \partial_zF_y,\ \partial_zF_x - \partial_xF_z,\ \partial_xF_y - \partial_yF_x\big)\). If the curl vanishes everywhere in a region with no holes, the work around every loop in it vanishes, and the force is conservative. Conversely, if \(\mathbf F = -\nabla U\) then \(\nabla\times\mathbf F = 0\) automatically, because mixed partial derivatives commute.

The program checks three forces. The first, \((2xy, x^2, 0)\), is the gradient of \(-x^2y\), so its curl is zero and the work around a closed square vanishes. The second, \((-y, x, 0)\), circulates around the \(z\) axis; its curl is \(2\hat{\mathbf z}\) and the work around the unit square equals 2, the curl times the area, exactly as Stokes' theorem says. The third is an inverse-square central force, whose curl vanishes away from the origin. Every central force whose magnitude depends only on \(r\) is conservative, with \(U(r) = -\int F(r)\,dr\).

import math

# 1. Is a force conservative? Check the curl numerically and the work around a closed square loop.
def curl(F, x, y, z, h=1e-5):
    d = lambda i, j: (F(*[c + h * (k == j) for k, c in enumerate((x, y, z))])[i]
                      - F(*[c - h * (k == j) for k, c in enumerate((x, y, z))])[i]) / (2 * h)
    return (d(2, 1) - d(1, 2), d(0, 2) - d(2, 0), d(1, 0) - d(0, 1))

def loop_work(F, n=4000):
    # counterclockwise unit square in the xy-plane, corner at the origin
    W, corners = 0.0, [(0, 0), (1, 0), (1, 1), (0, 1), (0, 0)]
    for (x1, y1), (x2, y2) in zip(corners, corners[1:]):
        for i in range(n):
            s = (i + 0.5) / n
            x, y = x1 + s * (x2 - x1), y1 + s * (y2 - y1)
            Fx, Fy, _ = F(x, y, 0.0)
            W += Fx * (x2 - x1) / n + Fy * (y2 - y1) / n
    return W

forces = {
    "F = (2xy, x^2, 0)": lambda x, y, z: (2 * x * y, x * x, 0.0),
    "F = (-y, x, 0)": lambda x, y, z: (-y, x, 0.0),
    "F = -r / |r|^3 at (1,2,2)": lambda x, y, z: tuple(-c / (x * x + y * y + z * z) ** 1.5 for c in (x, y, z)),
}
for name, F in forces.items():
    cx, cy, cz = (0.0 if abs(u) < 1e-7 else u for u in curl(F, 1.0, 2.0, 2.0))
    print(f"{name:26s} curl = ({cx:+.4f}, {cy:+.4f}, {cz:+.4f})")
for name in list(forces)[:2]:
    w = loop_work(forces[name]); print(f"work around the unit square for {name}: {0.0 if abs(w) < 1e-9 else w:+.4f}")

# 2. A proton in a magnetic field B = 0.10 T along z, with an electric field E = 1000 V/m along x
q, m, B, E = 1.602e-19, 1.673e-27, 0.10, 1000.0
wc = q * B / m
def acc(v):
    vx, vy, vz = v
    return ((q / m) * (E + vy * B), (q / m) * (-vx * B), 0.0)   # q (E + v x B) / m
x, v, T = [0.0, 0.0, 0.0], [0.0, 0.0, 2.0e3], 2 * math.pi / wc
dt = T / 2000
for _ in range(20 * 2000):                         # twenty cyclotron periods, RK4 on (r, v)
    def f(s): a = acc(s[3:]); return list(s[3:]) + list(a)
    s = x + v
    k1 = f(s); k2 = f([a + dt / 2 * b for a, b in zip(s, k1)])
    k3 = f([a + dt / 2 * b for a, b in zip(s, k2)]); k4 = f([a + dt * b for a, b in zip(s, k3)])
    s = [a + dt / 6 * (b1 + 2 * b2 + 2 * b3 + b4) for a, b1, b2, b3, b4 in zip(s, k1, k2, k3, k4)]
    x, v = s[:3], s[3:]
t = 20 * T
print(f"cyclotron frequency = {wc:.4e} rad/s, period = {T * 1e6:.4f} microseconds")
print(f"after 20 periods: drift speed in y = {x[1] / t:.1f} m/s (E/B = {E / B:.1f}, direction -y), z speed = {x[2] / t:.1f} m/s")

F = (2xy, x^2, 0)          curl = (+0.0000, +0.0000, +0.0000)
F = (-y, x, 0)             curl = (+0.0000, +0.0000, +2.0000)
F = -r / |r|^3 at (1,2,2)  curl = (+0.0000, +0.0000, +0.0000)
work around the unit square for F = (2xy, x^2, 0): +0.0000
work around the unit square for F = (-y, x, 0): +2.0000
cyclotron frequency = 9.5756e+06 rad/s, period = 0.6562 microseconds
after 20 periods: drift speed in y = -10000.0 m/s (E/B = 10000.0, direction -y), z speed = 2000.0 m/s

The caveat about holes

The curl test needs a region without holes. The field \(\mathbf F = (-y, x, 0)/(x^2 + y^2)\), which circulates around the \(z\) axis with strength falling as \(1/r\), has zero curl everywhere except on the axis itself, where it is undefined. Yet the work around any circle enclosing the axis is \(2\pi\), not zero. The field is conservative in any region that excludes the axis and cannot wrap around it, but not in a region that encircles it. This is the magnetic field of a long straight wire, up to constants, and the nonzero loop integral is Ampère's law: the line integral of \(\mathbf B\) around a wire measures the current through it.

Charged particles in magnetic fields

The Lorentz force on a charge \(q\) is \(\mathbf F = q(\mathbf E + \mathbf v\times\mathbf B)\). The magnetic part is always perpendicular to the velocity, so it does no work: \(\mathbf F\cdot\mathbf v = q(\mathbf v\times\mathbf B)\cdot\mathbf v = 0\). A magnetic field alone can bend a charged particle's path but never change its speed. It is the standard example of a velocity-dependent force that conserves kinetic energy without being the gradient of any potential.

In a uniform field \(\mathbf B = B\hat{\mathbf z}\), the velocity component along \(z\) is unaffected, and the perpendicular components obey \(\dot v_x = \omega_cv_y\), \(\dot v_y = -\omega_cv_x\), with the cyclotron frequency

The perpendicular velocity rotates at \(\omega_c\), independent of speed, and the particle moves on a helix with radius \(r_c = mv_\perp/|q|B\). The speed independence is what made Ernest Lawrence's cyclotron of 1931 possible: particles spiral outward as they gain energy, but each half-turn takes the same time, so a fixed-frequency voltage can kick them on every pass. The same physics traps charged particles in the Earth's Van Allen belts, sends them spiralling down field lines into the polar atmosphere to make the aurora, and lets a mass spectrometer sort ions by their radius of curvature.

Crossed electric and magnetic fields

Add a uniform electric field \(\mathbf E\) perpendicular to \(\mathbf B\). In a frame moving at velocity \(\mathbf v_d = \mathbf E\times\mathbf B/B^2\), the electric force is exactly cancelled by the magnetic force on the frame velocity, since \(q(\mathbf E + \mathbf v_d\times\mathbf B) = 0\) for perpendicular fields. In that frame the particle simply circles; in the original frame the circle drifts at \(\mathbf v_d\). A particle starting from rest traces a cycloid, the curve drawn by a point on the rim of a rolling wheel. Remarkably, the drift velocity is independent of the charge and mass: electrons and protons drift together in the same direction.

The numerical run in the program confirms this for a proton in a 0.10 T field with a 1000 V/m electric field. The cyclotron period is 0.656 μs; after twenty periods the average velocity across the fields is exactly \(E/B = 10^4\ \mathrm{m/s}\) in the \(-y\) direction, and the initial 2000 m/s along \(\mathbf B\) is untouched. The direction follows from \(\hat{\mathbf x}\times\hat{\mathbf z} = -\hat{\mathbf y}\). The same drift, with gravity or field gradients playing the role of \(\mathbf E\), governs the slow motion of plasma in fusion devices and in the magnetosphere, and the \(\mathbf E\times\mathbf B\) drift of electrons is how a magnetron in a microwave oven generates its oscillations.