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Exact relativistic helix in a uniform magnetic field

Physical scope

Exact global coordinate-time solutions of the adopted flat external-field charged test-particle model for a uniform constant magnetic field. The force law is conditional on the local RPM action postulate and its draft equivalence and mass-shell suppliers. The result describes model orbits only, with no self-field, radiation reaction, measurement, or empirical claim.

Statement

Use the flat SI charged-particle model and local external-action adoption of Charged and spinning test-body quantities and External-field charged-particle action, with the conditional equation and coordinate Lorentz force of Stationary charged action and Lorentz force and Charged test-body mass shell and laboratory work, adopt the flat massive-particle momentum model Adopted massive-particle momentum, observer measurement and isolated-collision conservation, and assume ${A C}_{ω}$ of The Axiom of Countable Choice ( ${A C}_{ω}$ ) for the canonical tangent-bundle smooth structure required by the current local IVP and action-equivalence carriers. Let $m > 0$ be in kilograms, $q \in R$ in coulombs, $c > 0$ in metres per second, and $B_{0} \in R$ in tesla; $E = 0$ , $B = B_{0} e_{3}$ . The smooth global potential $ϕ = 0$ , $A = (- B_{0} x^{2} / 2, B_{0} x^{1} / 2, 0)$ has $\nabla \times A = B_{0} e_{3}$ . Spatial coordinates are in metres, coordinate time $t$ in seconds, and initial momentum components in kilograms metres per second. Given initial position $x_{0}$ and momentum $p_{0} = (p_{01}, p_{02}, p_{03})$ at $t = 0$ , define

$S_{0} = \sqrt{m^{2} c^{2} + ∣ p_{0} ∣^{2}}, γ_{0} = S_{0} / (m c), Ω = \frac{q B_{0}}{γ_{0} m} .$

Here $S_{0}$ has momentum units, $γ_{0}$ is dimensionless, and the signed angular frequency $Ω$ has units of inverse seconds; all displayed positions are in metres and $E$ is in joules.

For $Ω \neq 0$ , the exact momentum and energy are

$p_{1} (t) = p_{01} \cos (Ω t) + p_{02} \sin (Ω t), p_{2} (t) = p_{02} \cos (Ω t) - p_{01} \sin (Ω t), p_{3} (t) = p_{03}, E (t) = c S_{0},$

and the exact position is

$x^{1} (t) = x_{0}^{1} + \frac{c}{S_{0} Ω} (p_{01} \sin (Ω t) + p_{02} (1 - \cos (Ω t))),$ $x^{2} (t) = x_{0}^{2} + \frac{c}{S_{0} Ω} (p_{02} \sin (Ω t) - p_{01} (1 - \cos (Ω t))), x^{3} (t) = x_{0}^{3} + \frac{c p_{03}}{S_{0}} t .$

The transverse projection is a circle of radius $∣ p_{⊥} ∣ / ∣ q B_{0} ∣$ when $p_{⊥} = (p_{01}, p_{02}) \neq 0$ ; it is a point when $p_{⊥} = 0$ . In that case the full spatial orbit is a straight line parallel to $e_{3}$ if $p_{03} \neq 0$ , and a stationary point if $p_{03} = 0$ . If $p_{⊥} \neq 0$ and $p_{03} = 0$ , the circular orbit is planar. The field does no work. If $Ω = 0$ (including $q = 0$ or $B_{0} = 0$ ), the solution is free motion $p (t) = p_{0}$ , $E = c S_{0}$ , $x (t) = x_{0} + c p_{0} t / S_{0}$ . These formulas give a global future-timelike orbit and are unique on every common connected time interval; they are conditional model deductions, not empirical claims.

Facts & Assumptions

Given: A named inertial SI chart, $m > 0$ , constant $q$ , $c > 0$ , the uniform field and explicit potential above, the adopted local action and flat massive-particle models, and ${A C}_{ω}$ for the canonical tangent-bundle smooth structure.

[F1]

Charged and spinning test-body quantities: the charged test-body field is an alternating two-form, $P = m U$ , $m > 0$ , and in the flat SI chart $F_{0 i} = - E_{i} / c$ , $F_{i j} = ϵ_{i j k} B_{k}$ ; local $C^{2}$ potentials represent $F = d A$ .

[A1]

External-field charged-particle action: model trajectories are stationary for the fixed-endpoint external-charge action in a local potential domain.

[F2]

Stationary charged action and Lorentz force: under the local hypotheses and ${A C}_{ω}$ , stationarity is equivalent to $m \nabla_{U} U = q F^{♯} (U)$ .

[F3]

Charged test-body mass shell and laboratory work: in the flat inertial specialization, $d p / d t = q (E + v \times B)$ and $d E / d t = q E \cdot v$ .

[A2]

Adopted massive-particle momentum, observer measurement and isolated-collision conservation: a flat-model massive particle has constant $m > 0$ and kinetic four-momentum $P = m U$ .

[A3]

The Axiom of Countable Choice ( ${A C}_{ω}$ ): assume only countable choice ${A C}_{ω}$ for the canonical tangent-bundle smooth structure, as stated by the current RPM suppliers.

[F4]

Energy, momentum, invariant mass and four-force of relativistic particles, and necessary collision thresholds: in an inertial frame, $E = γ m c^{2}$ , $p = γ m v$ , and $E^{2} = m^{2} c^{4} + c^{2} ∣ p ∣^{2}$ .

[F5]

Norms of four-velocity and four-acceleration, and the observer rest-space decomposition: future timelike chart motion has $∣ v ∣ < c$ .

[F6]

The derivatives of sine and cosine are cosine and minus sine: $(\sin s)^{'} = \cos s$ and $(\cos s)^{'} = - \sin s$ .

[F7]

A function continuous on an interval $I$ whose derivative vanishes at every interior point of $I$ is constant on $I$ ; consequently two such functions with the same derivative differ by a constant: a differentiable scalar function on an interval with zero derivative is constant.

[F8]

The cross product in $R^{3}$ : the right-handed coordinate convention gives $e_{2} \times e_{3} = e_{1}$ and $e_{1} \times e_{3} = - e_{2}$ .

[F9]

Sums, scalar multiples, products and quotients: $(f + g)^{'} (c) = f^{'} (c) + g^{'} (c)$ , $(α f)^{'} (c) = α f^{'} (c)$ , $(f g)^{'} (c) = f^{'} (c) g (c) + f (c) g^{'} (c)$ , and $(f / g)^{'} (c) = (f^{'} (c) g (c) - f (c) g^{'} (c)) / g (c)^{2}$ when $g (c) \neq 0$ : differentiable real components obey the sum and product rules, so $(d_{1}^{2} + d_{2}^{2})^{'} = 2 d_{1} d_{1}^{'} + 2 d_{2} d_{2}^{'}$ .

Proof

1.1givenF1A1A3F2F3algebra

The potential components satisfy $F_{12} = \partial_{1} A_{2} - \partial_{2} A_{1} = B_{0}$ and all electric components vanish, by the tensor convention [F1]. The potential is smooth on all of Minkowski space, so the local-potential assumption [A1] applies, and under [A3], [F2] supplies the covariant equation. The coordinate force and energy equations are therefore those of [F3].

2.1step 1.1A2A3F3F4F6F8F9algebra

With $E = 0$ , the actual current carrier [F3] under [A3] gives $\dot{E} = 0$ . The adopted massive-particle model [A2] and inertial relations [F4] give $E^{2} = c^{2} (m^{2} c^{2} + ∣ p ∣^{2})$ and $v = c^{2} p / E$ . Thus $E = c S_{0}$ and $v = c p / S_{0}$ . Using [F3], [F8], and $v = c p / S_{0}$ , one has $v \times B_{0} e_{3} = B_{0} (v_{2} e_{1} - v_{1} e_{2})$ , so ${\dot{p}}_{1} = Ω p_{2}$ , ${\dot{p}}_{2} = - Ω p_{1}$ , ${\dot{p}}_{3} = 0$ because $q B_{0} c / S_{0} = q B_{0} / (γ_{0} m) = Ω$ . Differentiation using [F6] verifies the stated sinusoidal momenta and their initial values. The sum and product rules [F9] show that the rotating components also satisfy $\frac{d}{d t} (p_{1}^{2} + p_{2}^{2}) = 2 Ω p_{1} p_{2} - 2 Ω p_{1} p_{2} = 0$ , so $∣ p (t) ∣ = ∣ p_{0} ∣$ and the displayed constant $E = c S_{0}$ obeys the shell and $\dot{E} = 0$ ; when $Ω = 0$ , the force is zero and momentum is constant.

3.1step 2.1F6algebra

For $Ω \neq 0$ , differentiating the proposed $x^{1}$ and $x^{2}$ with [F6] gives ${\dot{x}}^{1} = c p_{1} / S_{0}$ and ${\dot{x}}^{2} = c p_{2} / S_{0}$ ; differentiating $x^{3}$ gives $c p_{3} / S_{0}$ . These are exactly the velocity components from step 2.1, and all increments vanish at $t = 0$ . For $Ω = 0$ , the stated free position has constant derivative $c p_{0} / S_{0} = v$ .

4.1step 2.1step 3.1F3algebra

For $Ω \neq 0$ , set $A = c p_{01} / (S_{0} Ω)$ and $B = c p_{02} / (S_{0} Ω)$ . The transverse coordinates relative to the fixed centre $(x_{0}^{1} + B, x_{0}^{2} - A)$ are $(A \sin Ω t - B \cos Ω t, B \sin Ω t + A \cos Ω t)$ ; their squared distance from that centre is $A^{2} + B^{2}$ . Since $S_{0} = γ_{0} m c$ and $∣ Ω ∣ = ∣ q B_{0} ∣ / (γ_{0} m)$ , the radius is $c ∣ p_{⊥} ∣ / (S_{0} ∣ Ω ∣) = ∣ p_{⊥} ∣ / ∣ q B_{0} ∣$ . If $p_{⊥} = 0$ , $x^{1} = x_{0}^{1}$ and $x^{2} = x_{0}^{2}$ , so the transverse projection is a point. Then $x^{3} = x_{0}^{3} + c p_{03} t / S_{0}$ makes the full spatial orbit an axial straight line when $p_{03} \neq 0$ and a stationary point when $p_{03} = 0$ . If $p_{⊥} \neq 0$ and $p_{03} = 0$ , $x^{3}$ is constant and the circle is planar. The energy constancy in step 2.1 proves zero magnetic work.

5.1step 2.1step 3.1F3F4F5F7F9algebra∎

The displayed curves satisfy the force equation and initial data by steps 2.1 and 3.1. Any other solution with the same data has constant energy $E = c S_{0}$ by [F3], so step 2.1 puts its momentum in the same constant-coefficient system ${\dot{p}}_{1} = Ω p_{2}$ , ${\dot{p}}_{2} = - Ω p_{1}$ , ${\dot{p}}_{3} = 0$ . For the difference $d$ of two such momenta, the product rule [F9] gives $\frac{d}{d t} (d_{1}^{2} + d_{2}^{2}) = 2 Ω d_{1} d_{2} - 2 Ω d_{2} d_{1} = 0$ , and ${\dot{d}}_{3} = 0$ . With zero initial difference, [F7] makes $d_{1}^{2} + d_{2}^{2} = 0$ and $d_{3} = 0$ on every common connected interval, hence both momenta agree, including $Ω = 0$ . Their velocities then agree by [F4], so each position difference has zero derivative and zero initial value; [F7] gives equality. The formulas are smooth for all real $t$ ; $S_{0} \geq m c > 0$ , and in the rotating case the denominator $Ω$ is nonzero. The resulting worldline is timelike by [F5].

Exact relativistic helix in a uniform magnetic field

Statement

Facts & Assumptions

Proof

Depends on

Dependency tree · 2 levels

Sources