Exploring momentum redirection - Understanding Inertial Displacement

 Foundations, NASA’s Helical Engine, and Newton’s Legacy

1. Introduction: The Quest for Reactionless Propulsion

 Today we embark on an in‐depth exploration of advanced propulsion systems that—at first glance—seem to defy conventional intuition. According to Sir Isaac Newton, every action has an equal and opposite reaction. In his Philosophiæ Naturalis Principia Mathematica, Newton established that the momentum of an isolated system must be conserved. Later, Emmy Noether elegantly formalized this idea: every differentiable symmetry of the action of a physical system has a corresponding conservation law. For example, spatial translation symmetry guarantees conservation of linear momentum.

Yet here we stand, contemplating systems that propose to generate thrust by only rearranging internal momentum—without expelling mass. How can a vehicle “move” if, by Newton’s and Noether’s words, its center of mass must remain unchanged in the absence of an external force? Today, we analyze several proposed systems—including NASA’s Helical Engine, Thornson’s Inertial Propulsion, the Lewis Ratchet, and our own Closed Loop Pulse Propulsion (CLPP)—to see how, through subtle internal dynamics and time‐dependent effects, net external motion might be achieved.

Remember, as Newton famously stated, “To every action there is always opposed an equal reaction.” And Noether’s theorem assures us that if a system’s laws are invariant under spatial translation, then its total momentum is conserved. Our challenge is to understand how internal manipulations might lead to net displacement while honoring these sacred principles.

2. NASA’s Helical Engine: Principles and Mathematics

Let us begin with NASA’s Helical Engine, a concept advanced by Dr. David M. Burns and colleagues.

Concept Overview:
The Helical Engine is a theoretical system that accelerates ions in a closed, helical path. The idea is to manipulate the momentum of these ions—by using relativistic effects—to generate thrust without expelling mass. In essence, ions are accelerated to speeds where their relativistic mass increases, and then, through a controlled “contraction” of their helical path, the tangential speed is amplified.

Relativistic Momentum:
As Newton taught us, momentum is given by $p = mv$ in classical mechanics. However, at speeds approaching the speed of light, we must use the relativistic momentum:

$$p = \gamma m v, \quad \text{with } \gamma = \frac{1}{\sqrt{1-(v/c)^2}}.$$

Newton’s laws remain valid, but now the factor $\gamma$ captures the increased inertia of the moving ion. In the words of Newton, “The quantity of motion is the product of the mass and the velocity,” and here that product is modified by $\gamma$.

Impulse and Thrust:
If an ion’s velocity changes by an amount $\Delta v$ over a time interval $\Delta t$, the impulse per ion is

$$\Delta p = \gamma m \Delta v.$$

Summing over many ions, the thrust $F$ is

$$F = \frac{\Delta p}{\Delta t}.$$

Helical Contraction and Amplification:
Assume an ion is initially traveling in a circle of radius $r$ with angular velocity $\omega$, so that its tangential speed is

$$v = \omega r.$$

During the contraction phase, the ion’s orbit is forced to a smaller radius $r'$. Conservation of angular momentum (echoing Noether’s insight on translational symmetry) tells us:

$$m r^2 \omega = m {r'}^2 \omega' \quad \Longrightarrow \quad \omega' = \omega \left(\frac{r}{r'}\right)^2.$$

The new tangential speed then becomes

$$v' = \omega' r' = \omega \frac{r^2}{r'}.$$

Thus, the velocity is amplified by a factor of

$$\frac{r}{r'}.$$

The net change in velocity per ion is

$$\Delta v_{\text{ion}} = v' - v = \omega \left(\frac{r^2}{r'} - r\right),$$

and the impulse per ion is

$$J_{\text{ion}} = m \Delta v_{\text{ion}}.$$

If many ions undergo this process, their collective impulse yields a net thrust.

Newton and Noether on the Helical Engine:
Newton’s Third Law demands that for every change in momentum, an equal and opposite reaction is imparted to the system. In the Helical Engine, while the ions’ momentum is manipulated, the engine’s structure must absorb the reaction impulse. Noether’s theorem guarantees that if the closed system is isolated, the sum total of momentum remains conserved—even though cyclic internal exchanges produce bursts of thrust.

---

3. Summary so far:

• Revisited Newton’s foundational laws and Noether’s theorem as they relate to conservation of momentum.
• Explored NASA’s Helical Engine, outlining its reliance on relativistic momentum and geometric contraction to amplify thrust.
• Understood that, per Newton, the momentum change of the ions must be balanced by an opposite change in the platform, and per Noether, the overall system remains symmetrical even as we create transient forces.

In-Depth Analysis of Thornson’s Inertial Propulsion and the Lewis Ratchet

4. Roy Thornson’s Inertial Propulsion System

Concept Overview:
Roy Thornson’s system proposes that by using unbalanced, off-center rotators, one can generate a net force. Imagine a rotor that is not perfectly balanced; as it spins, the internal forces do not cancel perfectly, yielding a small net thrust.

Mathematical Foundations Using Lagrangian Mechanics:
Recall that the Lagrangian $L$ is defined as

$$L = T - V,$$

where $T$ is kinetic energy and $V$ is potential energy. For a rotating mass,

$$T = \frac{1}{2} I \omega^2,$$

with $I = m r^2$ for a point mass.
The dynamics are derived from the Euler–Lagrange equation:

$$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0,$$

where $q$ represents the generalized coordinate (which might be the angular displacement of the rotor).
If the rotor is unbalanced, a term representing the asymmetry appears in the equations of motion. In theory, this asymmetry produces a transient net force. However, if the system is truly isolated, then—as Newton would insist—the overall momentum must balance over a complete cycle.

Conservation of Momentum:
Newton reminds us, “To every action there is always opposed an equal reaction.” Thus, any net force generated internally must be counterbalanced. Thornson’s system claims that by carefully designing the rotor’s asymmetry and timing, one can create a cyclic net force. But detailed mathematical models show that unless there is a systematic bias (or energy loss in one phase), the full cycle’s impulse sums to zero. The subtlety lies in whether imperfections or deliberate design choices can yield a residual impulse.

Noether’s Perspective:
Noether would note that if the Lagrangian is invariant under spatial translation, then the net momentum must be conserved. Thus, Thornson’s device must either expend energy in a non-conservative way or leak momentum to the environment to generate thrust. In practice, experiments (e.g., on water) have demonstrated transient motion—but scaling such effects to reactionless propulsion in space remains unproven.

---

5. The Lewis Ratchet

Concept Overview:
The Lewis Ratchet is a concept in which internal masses are moved in a cyclic, asymmetric manner—effectively “ratcheting” the system forward with tiny hops. Unlike continuous systems, the Lewis Ratchet depends on precise timing differences between the impulses applied to different parts of the system.

Basic Impulse-Momentum Analysis:
Consider two masses $m_1$ and $m_2$ in a closed system with initial total momentum zero:

$$p_1 + p_2 = 0.$$

Now, suppose mass $m_1$ receives an impulse $J_1 = F t_1$ at time $t_1$, so its velocity changes by

$$\Delta v_1 = \frac{F t_1}{m_1}.$$

Then, after a delay $\Delta t = t_2 - t_1$, mass $m_2$ receives an impulse $J_2 = F t_2$ (with the opposite sign), so its change is

$$\Delta v_2 = -\frac{F t_2}{m_2}.$$

During the interval between $t_1$ and $t_2$, mass $m_1$ moves ahead relative to $m_2$. Although over a full cycle the center of mass remains fixed, the phase lag can produce a slight net displacement per cycle. If the net displacement per cycle is $\delta$, then after $N$ cycles, the total displacement is

$$\Delta X = N \delta.$$

In practice, the residual impulse per cycle is tiny; the challenge is whether such a mechanism can be made efficient enough to serve as propulsion in space.

Newton’s Words and Noether’s Theorem:
Newton’s third law insists that internal forces cancel; Noether’s theorem confirms that spatial translation symmetry demands zero net momentum change in an isolated system. Thus, the Lewis Ratchet must rely on minute imperfections or time delays that are “leaky” enough to allow a net impulse. In our joint theoretical work, we demonstrated mathematically that if the timing is not perfectly symmetric, a residual net impulse arises even though the full cycle conserves momentum.

---

6. Summary so far

In this section we covered:

• Thornson’s Inertial Propulsion: Using asymmetrical rotators and Lagrangian mechanics, we saw that while transient forces can be generated, conservation laws (per Newton and Noether) demand that over a full cycle, the net momentum remains zero unless an external exchange occurs.
• The Lewis Ratchet: We derived how staggered impulse application can lead to a slight “ratchet” or hop per cycle. Although the center of mass remains constant over a full cycle, the phase lag introduces a residual net displacement that, if accumulated, might produce thrust.

Both systems illustrate the subtle interplay between internal dynamics and external observables. As Newton stated, “Every action has an equal and opposite reaction,” and as Noether proved, symmetry enforces conservation. The challenge lies in harnessing these minute asymmetries to produce practical propulsion.

 Closed Loop Pulse Propulsion (CLPP) and the Connection to Conventional Rocket Physics

7. Closed Loop Pulse Propulsion (CLPP)

Concept Overview:
CLPP is our own contribution—a system that uses cyclic internal mass movements to produce a net thrust. It does so by accelerating a mass (often called a “slug”), redirecting its momentum (typically by converting between linear and angular momentum), and then reintroducing that momentum to the platform.

Phase 1: Linear Acceleration
Let $m_s$ be the mass of the slug and $M_p$ the mass of the platform. The slug is accelerated to an initial speed $v_i$ over time $t_a$. The impulse imparted is

$$J_1 = m_s v_i.$$

By Newton’s third law, the platform experiences an equal and opposite impulse:

$$\Delta V_p = -\frac{m_s v_i}{M_p}.$$

Phase 2: Angular Momentum Storage and Amplification
The slug is then redirected into a circular path of radius $r_i$. Its initial angular velocity is

$$\omega_i = \frac{v_i}{r_i},$$

so its angular momentum is

$$L_i = m_s r_i^2 \omega_i = m_s r_i v_i.$$

Next, the system “contracts” the circular path to a smaller radius $r_f$ (with $r_f < r_i$). Conservation of angular momentum gives:

$$m_s r_i v_i = m_s r_f v_f \quad \Rightarrow \quad v_f = v_i \frac{r_i}{r_f}.$$

Thus, the tangential speed is amplified by a factor of $\frac{r_i}{r_f}$.

Phase 3: Redirection and Recombination
When the slug is redirected back into the linear path, the net change in velocity is

$$\Delta v = v_f - v_i = v_i \left(\frac{r_i}{r_f} - 1\right).$$

The impulse delivered to the platform in this redirection phase is

$$J_2 = m_s \Delta v = m_s v_i \left(\frac{r_i}{r_f} - 1\right).$$

Thus, each cycle produces a net change in the platform’s velocity of

$$\Delta V_p = \frac{J_2}{M_p} = \frac{m_s v_i}{M_p} \left(\frac{r_i}{r_f} - 1\right).$$

Over $N$ cycles, the total velocity increment is

$$V_{p,\text{net}} = N \cdot \frac{m_s v_i}{M_p} \left(\frac{r_i}{r_f} - 1\right).$$

Newton and Noether in CLPP:

• Newton: “Every action has an equal and opposite reaction.” Here, every time the slug is accelerated and redirected, the platform experiences a reaction force.
• Noether: “For every continuous symmetry of the action, there is a corresponding conserved quantity.” Even though the internal motions conserve momentum on a full cycle, the controlled asymmetry and precise timing allow us to stack small impulses to generate net motion relative to an external inertial frame.

Connection to Rocket Physics:
Traditional rockets obey the Tsiolkovsky rocket equation:

$$\Delta v = v_e \ln \frac{m_0}{m_f},$$

where thrust is produced by expelling mass. In CLPP, no mass is expelled; instead, internal momentum is redirected. The net thrust per cycle is

$$F_{\text{CLPP}} \approx \frac{m_s v_i}{M_p} \left(\frac{r_i}{r_f} - 1\right) \frac{1}{t_{\text{cycle}}},$$

and over many cycles, this can accumulate to produce significant velocity change. The efficiency of CLPP hinges on minimizing energy losses during the conversion phases.

---

8. Synthesis and Final Thoughts

Let us now consolidate our findings from the four systems:

Consolidated Mathematical Summary

1. NASA’s Helical Engine:

   • Relativistic momentum: $$p = \gamma m v, \quad \gamma = \frac{1}{\sqrt{1 - (v/c)^2}}$$
   • Contraction amplification: $$v' = \omega \frac{r^2}{r'} \quad \Rightarrow \quad \Delta v = \omega \left(\frac{r^2}{r'} - r\right)$$
   • Thrust per ion: $$F = \frac{m \Delta v}{\Delta t}$$

2. Roy Thornson’s Inertial Propulsion:

   • Lagrangian: $$L = T - V, \quad T = \frac{1}{2} I \omega^2, \quad I = m r^2$$
   • Euler–Lagrange Equation: $$\frac{d}{dt}\left(\frac{\partial L}{\partial \dot{q}}\right) - \frac{\partial L}{\partial q} = 0$$
   • Asymmetry in rotor motion leads to transient forces that, if not perfectly canceled, yield a residual impulse.

3. The Lewis Ratchet:

   • Time-staggered impulse on two masses: $$v_{1,\text{final}} = v_{1} + \frac{F t_1}{m_1}, \quad v_{2,\text{final}} = v_{2} - \frac{F t_2}{m_2}$$
   • Net displacement per cycle: $$\delta \approx \text{(small imbalance)} \quad \text{and over } N \text{ cycles, } \Delta X = N \delta$$
   • Residual impulse accumulation emerges from phase differences.

4. Closed Loop Pulse Propulsion (CLPP):

   • Phase 1 – Linear acceleration: $$J_1 = m_s v_i, \quad \Delta V_p = -\frac{m_s v_i}{M_p}$$
   • Phase 2 – Angular conversion: $$\omega_i = \frac{v_i}{r_i}, \quad L_i = m_s r_i v_i$$ Contracting to radius $r_f$ gives: $$v_f = v_i \frac{r_i}{r_f}$$
   • Phase 3 – Recombination: $$\Delta v = v_f - v_i = v_i \left(\frac{r_i}{r_f} - 1\right)$$ Impulse: $$J_2 = m_s v_i \left(\frac{r_i}{r_f} - 1\right)$$ Net platform velocity change per cycle: $$\Delta V_p = \frac{J_2}{M_p} = \frac{m_s v_i}{M_p} \left(\frac{r_i}{r_f} - 1\right)$$ And over $N$ cycles: $$V_{p,\text{net}} = N \cdot \frac{m_s v_i}{M_p} \left(\frac{r_i}{r_f} - 1\right)$$

Bridging to Conventional Rocket Physics

In a conventional rocket, thrust is produced by expelling mass. The Tsiolkovsky rocket equation,

$$\Delta v = v_e \ln \frac{m_0}{m_f},$$

is derived from conservation of momentum for a system losing mass. By contrast, all the systems we discussed rely on internal momentum redistribution. Newton’s third law still applies; every internal impulse is matched by a reaction within the system. Noether’s theorem guarantees that in the absence of external forces, the overall center of mass remains fixed over a complete cycle. Yet, by engineering precise asymmetries or timing delays, these systems can accumulate a net external displacement over many cycles.

Newton and Noether – Their Words in Context

• Newton said in the Principia:
  “To every action, there is always opposed an equal reaction.”
  This means any thrust generated internally must be counterbalanced by an opposite impulse. Our systems do not violate this; they merely redistribute momentum within the system.

• Emmy Noether proved that:
  “Every differentiable symmetry of the action of a physical system has a corresponding conservation law.”
  Spatial translation symmetry leads directly to the conservation of linear momentum. Even if our devices create transient imbalances, the overall system obeys this conservation over a full cycle. The trick is in the timing—the transient imbalances, if not perfectly symmetric, can accumulate to yield measurable motion relative to an external frame.

---

9. Final Synthesis and Outlook

Over these three hours, we have:

• Explored NASA’s Helical Engine, which leverages relativistic effects and helical contraction to amplify thrust.
• Delved into Thornson’s Inertial Propulsion, employing asymmetrical rotators modeled through Lagrangian mechanics.
• Examined the Lewis Ratchet, where time‐staggered impulses produce tiny “hops” that may accumulate over many cycles.
• Developed the mathematics behind CLPP, our cyclic system that converts linear momentum to angular momentum (and back) to stack impulse over time.

All these systems adhere to the inviolable principles set forth by Newton and formalized by Noether. While they use internal dynamics to produce what appears to be net thrust, none can truly “create” momentum—they simply reallocate it. The challenge—and the beauty—lies in harnessing minute asymmetries, phase delays, and energy conversion efficiencies to achieve reactionless propulsion.

This exploration shows that, mathematically, every proposed mechanism obeys conservation laws, even if the practical engineering of such systems remains extraordinarily challenging. As we push forward in the exploration of advanced propulsion, these mathematical foundations remind us that innovation must always coexist with the fundamental principles of physics.