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A Minimal Derivation of the Mass-Energy Equation E = mc^2 – Dual Application of Definite Integral and the Principle of Least Action

Zhang Suhang

(Luoyang, Independent Researcher)

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Abstract

Traditional derivations of the mass-energy equation often rely on cumbersome coordinate transformation tensor operations or specific radiation models, which pose high pedagogical barriers. Departing from established frameworks, this paper takes the fundamental relativistic momentum form and the Lagrangian action principle as physical inputs, and employs only two elementary mathematical tools—definite integral energy accumulation and the variational principle of least action—to accomplish a minimal derivation of the mass-energy equation. By restricting the integration limit to the subluminal regime, singularities are naturally avoided and the relativistic kinetic energy is derived; by means of the time-independent Hamiltonian conserved quantity, the rest energy term is naturally separated. The entire derivation requires no involvement of high-dimensional geometric constructions. Through the clever selection of integration limits and variational construction, the intrinsic self-consistency of physical laws within an elementary mathematical framework is highlighted, offering a low-threshold, highly intuitive lightweight paradigm for teaching and popularizing special relativity.

Keywords: Mass-energy equation; Relativistic kinetic energy; Definite integral; Principle of least action; Hamiltonian conservation

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1 Introduction

The mass-energy equivalence reveals the profound identity between mass and energy and stands as a cornerstone of modern physics. In mainstream pedagogy and literature, derivations of the mass-energy equation generally fall into two categories. One follows Einstein's early light-radiation thought experiments, which offer vivid physical imagery but depend on specific electromagnetic processes and lack universality. The other relies on Lorentz transformations and four-dimensional spacetime tensors, which are logically rigorous but mathematically heavy, making it difficult for beginners to transition naturally from basic mechanics.

For a long time, there has been a demand for a derivation path that starts purely from work and action extremization and is carried through entirely with elementary calculus. This paper responds to this demand by constructing a three-tier progressive derivation system:

1. Using convergent subluminal definite integrals to rigorously derive relativistic kinetic energy;

2. Using the principle of least action to construct the Lagrangian for a free particle, and deriving the total energy via Hamiltonian conservation;

3. Extending to a general energy-difference form for arbitrary initial and final velocities.

The core "cleverness" of this paper lies in the following: without introducing high-dimensional geometric language, and merely by the judicious choice of integration limits (subluminal constraint) and the symmetry construction of the Lagrangian, the mathematical skeleton of the mass-energy relation emerges naturally.

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2 Fundamental Dynamical Relations

The fundamental definition of mechanical energy accumulation states that the work done by an external force equals the increase in the system's energy:

dE = F \, dx .

Using the dynamical definition F = dP/dt and the kinematic relation dx = v\,dt, we obtain the universal energy differential:

dE = v\,d(mv) . \tag{1}

Within the framework of relativistic dynamics, to ensure momentum conservation, the high-velocity momentum takes the following standard form (where \gamma is the Lorentz factor):

P = \gamma m_0 v,\qquad \gamma = \frac{1}{\sqrt{1 - v^2/c^2}} , \tag{2}

where m_0 is the rest mass. Differentiating with respect to velocity gives:

dP = \frac{m_0}{\left(1 - v^2/c^2\right)^{3/2}}\,dv . \tag{3}

Equations (1)–(3) serve as the fundamental dynamical inputs, and all subsequent derivations are based on them.

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3 Physical Prelude to the Speed-of-Light Limit

If we formally extend the velocity interval to 0 \to c, the energy integral becomes an improper integral:

E = \int_0^c v\,dP \to \infty .

Although mathematically crude, this divergent result conveys a clear physical message: infinite energy is required to push a massive particle to the speed of light. This provides an intuitive transition to the rigorous convergent subluminal integrals in Section 4.

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4 Rigorous Core Derivation: Subluminal Regime (0 \to v,\ v < c)

4.1 Definite Integral Method for Relativistic Kinetic Energy

Real physical processes always satisfy v < c, so the integration domain has no singularities. The kinetic energy increment required to accelerate a particle from rest to velocity v is:

E_k = \int_0^v v\,dP

= \int_0^v \frac{m_0 v}{\left(1 - v^2/c^2\right)^{3/2}}\,dv .

Performing the integration yields the standard relativistic kinetic energy formula:

E_k = \frac{m_0 c^2}{\sqrt{1 - v^2/c^2}} - m_0 c^2 . \tag{4}

This step clearly reveals that high-speed kinetic energy naturally appears as "total energy minus rest energy," rather than the classical \frac12 mv^2.

4.2 Principle of Least Action and Hamiltonian Total Energy

For a free particle, the Lagrangian is uniquely determined by two core constraints:

· Symmetry constraint: The action must be a Lorentz scalar;

· Classical limit: In the low-velocity limit, it should reduce to the classical form L \approx \frac12 m_0 v^2.

The unique construction satisfying these conditions is:

L = -m_0 c^2 \sqrt{1 - v^2/c^2} . \tag{5}

By the variational principle \delta \int L\,dt = 0, the true path automatically satisfies the Euler–Lagrange equation. Since the Lagrangian does not depend explicitly on time, the system possesses a conserved Hamiltonian:

H = v\frac{\partial L}{\partial v} - L

= \frac{m_0 c^2}{\sqrt{1 - v^2/c^2}} . \tag{6}

Physically, the Hamiltonian is the total energy of the system:

E = \gamma m_0 c^2 . \tag{7}

4.3 Closure of the Mass-Energy Relation

Combining (4) and (6), we obtain the energy decomposition:

E = E_k + m_0 c^2 .

Defining m = \gamma m_0, the standard mass-energy equation emerges:

\boxed{E = m c^2} .

This paper adopts the modern physical convention: m_0 is the intrinsic rest mass (a Lorentz invariant), and m is merely a compact expression for the energy structure.

4.4 Quantitative Description of the Light-Speed Barrier

From E_k = m_0 c^2 (\gamma - 1), as v \to c^-, we have \gamma \to \infty, which necessarily implies E_k \to \infty. This limiting behaviour rigorously proves that finite energy input cannot accelerate a massive particle to the speed of light.

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5 General Extension: Energy Difference for Arbitrary Initial and Final Velocities

Let the initial and final velocities be u and v, both satisfying the subluminal constraint. The energy change is:

\Delta E = \int_u^v v\,dP

= \gamma_v m_0 c^2 - \gamma_u m_0 c^2 . \tag{8}

Equation (8) is the most general dynamical difference form of the mass-energy relation, independent of any specific reference frame. When u = 0, it naturally reduces to the core derivation above, demonstrating full self-consistency.

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6 Conclusion

This paper, through the dual pathways of definite integrals and the principle of least action, has reduced the mathematical derivation of the mass-energy equation to the realm of elementary calculus while keeping the physical inputs standard. The core cleverness is twofold: first, by physically restricting the integration upper limit to the subluminal regime, mathematical rigour is ensured while simultaneously giving rise to the energy divergence and the light-speed limit; second, by uniquely determining the Lagrangian via symmetry and skilfully separating kinetic and rest energies through the Hamiltonian conservation mechanism, the need for complex spacetime geometry operations is avoided.

The derivation paradigm presented here combines clarity of physical intuition with conciseness of mathematical execution, and may serve as an efficient, self-consistent supplementary path for introductory teaching of special relativity.

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References

[1] A. Einstein, Relativity: The Special and the General Theory. Peking University Press.

[2] L. D. Landau and E. M. Lifshitz, Mechanics, 5th ed. Higher Education Press.

[3] Y. Yu, Introduction to General Relativity. Peking University Press.

[4] V. A. Zorich, Mathematical Analysis. Higher Education Press.