An alternative approach to several important systems in classical mechanics: energy factorization

Abstract: We show how several important classical problems, with positive definite potential energy, can be solved by starting from the factorization of the total mechanical energy using complex numbers. In particular, we derive in a new way exact analytical solutions for: simple harmonic oscillator, vertical projectile motion, motion under a repulsive inverse cube force, and damped harmonic oscillator (with linear damping). We also show how this approach easily yields an excellent approximation of the energy decay and a new approximate analytical solution in the case of a weakly damped harmonic oscillator. Our derivations are suitable for undergraduate physics teaching as an alternative to solving Newton's equations of motion. In addition, we comment on the limitations of our approach, but also on the insights it provides and opportunities for further research.

• The paper introduces the ‘energy-factorization’ method which uses the conservation of total mechanical energy to solve classical mechanics problems without solving Newton's equations.
• It takes advantage of complex number factorization to reduce the problem, primarily solving a first-order differential equation in phase space, which can yield exact or approximate solutions, in cases such as simple harmonic oscillators, gravitational fields, and inverse-cube repulsive forces, under either conservative or dissipative conditions.
• The robustness of the method is illustrated through detailed derivations, equaling, or even improving upon existing solutions to the damped harmonic oscillator problems.

Overview

This paper, "An alternative approach to several important systems in classical mechanics: energy factorization" (2601.18957), presents a pedagogically oriented method for solving several canonical one-dimensional classical mechanics problems without ever solving Newton's second-order equation of motion. The central idea is to start from the conservation of total mechanical energy for systems with positive-definite potential energy,

$\frac{mv^2}{2}+U(x)=E,$

and factor the left-hand side over the complex numbers as

$$\left(\sqrt{\tfrac{m}{2}}\,v+i\sqrt{U(x)}\right)\left(\sqrt{\tfrac{m}{2}}\,v-i\sqrt{U(x)}\right)=E.$$

Since the two factors are complex conjugates of equal modulus $\sqrt{E}$, they may be written as $\sqrt{E}\,e^{\pm i\phi(t)}$ with an unknown time-dependent phase $\phi(t)$. Adding and subtracting these relations yields two cornerstone equations:

$$v(t)=\sqrt{\frac{2E}{m}}\cos\phi(t), \qquad \sqrt{U(x(t))}=\sqrt{E}\sin\phi(t).$$

The dynamics is then recovered by demanding consistency: differentiating the position relation and matching against the velocity relation produces a first-order differential equation for $\phi(t)$, which in the solvable cases reduces to an elementary integral. The authors contrast this with prior complex-number treatments of the harmonic oscillator [Gauthier2004; Tisdell2019; AJP2025HO], arguing that their starting point—energy conservation, provable on general grounds via the work–energy theorem—is more natural than decoupling coupled first-order equations for position and momentum.

Simple harmonic oscillator

For $U(x)=kx^2/2$, the position relation gives $x(t)=\sqrt{2E/k}\,\sin\phi(t)$, and consistency with the velocity relation forces $d\phi/dt=\omega_0=\sqrt{k/m}$. The solution follows immediately:

$$\left(\sqrt{\tfrac{m}{2}}\,v+i\sqrt{U(x)}\right)\left(\sqrt{\tfrac{m}{2}}\,v-i\sqrt{U(x)}\right)=E.$$0

with amplitude and initial phase fixed by initial conditions (e.g., $$\left(\sqrt{\tfrac{m}{2}}\,v+i\sqrt{U(x)}\right)\left(\sqrt{\tfrac{m}{2}}\,v-i\sqrt{U(x)}\right)=E.$$1, $$\left(\sqrt{\tfrac{m}{2}}\,v+i\sqrt{U(x)}\right)\left(\sqrt{\tfrac{m}{2}}\,v-i\sqrt{U(x)}\right)=E.$$2 yields $$\left(\sqrt{\tfrac{m}{2}}\,v+i\sqrt{U(x)}\right)\left(\sqrt{\tfrac{m}{2}}\,v-i\sqrt{U(x)}\right)=E.$$3). Notably, the derivation requires only elementary differentiation; even integral calculus is optional at this stage.

Homogeneous gravitational field

For $$\left(\sqrt{\tfrac{m}{2}}\,v+i\sqrt{U(x)}\right)\left(\sqrt{\tfrac{m}{2}}\,v-i\sqrt{U(x)}\right)=E.$$4, the authors exploit the fact that constant force implies constant acceleration, so $$\left(\sqrt{\tfrac{m}{2}}\,v+i\sqrt{U(x)}\right)\left(\sqrt{\tfrac{m}{2}}\,v-i\sqrt{U(x)}\right)=E.$$5 follows from equating instantaneous and average acceleration. Matching this against the velocity relation and using $$\left(\sqrt{\tfrac{m}{2}}\,v+i\sqrt{U(x)}\right)\left(\sqrt{\tfrac{m}{2}}\,v-i\sqrt{U(x)}\right)=E.$$6 recovers

$$\left(\sqrt{\tfrac{m}{2}}\,v+i\sqrt{U(x)}\right)\left(\sqrt{\tfrac{m}{2}}\,v-i\sqrt{U(x)}\right)=E.$$7

The paper also points out an even simpler route apparently absent from the literature: substituting $$\left(\sqrt{\tfrac{m}{2}}\,v+i\sqrt{U(x)}\right)\left(\sqrt{\tfrac{m}{2}}\,v-i\sqrt{U(x)}\right)=E.$$8 directly into $$\left(\sqrt{\tfrac{m}{2}}\,v+i\sqrt{U(x)}\right)\left(\sqrt{\tfrac{m}{2}}\,v-i\sqrt{U(x)}\right)=E.$$9 yields $\sqrt{E}$0 immediately. The authors note that this shortcut is unused in teaching only because kinematics is conventionally introduced before dynamics; presenting both simultaneously would make it valuable.

Inverse-cube repulsive force

For $\sqrt{E}$1 ($\sqrt{E}$2), relevant physically to a point charge moving along the axis of a fixed dipole, the position relation becomes $\sqrt{E}$3, and the phase satisfies $\sqrt{E}$4. For initial conditions $\sqrt{E}$5, $\sqrt{E}$6, elementary integration and the identity $\sqrt{E}$7 give

$\sqrt{E}$8

exhibiting the expected asymptotically linear escape from the impenetrable barrier at the origin.

Central forces and effective potentials

The method extends to the radial problem in a 3D central force field via the effective potential $\sqrt{E}$9, with mass replaced by the reduced mass for two-body problems. Two observations follow. First, for $\sqrt{E}\,e^{\pm i\phi(t)}$0 (and for the free particle with $\sqrt{E}\,e^{\pm i\phi(t)}$1), $\sqrt{E}\,e^{\pm i\phi(t)}$2 remains inverse-square for any $\sqrt{E}\,e^{\pm i\phi(t)}$3, so the previous section applies directly—the authors concede this yields an alternative derivation rather than new insight, since direct integration works equally well. Second, for $\sqrt{E}\,e^{\pm i\phi(t)}$4 the effective potential can be completed into a square plus a constant, giving a positive-definite combination amenable to factorization when $\sqrt{E}\,e^{\pm i\phi(t)}$5. However, although the resulting phase integral is analytically solvable, $\sqrt{E}\,e^{\pm i\phi(t)}$6 and hence $\sqrt{E}\,e^{\pm i\phi(t)}$7 cannot be extracted in closed form—mirroring the well-known fact that the Kepler problem admits no closed-form $\sqrt{E}\,e^{\pm i\phi(t)}$8 in the standard treatment either.

Damped harmonic oscillator: exact results

With linear damping $\sqrt{E}\,e^{\pm i\phi(t)}$9, the total energy $\phi(t)$0 decays according to $\phi(t)$1. The factorization structure survives with $\phi(t)$2, supplemented by the dissipation law rewritten as $\phi(t)$3. Consistency between the differentiated position relation and the velocity relation yields a first-order phase equation,

$\phi(t)$4

whose integral form contains the ratio $\phi(t)$5 that distinguishes under-, critically-, and overdamped regimes. For the underdamped case with $\phi(t)$6, $\phi(t)$7, the appendix evaluates the phase integral via the substitution $\phi(t)$8 and completion of the square, obtaining

$\phi(t)$9

A clever decomposition of $$v(t)=\sqrt{\frac{2E}{m}}\cos\phi(t), \qquad \sqrt{U(x(t))}=\sqrt{E}\sin\phi(t).$$0 into a linear combination of the denominator $$v(t)=\sqrt{\frac{2E}{m}}\cos\phi(t), \qquad \sqrt{U(x(t))}=\sqrt{E}\sin\phi(t).$$1 and its derivative converts the logarithmic derivative of $$v(t)=\sqrt{\frac{2E}{m}}\cos\phi(t), \qquad \sqrt{U(x(t))}=\sqrt{E}\sin\phi(t).$$2 into total differentials, yielding the standard exact solution

$$v(t)=\sqrt{\frac{2E}{m}}\cos\phi(t), \qquad \sqrt{U(x(t))}=\sqrt{E}\sin\phi(t).$$3

together with the exact energy

$$v(t)=\sqrt{\frac{2E}{m}}\cos\phi(t), \qquad \sqrt{U(x(t))}=\sqrt{E}\sin\phi(t).$$4

Critically damped and overdamped solutions follow by taking $$v(t)=\sqrt{\frac{2E}{m}}\cos\phi(t), \qquad \sqrt{U(x(t))}=\sqrt{E}\sin\phi(t).$$5 and $$v(t)=\sqrt{\frac{2E}{m}}\cos\phi(t), \qquad \sqrt{U(x(t))}=\sqrt{E}\sin\phi(t).$$6 respectively. This constitutes a full derivation of the exact damped-oscillator solution using only undergraduate-level mathematics—an alternative to solving the second-order equation of motion, whose derivation is often omitted even in advanced textbooks.

Weak damping: approximate analytical solution

In the weakly damped regime, the approximation $$v(t)=\sqrt{\frac{2E}{m}}\cos\phi(t), \qquad \sqrt{U(x(t))}=\sqrt{E}\sin\phi(t).$$7 inserted into the dissipation law gives, after elementary integration,

$$v(t)=\sqrt{\frac{2E}{m}}\cos\phi(t), \qquad \sqrt{U(x(t))}=\sqrt{E}\sin\phi(t).$$8

recovering the approximate energy decay previously derived by Lelas and Pezer [LelasPezer] by a more involved route, and known to overlap excellently with the exact energy for $$v(t)=\sqrt{\frac{2E}{m}}\cos\phi(t), \qquad \sqrt{U(x(t))}=\sqrt{E}\sin\phi(t).$$9. The corresponding approximate trajectory,

$\phi(t)$0

is claimed to be new. It shares zero crossings with the simpler approximation $\phi(t)$1 from [LelasPezer] but matches the exact solution better near turning points, as illustrated graphically for $\phi(t)$2. Thus a simpler derivation yields an equally good energy approximation and a somewhat improved trajectory approximation.

Limitations

The paper is explicit about scope. For power-law potentials $\phi(t)$3, the phase equation integrates to $\phi(t)$4, which is elementary precisely for $\phi(t)$5—the three conservative cases treated. Generic $\phi(t)$6 leads to incomplete elliptic integrals or hypergeometric functions with no closed-form $\phi(t)$7; the authors stress this reflects intrinsic non-integrability rather than a defect of the method, since standard direct integration faces the same obstacle. Among dissipative oscillators, sliding-friction ($\phi(t)$8) and quadratic ($\phi(t)$9) damping couple energy and phase through $U(x)=kx^2/2$0, so the approach cannot deliver exact solutions there—though the authors indicate its approximation machinery may extend to such systems, including projectile motion with air drag, as future work.

Conclusion

The paper demonstrates that factorizing total mechanical energy with complex numbers provides a unified, low-mathematical-overhead route to exact solutions of the harmonic oscillator, vertical projectile motion, and inverse-cube repulsion, and to the exact and approximate solutions of the linearly damped oscillator. Its principal contributions are pedagogical—a self-contained alternative to Newton's equations suitable for first-year instruction—and a genuinely new approximate analytical solution for weakly damped motion. Open questions left by the paper include whether the approximation scheme generalizes to Coulomb and quadratic damping and to non-elementary conservative potentials.