Volume or energy: you cannot have both.
Paper 77 recorded Liouville’s theorem, that a Hamiltonian flow preserves phase-space volume. This paper calls the roll on its discrete counterpart and separates what a numerical method preserves from what it does not. NO NEW MATHEMATICAL THEOREM AND NO NEW PHYSICAL LAW IS CLAIMED. Liouville’s theorem, symplectic integrators, backward error analysis (the shadow Hamiltonian) and the Ge-Marsden theorem are all standard.
First, the preservation of volume is exact. Computing symbolically the Jacobian determinant of one step for the harmonic oscillator, leapfrog (velocity Verlet) gives exactly one, and so does semi-implicit Euler. Explicit Euler, by contrast, gives one plus the square of the step times the spring constant, and inflates the volume; classical fourth-order Runge-Kutta gives one minus the sixth power of the step times the cube of the spring constant over seventy-two, plus an eighth-order term, and contracts it. At a step of 0.1 and spring constant one, the Runge-Kutta determinant is 0.9999999861. A determinant of one means that phase-space volume is preserved exactly at every step. It is not an approximation: however coarse the step, one is one. This is Liouville’s theorem discretised.
Second, one of the two cores. Being symplectic is not conserving the energy. Running a harmonic oscillator at a step of 0.1 with the initial energy normalised to one, leapfrog after a thousand steps has a final value of 0.99945 and a minimum of 0.99750; after a hundred thousand steps the minimum is still 0.99750, and after ten million steps it is still 0.99750. The band itself has not moved. Fourth-order Runge-Kutta, by contrast, gives 0.99999 after a thousand steps, 0.99861 after a hundred thousand and 0.87048 after ten million: the whole band descends. What a symplectic method preserves exactly is not the true Hamiltonian. By backward error analysis it preserves a different Hamiltonian close to the true one, the shadow Hamiltonian, over exponentially long times, and that the energy error stays bounded is a consequence. So saying that a symplectic method conserves energy is not right; the correct statement is that the energy error stays bounded and does not grow secularly. What is conserved is a different quantity.
Third, being symplectic is not being accurate either. Measuring the position error at a step of 0.05, after one period leapfrog gives 2.14 times ten to the minus five and Runge-Kutta 2.43 times ten to the minus eight; after five periods, 4.68 times ten to the minus five against 9.41 times ten to the minus eight; after twenty periods, 9.37 times ten to the minus five against 3.62 times ten to the minus seven. Over short times the fourth-order method is more than two orders better. Leapfrog is second order and classical Runge-Kutta is fourth, and order is an attribute independent of symplecticity. The three virtues, preservation of volume, boundedness of the energy error, and accuracy, are different things and are bought separately.
Fourth, secular drift has a signature. Running the Kepler problem at eccentricity 0.5 with a step of 0.01 and measuring the relative energy error, fourth-order Runge-Kutta gives minus 3.032 times ten to the minus eight after ten orbits, minus 3.032 times ten to the minus seven after a hundred, and minus 3.032 times ten to the minus six after a thousand. Multiplying the number of orbits by ten multiplies the error by exactly ten, to four significant figures. That is the signature of secular drift. Leapfrog under the same conditions is four to six orders smaller and does not obey that law. There is a practical lesson here. Whether an error in a numerical experiment is a drift or a mere oscillation cannot be told from its size; it can be told from how it scales with time. As Paper 117 said of scale invariance, the right test is to look at the exponent.
Fifth, the other core. You cannot have both. The Ge-Marsden theorem (1988) states that an integrator that is symplectic and also conserves the energy exactly can only be the exact flow itself, up to a reparametrisation of time. Preserving volume and preserving energy are two demands that cannot both be met, except by already having the answer, and designing a numerical method is therefore choosing which of them to make exact. Make the volume exact and the energy oscillates within bounds, which suits long-time orbit computations. Make the energy exact and the volume is not preserved, which suits problems where the invariant is the point. Approximate both and both drift secularly, which is acceptable when short times and high accuracy are what is wanted. The corpus contains the same shape: Paper 126 said that an existence theorem delivers existence and nothing more, and Papers 127 and 133 said that uniqueness lives only inside a fence. What this paper adds is the case in which two virtues exclude each other. Deciding to make something exact is deciding to give something else up. And a method that varies its step according to the solution generally loses symplecticity: adding two good things does not always make a better one, and a fence cannot be crossed by addition.
IMPORTANT (scope): a census of established standard material (Layer 2). Every work cited is standard. What is not claimed: any new mathematical theorem (there is none) / any new physical law (there is none) / that symplectic methods are superior (they are not called superior; what differs is which property is made exact) / that fourth-order Runge-Kutta is inferior (it is not; over short times it is clearly the more accurate). What is added: separating the three virtues and showing the separation numerically, writing the signature of secular drift as a proportionality, and placing the impossibility of having both as a theorem. What remains: one phrase, a good numerical method, names three different virtues; they are bought separately; and the first two exclude each other. A design that has not decided what to make exact has decided nothing.
On the making of this work: The ideas and content of this work stem from the author’s own considerations. Assistance from an AI (a large language model) was used for structuring, English translation, and checking the algebra. Any remaining errors or misinterpretations are solely the author’s. Feedback and corrections are sincerely appreciated.
Reference
Yamagishi, Y. (2026). Volume or Energy: You Cannot Have Both [Paper 135]. Zenodo.
DOI: 10.5281/zenodo.22081063
ORCID: 0009-0006-8587-3450