Paine's 1982 Lotka-Volterra Lagrangian

Article Preface:

There are quite a few notes about this article:

1. This article is technical. In order to understand this article, it would help to have some comfort in Lagrangian Mechanics and Hamiltonian Mechanics. You should know what a Lagrangian is, what a Hamiltonian is and how to get it from a Lagrangian, and what the Euler-Lagrange Equations are. It would also help if you are vaguely aware of what the Lotka-Volterra Equations are. We will also lightly refer to terms that are found in the field of mathematics called Differential Geometry.
2. This article discusses the paper entitled The development of largangians for biological models. Note that there is a typo in the spelling of “Lagrangians” in the title of the paper—I’ve emailed Springer about it. On the other hand, the article hosted here has no typo, but it appears its DOI is broken…
3. This article makes frequent reference to “our work.” That is because we (my friend/colleague and I) have added a contribution to this research program ourselves, and it is entitled An Update on the Lagrangian of the Two-Dimensional Lotka-Volterra System, with a link soon to come.
4. Try the “problems” below yourself! Make sure you get the same things that I get.

A Review of Paine’s 1982 Two-Dimensional Lotka-Volterra Lagrangian

Impetus

The present investigation constitutes part of a literature review on the problem of using Lagrangian Mechanics to describe the Lotka-Volterra (hereafter “LV”) equations. The objective of this investigation is to determine if the work presented in An Update on the Lagrangian of the Two-Dimensional Lotka-Volterra System truly has not been seen in the literature before. Dr. Paine is one of many physicists who has offered a Lagrangian that generates the dynamics of the two-dimensional LV system. Thus, it is pertinent to assess the method by which he obtained it and contrast it with the approach taken in the new work. What we will do below is review only a section of his paper, which is the relevant section for us: the derivation of a Lagrangian for the two-dimensional LV system.

Introduction

There has been substantial prior work on formulating Lagrangians that generate the two-dimensional Lotka–Volterra equations. What we wish to learn about now in this article is the (dis)similarity between the Lagrangians derived in the previous literature and the Lagrangian we proposed in our work. Since there has been a handful of approaches to this problem in the past, we must scrutinize each one to determine how it relates to our result.

As a first step, let us pin down the precise form of the 2D LV equations that Paine wishes to generate. According to his Eq. (34), the dynamics he wants to generate are:

$$\begin{array}{rl}{\dot{x}}_1& =x_1-x_1{x}_2\\ {\dot{x}}_2& =-x_2+x_1{x}_2.\end{array}$$

Above, $x_1$ and $x_2$ stand for the prey and predator population’s spatial density respectively. It is essential to note that the LV equations written above do not contain any model parameters. Typically, the most general 2D LV equations are written according to

$$\begin{array}{rl}{\dot{x}}_1& =\alpha x_1-\beta x_1{x}_2\\ {\dot{x}}_2& =-\gamma x_2+\delta x_1{x}_2,\end{array}$$

where $\beta \ne \delta$ , describing non-reciprocal coupling/interactions between the species. (If $\beta =\delta$ , this is called the reciprocal coupling case.) Perhaps Paine has left out these parameters to draw attention more to the mathematics presented in the work. All of that is to say that we ought to now anticipate that our Lagrangian will not appear similar to the one he derives because our approach retains all of the model parameters. We may be able to draw a comparison only after we allow $\alpha ,\beta ,\gamma ,\delta \to 1$ in our approach.

We also remark that it is his equations above—the ones in which all the model parameters are set to $1$ —that he wishes to derive directly from the formalism of Lagrangian Mechanics. In other words, he wants to find a Lagrangian that is directly capable of producing the LV equations via the Euler-Lagrange equations (hereafter EL equations). We make this comment to emphasize that, even though the LV equations are first-order ordinary differential equations (ODEs), they are not derived through Hamiltonian Mechanics but rather directly through Lagrangian Mechanics (specifically via the calculus of variations). To belabor the point one more step further, we usually anticipate that Hamilton’s equations produce first-order ODEs while the EL equations produce second-order ODEs; Paine’s goal is to derive the 2D LV system using the EL equations rather than with Hamilton’s equations.

Paine’s Lagrangian

After following his procedure—which we do not go into in this article—Paine arrives at the following Lagrangian:

$$L_{\text{Paine}}=e^{-(x_1+x_2)}\{\frac{x_1{\dot{x}}_2-x_2{\dot{x}}_1}{x_1+x_2}+\frac{x_1{\dot{x}}_2-x_2{\dot{x}}_1}{{(x_1+x_2)}^2}-x_1{x}_2\}+\frac{x_2{\dot{x}}_1-x_1{\dot{x}}_2}{{(x_1+x_2)}^2}.$$

The Lagrangian above is his Eq. (38). It is also the central focus of this article. Our objective now is to study this Lagrangian, and identify to what degree it is related to other work on the same idea.

Let us first remark that the Lagrangian that is written above looks nothing like the one reported in An Update on the Lagrangian of the Two-Dimensional Lotka-Volterra System. The Lagrangian offered above contains two “generalized coordinates,” $x_1$ and $x_2$ , and their associated generalized velocities, ${\dot{x}}_1$ and ${\dot{x}}_2$ . The configuration space of this mechanical system is two-dimensional; if we were to pass to the Hamiltonian formulation, phase space (the cotangent bundle) would be four-dimensional. By contrast, in our approach, the configuration space is one-dimensional.

Rewriting the Lagrangian

Secondly, a glance at the structure of the Lagrangian above ought to make us wary of possible redundancies encoded into it. As we recall from Lagrangian Mechanics, Lagrangians are only unique up to a time derivative of a function $f(q;t)$ . That is,

$$L^{\prime }(q,\dot{q};t)\sim L(q,\dot{q};t)+\frac{d}{dt}f(q;t),$$

where $\sim$ expresses an equivalence relation, the equivalence being the derived dynamics from the calculus of variations. In fact, there is a high degree of redundancy in the reported Lagrangian. Paine does not attempt to eliminate these redundancies in his 1982 paper. Again, we suspect that the reason is because he wished to emphasize the mathematical techniques rather than the (classical) mechanics of the problem. So, let us follow up on his work by conducting this exercise now.

Our objective is to identify all of the redundancies in Paine’s Lagrangian. This is not merely an analytical exercise, but rather an indispensable part of determining if the Lagrangian we reported in our work is in any way, shape, or form similar to the one Paine has reported. Remember: Lagrangians are not unique.

---

Exercise: Prove that the final term in Paine’s Lagrangian,

$$\frac{x_1{\dot{x}}_2-{\dot{x}}_1{x}_2}{{(x_1+x_2)}^2},$$

is a total time derivative, and thus can be eliminated.

Solution: One can show that

$$\frac{d}{dt}\frac{x_2}{x_1+x_2}=\frac{x_1{\dot{x}}_2-{\dot{x}}_1{x}_2}{{(x_1+x_2)}^2}.$$

All we do is apply the product rule:

$$\frac{{\dot{x}}_2(x_1+x_2)-x_2({\dot{x}}_1+{\dot{x}}_2)}{{(x_1+x_2)}^2}=\frac{x_1{\dot{x}}_2-{\dot{x}}_1{x}_2}{{(x_1+x_2)}^2},$$

because the funky term $x_2{\dot{x}}_2$ cancels out.

---

Excellent! We’ve eliminated a bulky term from the original Lagrangian. But there is actually more that we can do. Now, the idea is the following: Since we have shown that one of the major terms in the curly brackets is a total time derivative, we are now starting at a term of the form

$$e^{-(x_1+x_2)}\frac{d}{dt}\left(\frac{x_2}{x_1+x_2}\right).$$

which looks like one term in the product rule applied to the product of $e^{-(x_1+x_2)}$ and $x_2/(x_1+x_2)$ . It is this term that we target next.

---

Exercise: Use the rules of differentiation to determine how to rewrite the term below, taking advantage of the removal of total time derivatives from Lagrangians without changing the dynamics:

$$e^{-(x_1+x_2)}\frac{d}{dt}\left(\frac{x_2}{x_1+x_2}\right).$$

Solution: The main idea here is to focus on computing the total derivative of the product below:

$$\frac{d}{dt}\left[e^{-(x_1+x_2)}\left(\frac{x_2}{x_1+x_2}\right)\right].$$

Without actually evaluating anything and using the product rule immediately, we find

$$\left(\frac{x_2}{x_1+x_2}\right)\frac{d}{dt}{e}^{-(x_1+x_2)}+e^{-(x_1+x_2)}\frac{d}{dt}\left(\frac{x_2}{x_1+x_2}\right).$$

So, of course, the term we focused on before was the second one above. The strategy is then to move the time derivative from the rational expression of $x_1$ and $x_2$ onto the exponential factor, and disregard the total time derivative because it will not affect the dynamics. Now, in one fell swoop, let’s actually evaluate the time derivative of the exponential.

$$\left(\frac{x_2}{x_1+x_2}\right)\frac{d}{dt}{e}^{-(x_1+x_2)}=e^{-(x_1+x_2)}\frac{x_2({\dot{x}}_2+{\dot{x}}_1)}{x_1+x_2}.$$

---

Let’s now examine where Paine’s Lagrangian stands after all these manipulations:

$$L_{\text{Paine}}=e^{-(x_1+x_2)}\{\frac{x_1{\dot{x}}_2-x_2{\dot{x}}_1}{x_1+x_2}+\frac{x_2({\dot{x}}_2+{\dot{x}}_1)}{x_1+x_2}-x_1{x}_2\}.$$

It is evident that we can combine the two terms with the same denominator. After doing so, we find that the denominator in fact wholly cancels out. After the dust settles, we find Paine’s Reduced Lagrangian:

$$\boxed{{\displaystyle L=e^{-(x_1+x_2)}({\dot{x}}_2-x_1{x}_2)}}.$$

While we have done a lot of work reducing Paine’s original behemoth of a Lagrangian, we need to show that indeed this simplified Lagrangian generates the same dynamics.

---

Exercise: Show that Paine’s Reduced Lagrangian generates the desired dynamics.

Solution: Simply apply the Euler-Lagrange Equations to find the desired dynamics. There are two dynamical variables in the Lagrangian, which are $x_1$ and $x_2$ . So, we will have two Euler-Lagrange equations. The various components of each of the EL equations are readily computed:

$$\begin{array}{rl}\frac{\partial L}{\partial x_1}& =-e^{-(x_1+x_2)}(x_2-x_1{x}_2+{\dot{x}}_2),\frac{\partial L}{\partial {\dot{x}}_1}=0,\\ \frac{\partial L}{\partial x_2}& =-e^{-(x_1+x_2)}(x_1-x_1{x}_2+{\dot{x}}_2),\frac{\partial L}{\partial {\dot{x}}_2}=e^{-(x_1+x_2)}.\end{array}$$

Great! Now, we just need to take the time derivative of those ${\partial }_{{\dot{x}}_i}L$ terms and combine it appropriately (according to the EL equations) with the ${\partial }_{x_i}L$ terms. For reference, remember what the EL equations are: for every dynamical variable $x_i$ ,

$$\frac{d}{dt}\frac{\partial L}{\partial {\dot{x}}_i}-\frac{\partial L}{\partial x_i}=0.$$

In our case, that leads to the two equations below:

$$\begin{array}{rl}0+e^{-(x_1+x_2)}(x_2-x_1{x}_2+{\dot{x}}_2)& =0\\ -e^{-(x_1+x_2)}({\dot{x}}_1+{\dot{x}}_2)+e^{-(x_1+x_2)}(x_1-x_1{x}_2+{\dot{x}}_2)& =0.\end{array}$$

It is evident that the first equation above is one of the two Lotka-Volterra equations after we eliminate the exponential factor. It is not as evident that the second equation above is the second Lotka-Volterra equation that Paine originally started with—but in fact it is. It is the presence of the minus sign in front of the “kinetic piece” that eliminates the ${\dot{x}}_2$ and thus defines the dynamics for ${\dot{x}}_1$ .

---

Paine’s Comments

We should remark that one of the main reasons behind Paine’s mathematical approach is because its central objective is actually to find integrals of the motion in dynamical systems. That objective stands in contrast the one that is anticipated upon reading so much about “Hamiltonians” and “Lagrangians” of the Lotka-Volterra system in his article—this is not just another application of Classical Mechanics. In Classical Mechanics, passing from the Lagrangian to the Hamiltonian formulation is relatively straightforward: we simply need to compute the conjugate momentum according to $p:={\partial }_{\dot{q}}L$ , and then compute

$$H=\sum _{i=1}^N{p}_i{\dot{q}}^i-L.$$

But this Hamiltonian does not generate any sort of recognizable dynamics that have been featured in this problem. In fact, as we commented on earlier, the 2D LV system would actually be explicable only in a four-dimensional phase space should we pass from its Lagrangian description to its Hamiltonian description. Paine is aware of this fact, and addresses this issue on pg. 759:

One would be tempted to call the integral of motion for systems (21), (34) and (40) [which are the three 2D dynamical systems he considers in the text] the Hamiltonian. We have refrained from doing so because the function $H(t,x)$ for all systems found by the Legendre transformation does not return the appropriate dynamical equations by the normal canonical relations… The Jacobian of this transformation vanishes everywhere

$$\left|\left|\frac{{\partial }^{2}L}{\partial {\dot{x}}_j\partial {\dot{x}}_i}\right|\right|=0;$$

on the other hand, normal Hamiltonian theory requires the Jacobian to be everywhere nonvanishing.

The condition he discussed above is loosely called the Hessian Condition of Lagrangian Mechanics; it is more technically called the nondegeneracy (or regularity) condition on the Lagrangian Hessian. The basic idea is the following: To pass to a Hamiltonian, you must be able to solve for the ${\dot{q}}^i$ s in terms of the $p_i$ s. Note that if

$$\frac{{\partial }^{2}L}{\partial {\dot{q}}^2}=\frac{\partial }{\partial \dot{q}}\left(\frac{\partial L}{\partial \dot{q}}\right)=0,$$

for the one-dimensional case, then clearly $p={\partial }_{\dot{q}}L$ does not depend on $\dot{q}$ in the first place! So, if there is no dependence of $p$ on $\dot{q}$ , then we simply cannot solve the equation for $\dot{q}$ in terms of $p$ . The multidimensional case is the extension from solving a single equation to solving a system of equations, where the defining matrix is the Hessian matrix, formed by the components

$$W_{ij}:=\frac{{\partial }^{2}L}{\partial {\dot{q}}^i\partial {\dot{q}}^j}.$$

Paine now readily generates the “Hamiltonian” corresponding to his Lagrangian (not his Reduced Lagrangian!). The constant of the motion (his Eq. (39)) is

$$H=x_1{x}_2{e}^{-(x_1+x_2)}.$$

---

Exercise: Prove that Paine’s constant of the motion above (his Eq. (39)) is indeed a constant of the motion.

Solution: Let’s take the time derivative of the proposed Hamiltonian. The first step uses the product rule, and gives

$$\dot{H}=({\dot{x}}_1{x}_2+x_1{\dot{x}}_2)e^{-(x_1+x_2)}-x_1{x}_2{e}^{-(x_1+x_2)}({\dot{x}}_1+{\dot{x}}_2).$$

We must note that ${\dot{x}}_1+{\dot{x}}_2=x_1-x_2$ based on the original 2D LV dynamical equations. Then, after inserting the dynamics into the equation above, we find

$$\begin{array}{rl}\dot{H}& =e^{-(x_1+x_2)}[{\dot{x}}_1{x}_2+x_1{\dot{x}}_2-x_1{x}_2(x_1-x_2)]\\ & =e^{-(x_1+x_2)}[(x_1-x_1{x}_2)x_2+x_1(-x_2+x_1{x}_2)-x_1^2{x}_2+x_1{x}_2^2]\\ & =e^{-(x_1+x_2)}(x_1{x}_2-x_1{x}_2^2-x_1{x}_2+x_1^2{x}_2-x_1^2{x}_2+x_1{x}_2^2)\\ & =0.\end{array}$$

---

What we are now interested in is the corresponding integral of the motion based on Paine’s Reduced Lagrangian. We can perform the same Legendre transform as before, which starts by writing

$$H=\frac{\partial L}{\partial {\dot{x}}_1}{\dot{x}}_1+\frac{\partial L}{\partial {\dot{x}}_2}{\dot{x}}_2-L.$$

We already computed each of the partial derivatives of the Reduced Lagrangian before, so we can easily substitute the relevant terms above:

$$H={\dot{x}}_2{e}^{-(x_1+x_2)}-e^{-(x_1+x_2)}({\dot{x}}_2-x_1{x}_2)$$

After some quick simplification, we find that we have recovered the same $H$ as Paine did previously:

$$H=x_1{x}_2{e}^{-(x_1+x_2)}.$$

Conclusion

We showed that Paine’s method of finding a Lagrangian for the 2D LV equations, which involves an appeal to a particular mathematical formalism that we did not discuss here, initially yields a large Lagrangian with several redundancies. We also briefly commented on its interpretation within the standard formalism of Lagrangian Mechanics. We also briefly commented on how both Paine’s original Lagrangian and his Reduced Lagrangian are fundamentally different from the one that we introduced in our work.

We plan to use the analysis in this article to compare similar approaches to the problem of finding a Lagrangian for the two-dimensional LV system. What will (likely) come next is an analysis of the work by M. C. Nucci and K. M. Tamizhmani entitled Lagrangians for Biological Models in which they also offer a Lagrangian for the 2D LV system.

Notes/References:

1. This property is called the Non-uniqueness of Lagrangians. The Wikipedia article on the subject does an excellent job at the brief introduction, but absolutely every single Classical Mechanics textbook will discuss this property of Lagrangians.

2. I am still not actually all that clear on what the semicolon notation really indicates even though we see it all the time our work nowadays. Somehow, somewhere, at sometime, I decided that what I mean by the notation $f(q;t)$ above is “a function $f$ of $q$ and maybe of $t$ as well.”

3. The Hessian Condition on the conjugate momenta is not currently discussed on the Wikipedia pages for Lagrangian mechanics or Hamiltonian mechanics. To know about this condition requires that we consult the very serious textbooks on Classical Mechanics, such as Classical Dynamics: A Contemporary Approach, for example.

Updates:

1. 2026-08-23: Found another website that hosts the article in question where the title is spelled correctly. See the preface.