## Abstract

A class of generally nonlinear dynamical systems is considered, for which the Lagrangian is represented as a sum of homogeneous functions of the displacements and their derivatives. It is shown that an energy partition as a single relation follows directly from the Euler–Lagrange equation in its general form. The partition is defined solely by the homogeneity orders. If the potential energy is represented by a single homogeneous function, as well as the kinetic energy, the partition between these energies is defined uniquely. For a steady-state solitary wave, where the potential energy consists of two functions of different orders, the Derrick–Pohozaev identity is involved as an additional relation to obtain the partition. Finite discrete systems, finite continuous bodies, homogeneous and periodic-structure waveguides are considered. The general results are illustrated by examples of various types of oscillations and waves: linear and nonlinear, homogeneous and forced, steady-state and transient, periodic and non-periodic, parametric and resonant, regular and solitary.

## 1. Introduction

For free linear oscillations and sinusoidal waves, it has long been recognized that kinetic and potential energies averaged over the period are equal to each other. In some partial cases, this is observed directly [1–4]. This fact is also confirmed by Whitham [5] for a non-specified linear sinusoidal wave. The statement is found as a ‘side effect’ of a version of the variation principle, which was introduced as the basis for the theory of slow-varying sinusoidal waves. In Whitham's considerations, the expression for a sinusoidal wave
*A*,** k** and

*ω*are the slow-varying parameters, is substituted in the Lagrangian, neglecting derivatives of these parameters. Then, the Lagrangian is averaged over the period. Thereafter, the equipartition directly follows as a result of the variation of the wave amplitude.

Along with this, the energy partition in different areas of nonlinear dynamics and with different meanings is of interest [2,6,7]. In addition, it seems that even the linear case is not fully explored. The cases of systems with time-dependent parameters, forced motions and solitary waves are not discussed.

We consider this topic in terms of classical mechanics and obtain the partition relations for a class of generally nonlinear oscillations and waves, where the Lagrangian is represented by a sum of homogeneous functions of the displacements and their derivatives, possibly not only of first order. The Lagrangian can also depend explicitly on time and spatial coordinates. Starting from the Euler–Lagrange equation of a general view and taking into account Euler's theorem on homogeneous functions we obtain both the energy partition relation and the conditions defining the regions of averaging (this could be the oscillation period, if it exists, and not only it). The partition relation is fully defined by the homogeneity orders, regardless of other parameters of the system and the dynamic process. In the case where the potential energy is represented by a single homogeneous function, as well as the kinetic energy, the partition between these energies is defined uniquely.

For a steady-state solitary wave, where the potential energy consists of two functions of different orders, there are three different energy terms, and the single equality is not sufficient to obtain the desired energy partition. At the same time, the wave is exponentially localized, which allows us to use the Derrick–Pohozaev identity [8–10] as an additional relation.

The identity is based on the representation of the steady-state solitary wave as a static state. Note that, in the steady-state mode, the kinetic energy can be considered as such, as a function of *u*′=∂*u*/∂*x*. In the latter case, it can be treated as an additional (negative) part of the potential energy. Different energy relations, one discussed in the present paper and another given by the Derrick–Pohozaev identity, follow from the static representation as a result of different variations. While the former corresponds to the displacement variation *δu*=*εu*, the latter is a uniform *x*-extension/compression. The use of both relations allows us to separate the kinetic and potential energies completely.

First, we consider general relations for finite discrete systems. The relations obtained for the latter are applicable, with minor additions, to a finite continuous-material body and to an infinite continuous or discrete waveguide as discussed below. Then, some examples of free and forced oscillations are presented, which evidence that the linearity is neither a necessary nor a sufficient condition for the equipartition. Next, some examples of linear and nonlinear oscillations are presented, where the energy partition and the regions of averaging are clearly seen. Homogeneous and forced, steady-state and transient, periodic and non-periodic, regular, parametric and resonant oscillations are examined.

Further, the general relations are presented to be applicable for a finite continuous body and for homogeneous, periodic-structure and discrete waveguides. Some examples of linear and nonlinear, periodic, steady-state, transient and solitary waves are presented, where the energy partition is also demonstrated.

## 2. The main relations

Consider the Euler–Lagrange equations
** u**=(

*u*

_{1},

*u*

_{2},…). Multiplying the equations by

*u*

_{i}and integrating over an arbitrary segment,

*t*

_{1}≤

*t*≤

*t*

_{2}, we obtain (with summation on repeated indices)

Regarding this formulation, we note that the relation (2.4) follows from Hamilton's principle of least action with the variation *δ*** u** replaced by

**. This, however, entails a change in the additional conditions. Namely, in the variational formulation, the integration limits,**

*u**t*

_{1,2}, are arbitrary under the condition

*δ*

**=0 at**

*u**t*=

*t*

_{1,2}, whereas, in the modified formulation, they are not arbitrary but must meet the condition (2.5).

We now suppose that the Lagrangian is a sum of homogeneous functions of *ν*_{n} is the homogeneity order. In this expression, *ν*_{n} can be not only an integer, and it is assumed that only a positive value can have a fractional exponent. For example, in the latter case, |*u*_{i}|^{ν} can be present in the Lagrangian, but not *u*_{i}|^{ν}≥0. Note that the homogeneity condition excludes possible non-uniqueness in the definition of the energies.

With reference to Euler's theorem on homogeneous functions

The latter relation can serve for the determination of the energy partition. For example, let *L* be a difference between the kinetic and potential energies, *μ* and *ν*, respectively. In this case, the relation between the averaged energies

In particular, in the linear case, where *ν*=*μ*=2, the averaged energies are equal. Cases *ν*>*μ* and *ν*<*μ* correspond to hardening and softening nonlinearity, respectively. The potential energy vanishes as

Note that *in statics*, *B*_{1,2}, vanish independently of *t*_{1,2}. It follows from (2.8) that
*c*(i)).

It is remarkable that the energy ratio in (2.10) is equal to the homogeneity order ratio, regardless of the other parameters of the system and the dynamic process. In turn, the homogeneity orders generally depend on the constitutive properties but not the material constants. In the case where the potential energy is represented by the sum of two or more homogeneous functions of different orders, the relation (2.8) is only one with respect to three or more terms. In this case, it is not sufficient for the determination of ratios between all terms, but it still allows one to draw some conclusions concerning the energy partition.

The condition in (2.5) is satisfied in periodic oscillations
** u**=0 at

*t*=

*t*

_{1,2}. In addition, if the kinetic energy, being a homogeneous function of order

*μ*with respect to

*u*

_{i},

*B*

_{1}(

*B*

_{2})=0 if

*t*=

*t*

_{1}(

*t*

_{2}). Therefore, the condition (2.5) is also satisfied in the case of non-periodic oscillations, where

**and/or**

*u**t*

_{1}<

*t*

_{2}<⋯<

*t*

_{m}<⋯ , the energy partition averaged over any interval between two such points is the same. So, the energy partition appears fixed for a large range of time.

Here, we have considered systems of a finite number of material points. The main relations presented here are applicable, with minor additions, to a finite continuous-material body and to an infinite continuous or discrete waveguide, as discussed in §§4 and 5.

While the energy partition is defined solely by the homogeneity orders, below we calculate the energy distribution function, *B*_{1}=0 at *t*=*t*_{1}=0. This function must vanish at *t*=*t*_{2} (2.8) if the condition (2.5) is satisfied at that point. This representation allows us to demonstrate how the partition varies during the (*t*_{1},*t*_{2})-interval and how

## 3. Discrete systems

### (a) The linearity and equipartition

Three examples are presented below which evidence that the linearity is neither a necessary nor a sufficient condition for the equipartition. First, consider a piecewise linear equation
*H* is the Heaviside step function and *u*(*t*), speed

It can be seen that the energies averaged over the period,

Next, we consider a nonlinear equation related to centrosymmetric oscillations of a bubble in unbounded, perfect, incompressible liquid, where the added mass is proportional to the bubble radius cubed. Let the (dimensionless) energies be
*r*(*t*) and the energy distribution function

It is seen that the equipartition also takes place in this strongly nonlinear situation.

Next, consider forced resonant oscillations as an inhomogeneous linear problem

It follows that under zero initial conditions
*P*=1.

Thus, there is no equipartition between the kinetic and potential energies in this inhomogeneous linear problem

### (b) Simplest example of unequal partition

Consider a ball of mass *M* thrown upwards with speed *g* is the acceleration of gravity. Here, *u*=0 at *t*=*t*_{1}=0 and *t*=*t*_{2}=*v*_{0}/*g*. It follows from (2.8) that the energies averaged over the (*t*_{1},*t*_{2})-segment must satisfy the relation

### (c) Linear and nonlinear oscillators

Consider oscillators with kinetic and potential energies
*u*, respectively, and can be different, *ν*/2. Plots of the displacements and energies for *ν*=6/5, 2 and 6 are shown in figures 4–7. To have the same period, 2*π*, for all these cases, we take *ν*=6/5, 2 and 6, respectively, and *m*=*u*_{0}=1,

### (d) Two-degrees-of-freedom nonlinear system

Consider a mass-spring chain (figure 8) with energies

Plots of *u*_{1}(*t*),*u*_{2}(*t*) and *m*_{1}=*m*_{2}=*u*_{0}=1, *ν*=4 based on the corresponding dynamic equations

### (e) Oscillators under variable stiffness

Consider oscillators whose potential energy depends explicitly on *t*. First, let the energies be
*t*_{1},*t*_{2}, where *B*_{1}=*B*_{2} (2.5). The corresponding dynamic equation is
*u*(*t*), under conditions *u*(0)=0,

Next, we consider parametric resonance based on Mathieu's differential equation. The energies are
*u*(*t*), under conditions

## 4. A finite continuous body

In this case, the Lagrangian depends, in addition, on derivatives with respect to the spatial variables. The corresponding Euler–Lagrange equation multiplied by ** u**(

**,**

*x**t*) is now integrated by parts over both the time-segment (

*t*

_{1},

*t*

_{2}) and the body material volume,

*Ω*. As a result, we obtain the relation (2.8), where

*L*is the Lagrangian incorporating the energy of the whole body including its boundaries, whereas the condition (2.5) becomes

*Ω*. Such terms reflect the energy located on body boundaries.

As an example consider the collision of a linearly elastic rod, 0<*x*<1, with a rigid obstacle. For this linear problem the (dimensionless) energies are
*μ*=*ν*=2. The corresponding one-dimensional wave equation is
*t*<2. Dependencies for the energies,

Next, we consider oscillations of a nonlinearly elastic rod, 0<*x*<*l*. Let the energies be
*u*(*x*,*t*)=*X*(*x*)*T*(*t*), which results in equations

## 5. Waves

In the case where the total energy of a wave is infinite, we have to choose a finite segment of the waveguide (*x*_{1},*x*_{2}), similar to (*t*_{1},*t*_{2}) in time, to avoid boundary terms in integration by parts over this segment. Namely, in the conversions
*x*_{1}<*x*<*x*_{2},*t*_{1}<*t*<*t*_{2}).

However, in some classes of waves, which are considered below, the averaging over one variable, *t* or *x*, appears to be sufficient.

### (a) Waves in a homogeneous waveguide

For waves depending only on one variable, *η*=*x*−*vt*, propagating in such a waveguide, averaging over the *t*- or *x*-period, that is, averaging over the *η*-period, is sufficient. So, in this case, the above relation (2.8) is valid with respect to any cross-section of the waveguide.

#### (i) Nonlinearly elastic beam

Consider a wave in a beam where the energies are
*L*(*η*) calculated based on this equation are presented in figure 16.

#### (ii) Linear and nonlinear Klein–Gordon equations

Consider periodic waves, *u*(*η*),*η*=*x*−*vt*, with the energies
*ν*=2 (the linear wave), 3 and 6 in figures 17–19, respectively. Note that, in the case of waves, the total energy varies during the period.

### (b) A wave in periodic structures

For a periodic structure, we consider a Floquet–Bloch wave, where the displacements at discrete points *x*=*an*+*x*_{0}, 0≤*x*_{0}<*a*, depend on *x*_{0} and *η*=*an*−*vt*, and possibly other variables, but not on integer *n* and time separately (here *a* is the spatial period). This means that each of the cross sections with the same *x*_{0} describes the same periodic trajectory, which differs only by a shift in time. We assume that (generalized) displacements of the cross sections corresponding to a fixed *x*_{0}, say, *x*_{0}=0, defined those for the other cross sections.

In the considered case of an infinite waveguide, the energy of a periodic wave is also infinite. However, in the case of a finite radius of the interaction, where the *n*th cross section is directly connected with a finite number of the others (with the same *x*_{0}), the displacement *u*_{n} appears only in a finite number of the infinite series representing the Lagrangian. Accordingly, the averaging (2.8) should correspond to the structure period
*a* is the spatial period of the structure. Note that the spatial integral becomes a sum in the case of a discrete structure.

As an example consider a linear mass-spring chain. The (non-dimensional) energies are
*μ*=*ν*=2 and there is equipartition. This result also follows from the dynamic equation

Waves in nonlinear chains are considered in §5*c*.

### (c) Solitary wave

In the case of a solitary wave, the partition of its total energy is considered, and the integration limits become infinite,

#### (i) Phi-four equation

Consider the so-called Phi-four equation [10,11]
*t* and *x*, respectively. Equation (5.12) is satisfied by a kink
*v* (*v*^{2}<1) and *ϕ*_{0}, are the kink's speed and the ‘initial’ phase, respectively. The corresponding energies are
*ν*=0) is chosen to limit the potential energy.

While the moving kink corresponds to the transition from one zero-energy state to another, the particle velocity,

The kink, the solitary wave and the energies as functions of *η* are shown in figures 20 and 21. Far away from the transition region, there is a static state, where, in accordance with (2.11) and (5.16),

#### (ii) Derrick–Pohozaev identity

In the problem (5.16), there are three unknown energy terms, *u*′=∂*u*/∂*x*,
*δu*=*εu*, the latter is a uniform *x*-extension. The use of both relations allows us to separate the kinetic and potential energies completely.

The Derrick–Pohozaev identity is found for the nonlinear Klein–Gordon equation with a general function of *u* (see [8,9] and [10], sect. 7.4). Derived for a multi-dimensional case, it reads in one dimension as the equality of two parts of the energy, one as a function of *u* and another as a function of *u*′. In our terms, the identity is

#### (iii) Fermi–Pasta–Ulam problem

Next, we consider a transient problem for a wave excited by a pulse applied to the same but ‘semi-infinite’ linear chain. No solitary wave can exist in the latter, and we study this problem to compare the result with that for a nonlinear chain first examined by Fermi *et al.* [12]. The displacements of the first 20 masses under unit pulse acting on the first left-hand mass and also the corresponding speeds are presented in figures 23 and 24, respectively.

The wave itself, especially the particle velocity wave, and the energy distribution function, *μ*=*ν*_{1}=2, *ν*_{2}=4 and

A fast-established strongly localized solitary wave appears in the nonlinear chain with quadratic terms in the expression of the potential energy (5.21) removed. In this case,

## 6. Conclusion

The energy partition obeys the relation derived for the case where the Lagrangian is represented as a sum of homogeneous functions of the displacements and their derivatives (explicit dependencies on time and spacial coordinates are not excluded). If the kinetic and potential energies are entirely homogeneous the homogeneity orders define the partition uniquely.

The equipartition corresponds to the case where the kinetic and potential energies are of the same homogeneity order. In particular, this is true for homogeneous linear problems where both energies are of second order. At the same time, the linearity is neither a necessary nor a sufficient condition for the equipartition.

The energy partition corresponds to the energies averaged over a region satisfying the condition (2.5). Thus, the integration can correspond not only to the period, if it exists, but also to the end points, where ** u**=0 and, in the case (2.14), where

The energy partition relation is valid as far as the related general conditions are satisfied. So, it is applicable to any specific problem of the corresponding class, and there is no need for an examination of any specific problem in more detail. Nevertheless, we have presented various examples to show how the partition varies during the time interval of averaging, and also to demonstrate the validity of the relation and the conditions relating to the averaging. These examples show that the relation is valid for various problems: linear and nonlinear, steady state and transient, conservative and non-conservative, homogeneous and forced oscillations, periodic and solitary waves.

In the case of the steady-state solitary wave, where the potential energy consists of two functions of different orders, the static Derrick–Pohozaev identity is involved along with the dynamic energy partition relation derived in this paper. The use of both of them allowed us to find the desired kinetic-to-potential energy relation.

## Funding statement

The author acknowledges the support provided by the FP7 Marie Curie grant no. 284544-PARM2.

## Conflict of interests

The author has no competing interests.

## Acknowledgements

I thank the member of the editorial board who drew my attention to the Derrick–Pohozaev identity.

- Received October 30, 2014.
- Accepted January 23, 2015.

- © 2015 The Author(s) Published by the Royal Society. All rights reserved.