Variational principle for bifurcation in Lagrangian mechanics

TL;DR

This paper develops a variational approach using derivatives of the action integral to identify and analyze bifurcations of periodic solutions in Lagrangian mechanics, emphasizing the role of the Hessian eigenvalues.

Contribution

It introduces a method leveraging higher derivatives of the action to detect and characterize bifurcations in Lagrangian systems, linking eigenvalues to bifurcation points.

Findings

Eigenvalues of the Hessian tend to zero at bifurcation points.

Higher derivatives of the action determine bifurcation properties.

The method provides a way to find bifurcations from known solutions.

Abstract

An application of variational principle to bifurcation of periodic solution in Lagrangian mechanics is shown. A few higher derivatives of the action integral at a periodic solution reveals the behaviour of the action in function space near the solution. Then the variational principle gives a method to find bifurcations from the solution. The second derivative (Hessian) of the action has an important role. At a bifurcation point, an eigenvalue of Hessian tends to zero. Inversely, if an eigenvalue tends to zero, the zero point is a bifurcation point. The third and higher derivatives of the action determine the properties of the bifurcation and bifurcated solution.

Equations

$$L=\sum _{k=1}^N\frac{m_k}{2}{\left(\frac{d{q}_k}{dt}\right)}^2+V(q_1,q_2,\dots ,q_N),S=\int dtL.$$

$$\delta S=0\iff -m_k\frac{d^2{q}_k}{d{t}^2}+\frac{\partial V}{\partial q_k}=0.$$

$$\kappa \to 0\text{mbox}for\xi \to {\xi }_0.$$

$$S(x)=S(0)+\frac{\kappa }{2}{x}^2-\frac{A_3}{3!}{x}^3+\text{Or}(x^4),A_3\ne 0,$$

$$S^{\prime }(x)=x\left(\kappa -\frac{A_3}{2}x+\text{Or}(x^2)\right)=0$$

$$x_{\text{b}}=\frac{2}{A_3}\kappa +\text{Or}({\kappa }^2)$$

$$S(x_{\text{b}})=S(x_{\text{o}})+\frac{2}{3A_3^2}{\kappa }^3+\text{Or}({\kappa }^4),S^{\prime \prime }(x_{\text{b}})=-\kappa +\text{Or}({\kappa }^2).$$

$$S(q+\delta q)=S(q)+\frac{1}{2}{\int }_0^{T}dt\sum _{ij}\delta q_i\left(-m_i{\delta }_{ij}\frac{d^2}{d{t}^2}+\frac{{\partial }^{2}V}{\partial q_i\partial q_j}\right)\delta q_j+\text{Or}(\delta q^3),$$

$$q(t+T)=q(t),\delta q(t+T)=\delta q(t).$$

$${\mathcal{H}}_{ij}=-m_i{\delta }_{ij}\frac{d^2}{d{t}^2}+\frac{{\partial }^{2}V}{\partial q_i\partial q_j}.$$

$$S(q+\delta q)=S(q)+\frac{1}{2}{\int }_0^{T}dt\delta q\mathcal{H}\delta q+\text{Or}(\delta q^3),\mathcal{H}=-m\frac{d^2}{d{t}^2}+V_{qq}$$

$$m\frac{d^{2}q}{d{t}^2}=V_q(q).$$

$$m\frac{d^2{q}_{\text{b}}}{d{t}^2}=V_q(q_{\text{b}}),m\frac{d^2{q}_{\text{o}}}{d{t}^2}=V_q(q_{\text{o}}).$$

$$Rm\frac{d^2\Phi }{d{t}^2}=V_q(q_{\text{o}}+R\Phi )-V_q(q_{\text{o}})=R{V}_{qq}\Phi +\text{Or}(R^2).$$

$$\mathcal{H}\Phi =\text{Or}(R).$$

$$\mathcal{H}\Phi \to 0.$$

$$q_{\text{b}}-q_o=r\varphi +r\sum _{\alpha }{ϵ}_{\alpha }{\psi }_{\alpha },\mathcal{H}\varphi =\kappa \varphi ,\mathcal{H}{\psi }_{\alpha }={\lambda }_{\alpha }{\psi }_{\alpha },$$

$${\int }_0^{T}dt{\varphi }^2=1,{\int }_0^{T}dt{\varphi }_{\alpha }{\psi }_{\beta }={\delta }_{\alpha \beta }\text{mbox}and{\int }_0^{T}dt\varphi {\psi }_{\alpha }=0.$$

$$r\mathcal{H}(\varphi +\sum _{\alpha }{ϵ}_{\alpha }{\psi }_{\alpha })=r(\kappa \varphi +\sum _{\alpha }{ϵ}_{\alpha }{\lambda }_{\alpha }{\psi }_{\alpha })=O(r^2).$$

$$\kappa \varphi +\sum _{\alpha }{ϵ}_{\alpha }{\lambda }_{\alpha }{\psi }_{\alpha }=O(r).$$

$$\kappa =O(r),{ϵ}_{\alpha }=O(r)\text{mbox}forr\to 0.$$

$$\delta q=q_{\text{o}}(t+r)-q_{\text{o}}(t)=r\frac{d{q}_{\text{o}}}{dt}+\sum _{n\ge 2}\frac{r^n}{n!}\frac{d^n{q}_{\text{o}}(t)}{d{t}^n}.$$

$$S(r,ϵ)$$

$$⟨f_1{f}_2\dots f_n⟩={\int }_0^{T}dt\frac{{\partial }^{n}V}{\partial q^n}{f}_1{f}_2\dots f_n.$$

$$⟨fg⟩=\int dt\frac{{\partial }^{2}V}{\partial q_i\partial q_j}{f}_i{g}_j,⟨f^2⟩=\int dt\frac{{\partial }^{2}V}{\partial q_i\partial q_j}{f}_i{f}_j,\dots .$$

$${\partial }_{r}S(r,ϵ)={\partial }_{{ϵ}_{\alpha }}S(r,ϵ)=0$$

$$S_{\text{LS}}(r)=S(r,ϵ(r)).$$

$$S_{\text{LS}}^{\prime }(r)=\frac{d{S}_{\text{LS}}(r)}{dr}={\partial }_{r}S(r,ϵ)+\sum _{\alpha }{{\partial }_{{ϵ}_{\alpha }}S(r,ϵ)|}_{ϵ={ϵ}_{\alpha }(r)}\frac{d{ϵ}_{\alpha }(r)}{dr},$$

$${\partial }_{{ϵ}_{\alpha }}S(r,ϵ)=r^2\left({\lambda }_{\alpha }{ϵ}_{\alpha }+\sum _{n\ge 2}\frac{r^{n-1}}{n!}⟨{\varphi }^n{\psi }_{\alpha }⟩+\sum _{\beta }{ϵ}_{\beta }\sum _{m\ge 1}\frac{r^m}{m!}⟨{\varphi }^m{\psi }_{\alpha }{\psi }_{\beta }⟩+\dots \right)=0.$$

$${ϵ}_{\alpha }=-\sum _{n\ge 2}\frac{r^{n-1}}{n!}\frac{⟨{\varphi }^n{\psi }_{\alpha }⟩}{{\lambda }_{\alpha }}+\sum _{m\ge 1,n\ge 2}\frac{r^{m+n-1}}{m!n!}\sum _{\beta }\frac{⟨{\varphi }^m{\psi }_{\alpha }{\psi }_{\beta }⟩⟨{\varphi }^n{\psi }_{\beta }⟩}{{\lambda }_{\alpha }{\lambda }_{\beta }}+\dots$$

Full text

Keywords: variational principle, bifurcation, Lagrangian, action, Hessian, Lyapunov-Schmidt

1 Variational principle

Variational principle demands that a function that gives stationary point of the action integral is a solution of motion [1, 2]. In this paper, we consider dynamical system described by the following Lagrangian $L$ and the action integral $S$ with $N$ degrees of freedom,

[TABLE]

The variables $q_{k}$ can be generalized coordinates. However, we assume the kinetic term is the quadratic form of $dq_{k}/dt$, and $m_{k}$ are constants. The variational principle produces the equation of motion,

[TABLE]

The variational method to find a solution of the equation of motion is an application of the variational principle. Namely, a minimum or maximum point of the action $S$ in function space is a solution of the equation of motion, since minimum or maximum satisfies $\delta S=0$ [3]. A proof of existence of minimum or maximum of the action gives a proof of existence of a solution [4].

The aim of this paper is to describe a method to find bifurcation point and bifurcated function emerged from a periodic solution based on the variational principle. Suppose we get a few derivatives of the action at a solution, $\delta^{2}S,\delta^{3}S,\dots$, we will see the values of the action varies with place to place in functional space around the solution. Then, if there is a stationary point other than the original solution, the point must be a new solution by the principle. Since, with a few derivatives, the region we can see around the original solution is very restricted, the other stationary point is usually out of the region. Fortunately, observation near the bifurcation point will give us a chance to find the original and other stationary point. Suppose the Lagrangian has a parameter $\xi$, and $\xi=\xi_{0}$ is a bifurcation point. If we observe the action in a range of $\xi$ that include $\xi_{0}$, we will see the bifurcated solution come into our region, and finally reaches to the original solution at $\xi_{0}$. Thus, bifurcations can be described by a few derivatives of the action.

How do we find the bifurcation point observing only the original solution? In other words, how do we know that another stationary point is approaching to the original? If the action $S$ were a function of single variable $x$, approaching two stationary points that satisfy $S^{\prime}(x_{\textrm{\scriptsize o}})=S^{\prime}(x_{\textrm{\scriptsize b}})=0$ makes the second derivative $S^{\prime\prime}(x_{\textrm{\scriptsize o}})$ zero for the limit $x_{\textrm{\scriptsize b}}\to x_{\textrm{\scriptsize o}}$. Indeed, in section 2, we will show that the second derivative of the action, described by Hessian operator, must have zero eigenvalue at the bifurcation point. Let $\kappa$ be the eigenvalue that tends to zero,

[TABLE]

This is a necessary condition for bifurcation. There are two possibilities for (3),

$\kappa\neq 0$ and $\kappa\to 0$ for $\xi\to\xi_{0}$, 2. 2.

$\kappa$ is always zero.

In this section, we just consider the case (1). The case (2) is irrelevant to bifurcations, that will be discussed later.

The answer of the question is the following, if an eigenvalue of the Hessian of the original solution tend to zero, a bifurcation may appear. Inversely, if no eigenvalue tends to zero, no bifurcation appears.

What will happen if $\kappa\neq 0$ and $\kappa\to 0$. Again, if the action were a function of single variable $x$, for example,

[TABLE]

then the condition for stationary points

[TABLE]

has two zeros. The solution $x_{\textrm{\scriptsize o}}=0$ stands for the original solution and

[TABLE]

for the bifurcated solution. Thus if the second derivative of the original solution $S^{\prime\prime}(0)=\kappa$ tends to zero, a bifurcation appears. We now know in the case (4) that the bifurcated solution exists both sides of $\kappa>0$ and $\kappa<0$. We also know that the action and the second derivative of the action at the bifurcated solution are

[TABLE]

Although the action is a function of infinitely many variables, Lyapunov-Schmidt reduction makes the action to be a function of one or a few variables. This procedure will be shown in section 3. Using the reduced action, we will show that if an eigenvalue of Hessian tends to zero, the point zero is a bifurcation point. This gives a sufficient condition for bifurcation.

The Lyapunov-Schmidt reduced action also describes in which side of $\kappa$ the bifurcated solution exists, and describes value of the action and the eigenvalue of Hessian at the bifurcated solution as shown in (7). It will be shown in section 4 and 5.

Summary and discussions are given in section 6. Applications of this method to individual periodic solutions in Lagrangian mechanics will be published separately.

2 The second derivative of the action

In this section, we first define Hessian operator that describes the second derivative of the action. Then, we describe the necessary condition for bifurcations.

2.1 Definition of Hessian

Now, let us calculate the second derivative of the action (1) at a periodic solution $q(t)$,

[TABLE]

where the symbol $\delta_{ij}$ is the Kronecker delta, and

[TABLE]

We restricted the function space to be periodic with period $T$ which is the same as that of the solution. The first order of $\delta q$ is zero, since $q$ is a solution of the equation of motion.

The second derivative defines Hessian operator $\mathcal{H}$,

[TABLE]

Hereafter, we use the vector and matrix notation, $m=(m_{i}\delta_{ij})$, $q=(q_{i})$, $V_{q}=(\partial V/\partial q_{i})$, and $\mathcal{H}=(\mathcal{H}_{ij})$ with $i,j=1,2,\dots,N$.

Then, the action up to the second order is

[TABLE]

and the equation of motion is

[TABLE]

2.2 Necessary condition for bifurcation

Let us assume the Lagrangian contains a parameter $\xi$, and at $\xi=\xi_{0}$ the solution $q_{\textrm{\scriptsize o}}$ bifurcates to make a bifurcated solution $q_{\textrm{\scriptsize b}}$. The period $T$ may or may not depend on the parameter $\xi$. At $\xi$ being very close to $\xi_{0}$ where two solutions exists, the following equations of motion are satisfied,

[TABLE]

Let the difference of the two solutions be $q_{\textrm{\scriptsize b}}-q_{\textrm{\scriptsize o}}=R\Phi$, $R\neq 0$ and $\int_{0}^{T}dt\Phi^{2}=1$. Difference of two equations yields

[TABLE]

Dividing by $R$, we obtain

[TABLE]

The limit $\xi\to\xi_{0}$ makes $R\to 0$ and

[TABLE]

This is a necessary condition for bifurcation. Namely, at the bifurcation point, the Hessian must have zero eigenvalue.

Now, we expand the difference $q_{\textrm{\scriptsize b}}-q_{o}$ by the eigenfunctions of Hessian,

[TABLE]

and consider the contents of the condition (16). We use Greek indexes $\alpha,\beta,\dots$ to distinguish functions, while roman $i,j,\dots$ to express the vector component of a function. The functions $\phi$ and $\psi_{\alpha}$ are normalized orthogonal functions

[TABLE]

The function $\phi$ is the eigenfunction that is shown to exist by the condition (16). Namely $\kappa\to 0$ for $\xi\to\xi_{0}$. The same arguments for (13) to (14) yields

[TABLE]

Dividing by $r\neq 0$, we obtain

[TABLE]

Let us take a small region of $\xi$ that contains $\xi_{0}$, where only $\kappa\to 0$ for $\xi\to\xi_{0}$ and all $|\lambda_{\alpha}|$ are grater than a positive number. (We exclude zero eigenvalues that are always zero for any parameter $\xi$. Discussion for zero eigenvalues will be given soon later.) Since, $\phi$ and $\psi_{\alpha}$ are orthogonal normalized functions and $|\lambda_{\alpha}|$ is grater than a positive number,

[TABLE]

This is the contents of (16) for necessary condition for bifurcation. Since the difference $q_{\textrm{\scriptsize b}}-q_{o}=r\phi+r\sum\epsilon_{\alpha}\psi_{\alpha}$ and $r\epsilon_{\alpha}=O(r^{2})$, the function $\phi$ is the primary part and $\psi_{\alpha}$ are secondary part for the difference.

Zero eigenvalues that always exists for any values of the parameter $\xi$ are irrelevant to any bifurcations. They are connected to the conservation laws. For example, the energy conservation law is connected to a shift of the origin of time, $q(t)\to q(t+r)$ by Noether’s theorem [1, 2]. The corresponding eigenfunction is proportional to $dq_{\textrm{\scriptsize o}}/dt$. Actually, the action is flat along the curve parametrized by $r$,

[TABLE]

This eigenvector $dq_{\textrm{\scriptsize o}}/dt$ does not make bifurcation. Moreover, we can exclude $dq_{\textrm{\scriptsize o}}/dt$ from the set $\{\psi_{\alpha}\}$ by fixing the origin of time. In general, eigenfunctions that belong to zero eigenvalues are irrelevant to bifurcations and can be excluded from the set $\{\phi,\psi_{\alpha}\}$.

In the next section, we will show that if an eigenvalue $\kappa\neq 0$ tends to zero, the zero point is a bifurcation point.

3 Lyapunov-Schmidt reduction of the action

and a sufficient condition for bifurcation

In this section, we will show that the action $S(q_{\textrm{\scriptsize o}}+r\phi+r\sum\epsilon_{\alpha}\psi_{\alpha})$ actually has stationary point if $\kappa\neq 0$ tends to zero. This gives the sufficient condition for bifurcation.

The eigenvalue $\kappa$ may be degenerated. We first consider non-degenerated case in subsection 3.1 and 3.2. Degenerated case will be considered in 3.3.

3.1 Reduced action for non-degenerated case

In this subsection, we assume the eigenvalue $\kappa$ is not degenerated. So, the eigenfunction $\phi$ belongs to $\kappa$ is unique. The action $S(q_{\textrm{\scriptsize o}}+r\phi+r\sum\epsilon_{\alpha}\psi_{\alpha})$ is given by the expansion series of $r$ and $\epsilon_{\alpha}$,

[TABLE]

where for functions $f_{1},f_{2},\dots,f_{n}$,

[TABLE]

The vector indexes of functions are understood to be properly contracted with the indexes of derivative of $V$, for example,

[TABLE]

By the variational principle, the solutions of stationary conditions

[TABLE]

are solutions of equation of motion. From among them, we choose solutions that satisfy $\epsilon_{\alpha}\to 0$ and $\kappa\to 0$ for $r\to 0$, since we are looking for bifurcated solutions.

Instead of solving the conditions (26) at once, we first solve $\epsilon_{\alpha}(r)$ for arbitrary $r$ and put them into the action to define the reduced action,

[TABLE]

This process is known as Lyapunov-Schmidt reduction [5, 6]. Since

[TABLE]

the condition $S^{\prime}_{\textrm{\scriptsize LS}}(r)=0$ with $\epsilon_{\alpha}=\epsilon_{\alpha}(r)$ is equivalent to the conditions (26). Therefore, if $S^{\prime}_{\textrm{\scriptsize LS}}(r)=0$ has solution $r=r_{\textrm{\scriptsize b}}\neq 0$ and $r_{\textrm{\scriptsize b}}\to 0$ for $\kappa\to 0$, a bifurcated solution $q_{\textrm{\scriptsize b}}=q_{\textrm{\scriptsize o}}+r_{\textrm{\scriptsize b}}\phi+r_{\textrm{\scriptsize b}}\sum\epsilon_{\alpha}(r_{\textrm{\scriptsize b}})\psi_{\alpha}$ exists. The value of the action is also correctly given by the reduced action, $S_{\textrm{\scriptsize LS}}(r_{\textrm{\scriptsize b}})=S(r_{\textrm{\scriptsize b}},\epsilon(r_{\textrm{\scriptsize b}}))$.

Now, let us solve the condition (26) for $\epsilon$.

[TABLE]

The solution that satisfy $\epsilon_{\alpha}\to 0$ for $r\to 0$ is uniquely determined recursively,

[TABLE]

The lowest order term is

[TABLE]

Substituting this solution into (23), we obtain the reduced action

[TABLE]

where

[TABLE]

Thus we obtain the reduced action $S_{\textrm{\scriptsize LS}}(r)$ that depends on single variable $r$. Other variables $\epsilon_{\alpha}$ are eliminated by the condition for stationary point .

3.2 Sufficient condition for bifurcation

Let the reduced action generally be

[TABLE]

Then the condition for stationary point is

[TABLE]

the solution $r_{\textrm{\scriptsize o}}=0$ stands for the original solution, and the solution of

[TABLE]

stands for a bifurcation, Then the action at the bifurcated solution is

[TABLE]

So far, we have shown that there are bifurcated solution given in series expansion of $r$, namely (30) and (36), which certainly satisfies conditions (21).

Now, we proceed to the last step for sufficient condition by solving the series (36) inversely. If $A_{3}\neq 0$, a bifurcated solution

[TABLE]

appears at $\kappa=0$. We call this a third order bifurcation. Whereas, if $A_{3}=0$ and $A_{4}\neq 0$, a bifurcated solution

[TABLE]

appears. We call this a fourth order bifurcation. In general, if $A_{3}=A_{4}=\dots=A_{n-1}=0$ and $A_{n}\neq 0$, a bifurcation

[TABLE]

will appear, that we call a $n$-th order bifurcation. The bifurcated solution $r_{\textrm{\scriptsize b}}$ exists both $\kappa<0$ and $\kappa>0$ for odd $n$. Whereas it exists only one side of $\kappa$ for even $n$, $\kappa>0$ side if $A_{n}>0$ and $\kappa<0$ side if $A_{n}<0$. Therefore, for almost all cases, a bifurcation appears at $\kappa=0$. So, $\kappa\neq 0$ for $\xi\neq\xi_{0}$ and $\kappa\to 0$ for $\xi\to\xi_{0}$ is the sufficient condition for bifurcation.

One possible exception is the case $A_{n}=0$ for all $n$. In this case, however, the reduced action is exactly $S_{\textrm{\scriptsize LS}}(r)=S(q_{\textrm{\scriptsize o}})+\kappa r^{2}/2$. This action behaves badly. Consider the behaviour of this reduced action at sufficiently large $r=M$. For the small interval of $-1/M<\kappa<1/M$, the change of action is huge, since $-M/2<\kappa r^{2}/2<M/2$. Although, we didn’t find a logic to exclude this case, this case unlikely exists.

Before leaving this subsection, let us calculate the distance between the bifurcated and the original solution that is defined by,

[TABLE]

because $\epsilon_{\alpha}^{2}=\Or(r^{2})$.

3.3 Degenerated case

Now, let us consider the case $\kappa$ is degenerated with degeneracy number $g$. In this case, there are $g$ linearly independent eigenfunctions for $\mathcal{H}\phi_{\gamma}=\kappa\phi_{\gamma}$, $\gamma=1,2,\dots,g$. In this case, any function in the space spanned by $\phi_{\gamma}$ can be expressed as $r\phi(\theta)$ using a polar coordinates $(r,\theta)=(r,\theta_{1},\theta_{2},\dots,\theta_{g-1})$. For example, $r\phi(\theta)=r(\cos(\theta)\phi_{1}+\sin(\theta)\phi_{2})$ if $g=2$.

Following the same arguments in section 3.1, we obtain the reduced action $S_{\textrm{\scriptsize LS}}(r,\theta)$,

[TABLE]

Where

[TABLE]

Then, the condition for stationary point for $\theta$ is

[TABLE]

Solving this equation, and substituting the solution $\theta(r)$ into $S_{\textrm{\scriptsize LS}}(r,\theta)$ yields one variable function $S_{\textrm{\scriptsize LS}}(r,\theta(r))$. Then, the same arguments for non-degenerated case in section 3.2 will be used to describe the sufficient condition for bifurcation.

There may several solutions $\theta(r)$ of equation (44). In such case, each solution yields each $S_{\textrm{\scriptsize LS}}(r,\theta(r))$. As a result, multiple bifurcated solutions corresponds to each solution $\theta(r)$ will emerge from the original at $\kappa=0$.

3.4 The second derivative of the reduced action

Before closing this section, let us consider a meaning of the second derivative of the reduced action. The second derivative of the reduced action by $r$ is

[TABLE]

Since this gives the curvature of $S_{\textrm{\scriptsize LS}}(r)$ to $r$ direction, we expect that this expression gives the eigenvalue $\kappa_{r}$ of the Hessian at $r$, $\mathcal{H}_{r}\Phi_{r}=\kappa_{r}\Phi_{r}$, where

[TABLE]

and $\Phi_{r}\to\phi$, $\kappa_{r}\to\kappa$ for $r\to 0$. Indeed, in A, we will show that the eigenvalue $\kappa_{r}$ to the order $r^{2}$ is correctly given by (45) for non-degenerated case.

4 Properties of third order bifurcation

If $A_{3}\neq 0$, a third order bifurcation appears. The reduced action is

[TABLE]

The stationary point of $r\neq 0$ is the solution of

[TABLE]

Inversely,

[TABLE]

The distance between $q_{\textrm{\scriptsize b}}$ and $q_{\textrm{\scriptsize o}}$ is

[TABLE]

The difference of the action is given by

[TABLE]

Since the coefficient for $\kappa^{3}$ is positive, the sign of $S(q_{\textrm{\scriptsize b}})-S(q_{\textrm{\scriptsize o}})$ is the same of $\kappa$. It is obviously shown in figure 1.

Combining (50) and (51), we obtain relation between the difference of the action and the distance,

[TABLE]

The second derivative of the reduced action $S^{\prime\prime}_{\textrm{\scriptsize LS}}(r)$ at $r=r_{\textrm{\scriptsize b}}$ gives the eigenvalue of the Hessian for the bifurcated solution, which we call $\kappa_{\textrm{\scriptsize b}}$,

[TABLE]

The leading term in (53) is $-\kappa$ that is independent from coefficients $A_{n}$. This is a general property of the graph $y=ax^{2}+bx^{3}$, which is point symmetric about the inflection point. Therefore, the second derivative at two stationary points has opposite sign and the same magnitude. See figure 1.

In this case, the bifurcated solution exists both side of $\kappa$, namely, $\kappa>0$ and $\kappa<0$. The bifurcation point is $\kappa=0$. This bifurcation is the most simplest case, however it turns out to describe two bifurcations that have completely different appearance. One is crossing of two solutions. The bifurcated solution approaches and crosses the original one at the bifurcation point. Another one is a pair annihilation or creation of two solutions at the bifurcation point.

In the following subsections, we will explain these two cases.

4.1 Crossing of two solutions

Suppose a case where the relation of $\kappa$ and the parameter $\xi$ is given by

[TABLE]

where $a$ is a nonzero constant. Then the correspondence of $\xi$ and $\kappa$ is one to one in a small region around the bifurcation point $\kappa=0$ or $\xi=\xi_{0}$. In this case, the figure 1 shows $S_{\textrm{\scriptsize LS}}(r)$ of the cases $a(\xi-\xi_{0})$ negative, zero, and positive. For both sides of $\xi$, both the original and bifurcated solution exists. We will observe the two solution crossing at the bifurcation point.

4.2 Pair annihilation or creation

Suppose a case where the relation of $\kappa$ and the parameter $\xi$ is given by

[TABLE]

where $a$ is a nonzero constant. Let us assume $a>0$ for simplicity. It is similar for the case $a<0$. In this case, $\xi$ exists only for $\xi\leq\xi_{0}$. And there are two values of $\kappa$,

[TABLE]

This situation will happen if there are almost identical two solutions $q_{1}$ and $q_{2}$ for parameter in $\xi<\xi_{0}$. Approaching $\xi\to\xi_{0}$, two solutions approach, then vanish at $\xi=\xi_{0}$. There are no solutions for $\xi>\xi_{0}$. This is a pair annihilation of two solutions at $\xi=\xi_{0}$. If we observe this process in inverse direction of $\xi$, this is a pair creation. A typical behaviour of $\xi$ and $\kappa$ for pair annihilation/creation is shown in figure 2.

To our knowledge, we can take alternative parameter $\tilde{\xi}$, by which we can “turn around” the pair annihilation point smoothly. The correspondence between $\kappa$ and $\tilde{\xi}$ is locally one-to-one. So, we can use $\kappa$ for the new parameter $\tilde{\xi}$. Starting from the solution $q_{1}$ whose eigenvalue $\kappa_{1}(\xi)<0$, changing the parameter $\kappa$ beyond zero, we will reach $\kappa_{2}(\xi)>0$. See figure 2. Therefore, there is no fundamental difference between $q_{\textrm{\scriptsize o}}$ and $q_{\textrm{\scriptsize b}}$ in this case. If we take $q_{\textrm{\scriptsize o}}=q_{1}$ at $\kappa=\kappa_{1}<0$, then $q_{\textrm{\scriptsize b}}=q_{2}$ at $\kappa_{2}$. With the same light, we can take $q_{\textrm{\scriptsize o}}=q_{2}$ at $\kappa=\kappa_{2}>0$, then $q_{\textrm{\scriptsize b}}=q_{1}$. Assuming the continuity in $\kappa$ of the coefficient $A_{n}$ in (47), we can pass through the bifurcation point $\kappa=0$ smoothly using the parameter $\kappa$. This is the situation that is consistent to our knowledge.

The situation might be clearly understood by changing the origin of $S_{\textrm{\scriptsize LS}}(r)$ to the inflection point $r_{\textrm{\scriptsize I}}$ defined by $S^{\prime\prime}_{\textrm{\scriptsize LS}}(r_{\textrm{\scriptsize I}})=0$,

[TABLE]

Then

[TABLE]

There are no $\tilde{r}^{2}$ term, since $r_{\textrm{\scriptsize I}}$ is the inflection point. Note that the sign of the coefficient of $\tilde{r}$ and $\tilde{r}^{3}$ are determined by the sign of $A_{3}$ for small $\kappa$. For both $\kappa>0$ and $\kappa<0$, there are two stationary points. See figure 3. There are no singularity at $\kappa=0$. While, by the parameter $\xi$, the behaviour (55) is equivalent to

[TABLE]

Then, if $a>0$, two stationary points exist for $\xi_{0}-\xi>0$, while no for $\xi_{0}-\xi<0$. This describes pair annihilation of solutions.

5 Properties of fourth order bifurcation

If there is some reason for $A_{3}=0$ and $A_{4}\neq 0$, the reduced action is given by

[TABLE]

The condition for stationary point

[TABLE]

has bifurcated solution

[TABLE]

In this case, the bifurcated solution exists only one side of $\kappa$.

If $A_{4}>0$ then it exists in $\kappa>0$, and if $A_{4}<0$ it exists in $\kappa<0$. See figure 4. The difference of the action and the distance are

[TABLE]

Since, the sign of $\kappa$ and $A_{4}$ is the same, The sign of $S_{\textrm{\scriptsize b}}-S_{\textrm{\scriptsize o}}$ is also the same. This is obviously shown in figure 4. Combining equations (63) and (64), we obtain,

[TABLE]

The second derivative of reduced action with respect to $r$ gives the eigenvalue of the Hessian at the bifurcated solution,

[TABLE]

The leading term does not depends on $A_{n}$. This is a general property of the function $y=ax^{2}+bx^{4}$. See figure 4.

6 Summary and discussions

An application of variational principle to bifurcation of periodic solution in Lagrangian mechanic is shown. The second derivative of the action integral defines linear operator, Hessian $\mathcal{H}$. Eigenvalues of the Hessian has important role for bifurcation. The necessary and sufficient condition for bifurcation is one of the eigenvalue tends to zero. For sufficient condition, we couldn’t exclude one exceptional case, that unlikely exists. The eigenvalues of $\mathcal{H}$ being always zero are connected to conservation laws. Therefore they are irrelevant to bifurcation. Eigenvalue $\kappa\neq 0$ that tends to zero yields bifurcation. The eigenfunction $\phi$ that belonging to the eigenvalue $\kappa$ is primary for bifurcated solution $q_{\textrm{\scriptsize b}}$.

Reduced action $S_{\textrm{\scriptsize LS}}$, that is constructed by Lyapunov-Schmidt reduction, is a function of one variable or a few variables. The reduced action $S_{\textrm{\scriptsize LS}}(r)$ for non-degenerated $\kappa$ or $S_{\textrm{\scriptsize LS}}(r,\theta)$ for degenerated $\kappa$ describes the behaviour of the action. Some derivatives of $S_{\textrm{\scriptsize LS}}$ at the original solution $q_{\textrm{\scriptsize o}}$ reveal the behaviour of $S_{\textrm{\scriptsize LS}}$ near $q_{\textrm{\scriptsize o}}$. A stationary point of $S_{\textrm{\scriptsize LS}}$ is a stationary point of full action, therefore it is a solution of equations of motion. Thus we can find a bifurcated solution $q_{b}$. The difference of $S_{\textrm{\scriptsize LS}}$ between the original and bifurcated solution gives the difference of full action. The second derivative of $S_{\textrm{\scriptsize LS}}(r)$ at $q_{b}$ is shown to give correct eigenvalue of Hessian at $q_{b}$ to the term $r^{2}$ for non-degenerated case. Thus the reduced action $S_{\textrm{\scriptsize LS}}$ is important and useful to investigate bifurcations.

Applications for individual Lagrangian mechanics needs information of symmetry of the solution. Therefore, they will be published separately. Muñoz et al[7], and Fukuda et al[8, 9, 10] found many bifurcations form figure-eight solutions under homogeneous potential $1/r^{a}$ or Lennard-Jones potential $1/r^{6}-1/r^{12}$. The description by $S_{\textrm{\scriptsize LS}}$ explains qualitative properties of bifurcations and bifurcated solutions observed by these authors. Quantitative comparison will be shown in elsewhere.

Period $K$ bifurcations, period doubling etc, will be treated by relaxing the boundary condition for variation to $\delta q(t+KT)=\delta q(t)$, $K=2,3,\dots$. This extends the function space of eigenvalue problem $\mathcal{H}\Psi=\kappa\Psi$, $\Psi(t+KT)=\Psi(t)$. Then, an eigenvalue $\kappa\to 0$ gives period $K$ bifurcation. Some of $5$-slalom solutions found by Sǔvakov et al[11, 12] can be described as bifurcated solutions of period $5$ bifurcation from the figure-eight.

Equality of the second derivative of $S_{\textrm{\scriptsize LS}}(r)$ at bifurcated solution and the eigenvalue of Hessian to $r^{3}$ or higher for non-degenerated case needs further investigation. The equality for degenerated case will be shown in a separate paper.

Relations between the behaviour of the action and linear stability of solutions $q_{\textrm{\scriptsize o}}$, $q_{\textrm{\scriptsize b}}$ needs further investigations. The relation is unclear at present.

In this paper we use the terms ‘third order bifurcation’, ‘fourth order bifurcation’, ‘crossing of solutions’ and ‘pair creation/annihilation of solutions’ instead of ‘trans-critical’, ‘saddle-node’, or ‘pitchfork’. The relations between the bifurcations described in this paper and ‘trans-critical’, ‘saddle-node’, or ‘pitchfork’ are unclear.

This work was supported by JSPS Grant-in-Aid for Scientific Research 17K05146 (HF) and 17K05588 (HO).

Appendix A Eigenvalue of Hessian at a bifurcated solution

In this section, we calculate the eigenvalue of

[TABLE]

at

[TABLE]

for non-degenerated case. Here $\phi$ is the eigenfunction of the Hessian at $q$ with eigenvalue $\kappa$,

[TABLE]

Expanding $V_{qq}(q_{r})$ by $r$, we obtain

[TABLE]

We calculate the eigenvalue problem by perturbation theory with parameter $\rho$,

[TABLE]

with the zeroth order (69). The first order term $\kappa_{1}$ is

[TABLE]

And the second order is

[TABLE]

Substituting the expression for $\epsilon_{\alpha}$ and using the fact $\kappa=\Or(r)$, we get the eigenvalue

[TABLE]

which is equal to the expression (45).

**ORCID iDs

Toshiaki Fujiwara: https://orcid.org/0000-0002-6396-3037

Hiroshi Fukuda: https://orcid.org/0000-0003-4682-9482

Hiroshi Ozaki: https://orcid.org/0000-0002-8744-3968

**