**Author details**

� *d dt <sup>G</sup> dF dx*\_ � �

<sup>2</sup> � <sup>1</sup> *xG*€ *dG dx <sup>F</sup>* � *<sup>x</sup>*\_

*d*2 *F dx*\_

1 *xG*€ *dG dx* ¼

> *d*2 *F dx*\_

*F*ð Þ¼ *x*\_ *α*2*x*\_ � *α*<sup>3</sup> *e*

0 @

Eq. (64) can be rewritten in the form

*Progress in Relativity*

depends only *x* variable. Then we may set

and the solution *F* is given by

*L*ð Þ¼ *x*\_, *x k*1*x*\_ � *k*<sup>2</sup> *e*

lim

Lagrangian.

**86**

Lagrangian Eq. (70) is now written in the form

*<sup>L</sup>λ*ð Þ¼ *<sup>x</sup>*\_, *<sup>x</sup> <sup>m</sup>λ*<sup>2</sup> *<sup>e</sup>*

2 4

obtain

<sup>þ</sup> *<sup>F</sup> dG*

Using equation of motion, we observe that the coefficient of the second term

*G dG dx* ¼ � *<sup>A</sup> m dV*

We find that it is not difficult to see that the function *G* that satisfies Eq. (66) is

where *α*<sup>1</sup> is a constant to be determined. Inserting Eq. (66) into Eq. (65), we

*dF dx*\_ � �

<sup>2</sup>*=*<sup>2</sup> <sup>þ</sup> *xA*\_

ð *x*\_

0

2

0

*dve*�*v*2*=*2*λ*<sup>2</sup>

1 A*e*

ð *x*\_

*dve*�*Av*2*=*<sup>2</sup>

1 A

3 5*e*

0

*dve*�*Av*2*=*<sup>2</sup>

def *<sup>A</sup>* ! <sup>1</sup>

*G x*ð Þ¼ *α*1*e*

<sup>2</sup> � *A F* � *x*\_

0 @

where *α*<sup>2</sup> and *α*<sup>3</sup> are constants. Then the multiplicative Lagrangian is

<sup>2</sup>*=*<sup>2</sup> <sup>þ</sup> *xA*\_

where *k*<sup>1</sup> ¼ *α*1*α*<sup>2</sup> and *k*<sup>2</sup> ¼ *α*1*α*<sup>3</sup> are new constants to be determined. We find that if we choose *<sup>k</sup>*<sup>1</sup> <sup>¼</sup> 0,*<sup>A</sup>* <sup>¼</sup> <sup>1</sup>*=λ*<sup>2</sup> which is a unit of inverse velocity squared and *<sup>k</sup>*<sup>2</sup> ¼ �*mλ*<sup>2</sup> which is in energy unit, Lagrangian Eq. (70) can be simplified to

the standard Lagrangian at the limit *λ* approaching to infinity. Therefore, the

which can be considered as the one-parameter extended class of the standard

�*Ax*\_

*<sup>λ</sup>*!<sup>∞</sup> *<sup>L</sup>λ*ð Þ� *<sup>x</sup>*\_, *<sup>x</sup> <sup>m</sup>λ*<sup>2</sup> � � <sup>¼</sup> *mx*\_

0 @ �*x*\_ <sup>2</sup>*=*2*λ*<sup>2</sup> þ *x*\_ *λ*2 ð *x*\_

�*Ax*\_

*dF dx*\_ � �

*dx* <sup>¼</sup> <sup>0</sup> (64)

¼ 0 (65)

*dx* (66)

�*AV x*ð Þ*=<sup>m</sup>* (67)

1

<sup>2</sup> � *V x*ð Þ¼ *LN*ð Þ *<sup>x</sup>*\_, *<sup>x</sup>* (71)

�*V x*ð Þ*=mλ*<sup>2</sup>

(72)

¼ 0 (68)

A (69)

�*AV x*ð Þ*=<sup>m</sup>* (70)

Sikarin Yoo-Kong The Institute for Fundamental Study (IF), Naresuan University, Phitsanulok, Thailand

\*Address all correspondence to: sikariny@nu.ac.th

© 2019 The Author(s). Licensee IntechOpen. This chapter is distributed under the terms of the Creative Commons Attribution License (http://creativecommons.org/licenses/ by/3.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
