Classification of Time-Optimal Paths Under an External Force Based on Jacobi Stability in Finsler Space

Abstract

Two-dimensional time-optimal paths of objects moving under the influence of an external force are discussed based on an analysis of Jacobi stability in Finsler space. When the external force on an object can be described by a function of only one variable, the deviation curvature tensor that determines the Jacobi stability of the object’s path can be obtained from the equation of the path. In such cases, the Jacobi stability of the path is represented by the trace of the deviation curvature tensor. The relationship between the Jacobi stability and the type of path is considered for a force that is described by a single-variable trigonometric function. This type of periodic external force induces a path that extends radially and a path along in a specific direction. Then, we consider the time-averaged eigenvalues of the deviation curvature tensor for each type. A large peak in these average values is observed when the type of path changes. Therefore, the Jacobi instability becomes very large at the boundaries between the path types, and the Jacobi stability analysis can be used as the basis of a classification of the path types.

Appendix A: Geometrical Quantities Obtained From Finsler Function with Periodic External Forces

In this Appendix, we describe the geometrical quantities obtained from Finsler function (2) with the external forces for cases (A) and (B).

1.1 Geometrical Quantities in Case of External Force (A)

First, the geometrical quantities are considered for case (A). The coefficients of the nonlinear connection are given by the functions $G^{1}$ and $G^{2}$:

$$\begin{aligned}{} & {} N^{1}_{1}=\frac{\gamma }{1-h^{2}}(\alpha f_{xx}+g_{xx}), \end{aligned}$$

(47)

$$\begin{aligned}{} & {} N^{1}_{2}=\frac{\gamma {\dot{x}}}{1-h^{2}}(\alpha f_{xy}+g_{xy}), \end{aligned}$$

(48)

$$\begin{aligned}{} & {} N^{2}_{1}=-\gamma {\dot{x}}(\alpha f_{yx}+g_{yx}), \end{aligned}$$

(49)

$$\begin{aligned}{} & {} N^{2}_{2}=\frac{\gamma }{1-h^{2}}(\alpha f_{xy}+g_{yy}), \end{aligned}$$

(50)

where we put the coefficients in the nonlinear connection as

$$\begin{aligned}{} & {} \gamma =\frac{h'}{2\alpha ^{2}({\dot{x}}^{2}+{\dot{y}}^{2})^{2}}, \end{aligned}$$

(51)

$$\begin{aligned}{} & {} f_{xx}=2h^{2}(h^{2}-1){\dot{x}}^{6}-h^{2}(5-4h^{2}){\dot{x}}^{4} {\dot{y}}^{2}-(1+h^{2}){\dot{x}}^{2}{\dot{y}}^{4}-{\dot{y}}^{6}, \end{aligned}$$

(52)

$$\begin{aligned}{} & {} f_{xy}={\dot{y}}\{(-2h^{4}+3h^{2}-2){\dot{x}}^{4}+(h^{2}-3) {\dot{x}}^{2}{\dot{y}}^{2}-{\dot{y}}^{4}\}, \end{aligned}$$

(53)

$$\begin{aligned}{} & {} f_{yx}={\dot{y}}\{h^{2}{\dot{x}}^{4}+(3h^{2}-1){\dot{x}}^{2} {\dot{y}}^{2}-{\dot{y}}^{4}\}, \end{aligned}$$

(54)

$$\begin{aligned}{} & {} f_{yy}=h^{2}(h^{2}-1){\dot{x}}^{6}-(h^{4}-2h^{2}+3){\dot{x}}^{4} {\dot{y}}^{2}+(h^{2}-5){\dot{x}}^{2}{\dot{y}}^{4}-2{\dot{y}}^{6}, \end{aligned}$$

(55)

$$\begin{aligned}{} & {} g_{xx}=h{\dot{y}}\{(h^{4}-3h^{2}+2){\dot{x}}^{6}+(3h^{4}-9h^{2}+7) {\dot{x}}^{4}{\dot{y}}^{2}\nonumber \\ {}{} & {} \qquad \quad \,+2(3-2h^{2}){\dot{x}}^{2}{\dot{y}}^{4}+{\dot{y}}^{6}\}, \end{aligned}$$

(56)

$$\begin{aligned}{} & {} g_{xy}=h\{(h^{4}-3h^{2}+2){\dot{x}}^{6}-(h^{4}+h^{2}-3) {\dot{x}}^{4}{\dot{y}}^{2}+2{\dot{x}}^{2}{\dot{y}}^{4}+{\dot{y}}^{6}\}, \end{aligned}$$

(57)

$$\begin{aligned}{} & {} g_{yx}=-2h{\dot{x}}^{2}\{(h^{2}-1){\dot{x}}^{2}-{\dot{y}}^{2}\} ({\dot{x}}^{2}+2{\dot{y}}^{2}), \end{aligned}$$

(58)

$$\begin{aligned}{} & {} g_{yy}=2h{\dot{y}}\{(h^{4}-3h^{2}+2){\dot{x}}^{6}+(4-3h^{2}) {\dot{x}}^{4}{\dot{y}}^{2}-(h^{2}-3){\dot{x}}^{2}{\dot{y}}^{4}+{\dot{y}}^{6}\}. \end{aligned}$$

(59)

The coefficients of the Berwald connection $G^{i}_{jk}$ are given by

$$\begin{aligned}{} & {} G^{1}_{11}=\mu {\dot{x}}(F_{xxx}+\alpha ^{3}G_{xxx}), \end{aligned}$$

(60)

$$\begin{aligned}{} & {} G^{1}_{12}=-\frac{\mu }{1-h^{2}}(F_{xxy}+\alpha ^{3}G_{xxy}), \end{aligned}$$

(61)

$$\begin{aligned}{} & {} G^{1}_{22}=\mu {\dot{x}}^{3}\left( F_{xyy}+\alpha ^{3}G_{xyy}\right) , \end{aligned}$$

(62)

$$\begin{aligned}{} & {} G^{2}_{11}=\mu (F_{yxx}+\alpha ^{3}G_{yxx}), \end{aligned}$$

(63)

$$\begin{aligned}{} & {} G^{2}_{12}=\mu {\dot{x}}^{3}(F_{yxy}+\alpha ^{3}G_{yxy}), \end{aligned}$$

(64)

$$\begin{aligned}{} & {} G^{2}_{22}=-\frac{\mu }{1-h^{2}}(F_{yyy}+\alpha ^{3}G_{yyy}). \end{aligned}$$

(65)

Here, the coefficients in the Berwald connection are defined by

$$\begin{aligned}{} & {} \mu =\frac{h'}{2\alpha ^{3}({\dot{x}}^{2}+{\dot{y}}^{2})^{3}}, \end{aligned}$$

(66)

$$\begin{aligned}{} & {} F_{xxx}=2h^{2}(h^{2}-1){\dot{x}}^{8}+3h^{2}(2h^{2}-3){\dot{x}}^{6}{\dot{y}}^{2} \nonumber \\{} & {} \hspace{3cm}+3(4h^{4}-8h^{2}+1){\dot{x}}^{4}{\dot{y}}^{4} +(6-17h^{2}){\dot{x}}^{2}{\dot{y}}^{6}+3{\dot{y}}^{8}, \end{aligned}$$

(67)

$$\begin{aligned}{} & {} F_{xxy}={\dot{y}}^{3}\{(-8h^{6}+20h^{4}-15h^{2}+4){\dot{x}}^{6} \nonumber \\{} & {} \hspace{3cm}+3(4h^{4}-6h^{2}+3){\dot{x}}^{4}{\dot{y}}^{2} -3(h^{2}-2){\dot{x}}^{2}{\dot{y}}^{4}+{\dot{y}}^{6}\}, \end{aligned}$$

(68)

$$\begin{aligned}{} & {} F_{xyy}=(-2h^{4}+3h^{2}-2){\dot{x}}^{6}+3(2h^{4}-2h^{2}-1){\dot{x}}^{4}{\dot{y}}^{2} -9h^{2}{\dot{x}}^{2}{\dot{y}}^{4}+{\dot{y}}^{6}, \end{aligned}$$

(69)

$$\begin{aligned}{} & {} F_{yxx}={\dot{y}}^{3}F_{xyy}, \end{aligned}$$

(70)

$$\begin{aligned}{} & {} F_{yxy}=h^{2}(h^{2}-1){\dot{x}}^{6}+3(2h^{4}-3h^{2}+1){\dot{x}}^{4}{\dot{y}}^{2} \nonumber \\{} & {} \hspace{2cm}-3(h^{4}+h^{2}-2){\dot{x}}^{2}{\dot{y}}^{4}+(3+5h^{2}){\dot{y}}^{6}, \end{aligned}$$

(71)

$$\begin{aligned}{} & {} F_{yyy}={\dot{y}}\{3(-2h^{6}+5h^{4}-5h^{2}+2){\dot{x}}^{8} +(2h^{6}+4h^{4}-21h^{2}+17){\dot{x}}^{6}{\dot{y}}^{2} \nonumber \\{} & {} \hspace{3.5cm}-3(h^{4}+3h^{2}-6){\dot{x}}^{4}{\dot{y}}^{4} -3(h^{2}-3){\dot{x}}^{2}{\dot{y}}^{6}+2{\dot{y}}^{8}\}, \end{aligned}$$

(72)

$$\begin{aligned}{} & {} G_{xxx}=-2h{\dot{y}}^{3}({\dot{x}}^{2}-3{\dot{y}}^{2}), \end{aligned}$$

(73)

$$\begin{aligned}{} & {} G_{xxy}=h\{(h^{2}-2){\dot{x}}^{6}+3(2h^{2}-3){\dot{x}}^{4} {\dot{y}}^{2}-3h^{2}{\dot{x}}^{2}{\dot{y}}^{4}-{\dot{y}}^{6}\}, \end{aligned}$$

(74)

$$\begin{aligned}{} & {} G_{xyy}=-2h{\dot{y}}(3{\dot{x}}^{2}-{\dot{y}}^{2}), \end{aligned}$$

(75)

$$\begin{aligned}{} & {} G_{yxx}=-2h{\dot{x}}^{2}({\dot{x}}^{4}+3{\dot{x}}^{2}{\dot{y}}^{2}+6{\dot{y}}^{4}), \end{aligned}$$

(76)

$$\begin{aligned}{} & {} G_{yxy}=8h{\dot{y}}^{3}, \end{aligned}$$

(77)

$$\begin{aligned}{} & {} G_{yyy}=2h\{(h^{2}-2){\dot{x}}^{6}-3h^{2}{\dot{x}}^{4}{\dot{y}}^{2} -3{\dot{x}}^{2}{\dot{y}}^{4}-{\dot{y}}^{6}\}. \end{aligned}$$

(78)

1.2 Geometrical Quantities in Case of External Force (B)

The geometrical quantities of the external force for case (B) can be obtained as case for (A). The coefficients of the nonlinear connection are given by the functions $G^{1}$ and $G^{2}$:

$$\begin{aligned}{} & {} N^{1}_{1}=\frac{\zeta {\dot{y}}}{\beta }({\tilde{f}}_{xx}+\beta {\dot{x}}{\tilde{g}}_{xx}), \end{aligned}$$

(79)

$$\begin{aligned}{} & {} N^{1}_{2}=\frac{\zeta }{\beta }({\tilde{f}}_{xy}+\beta {\tilde{g}}_{xy}), \end{aligned}$$

(80)

$$\begin{aligned}{} & {} N^{2}_{1}=\frac{\zeta }{1-h^{2}}({\tilde{f}}_{yx}-2\beta h{\dot{x}}{\tilde{g}}_{yx}), \end{aligned}$$

(81)

$$\begin{aligned}{} & {} N^{2}_{2}=\zeta {\dot{y}}({\tilde{f}}_{yy}-2h\beta {\tilde{g}}_{yy}), \end{aligned}$$

(82)

where we put the coefficients in the nonlinear connection as

$$\begin{aligned}{} & {} \zeta =\frac{h'}{2\beta (1-h^{2})({\dot{x}}^{2}+{\dot{y}}^{2})^{2}}. \end{aligned}$$

(83)

$$\begin{aligned}{} & {} {\tilde{f}}_{xx}\!=\!h\{2{\dot{x}}^{6}\!-\!(h^{2}\!-\!5){\dot{x}}^{4}{\dot{y}}^{2}\!-\!(h^{4} \!+\!3h^{2}\!-\!6){\dot{x}}^{2}{\dot{y}}^{4}\!+\!(h^{2}\!-\!3)(h^{2}\!-\!1){\dot{y}}^{6}\}, \end{aligned}$$

(84)

$$\begin{aligned}{} & {} {\tilde{f}}_{xy}=h{\dot{x}}\{2{\dot{x}}^{6}+(9-5h^{2}){\dot{x}}^{4}{\dot{y}}^{2} +(h^{2}-2)(3h^{2}-5){\dot{x}}^{2}{\dot{y}}^{4}\nonumber \\ {}{} & {} \qquad \quad \,+(h^{2}-3)(h^{2}-1){\dot{y}}^{6}\}, \end{aligned}$$

(85)

$$\begin{aligned}{} & {} {\tilde{f}}_{yx}=2(1+h^{2}){\dot{x}}^{6}-(h^{2}-5)(h^{2}+1) {\dot{x}}^{4}{\dot{y}}^{2} \nonumber \\{} & {} \hspace{2cm}-(h^{2}+1)(h^{4}-3){\dot{x}}^{2}{\dot{y}}^{4} +h^{2}(h^{2}-3)(h^{2}-1){\dot{y}}^{6}, \end{aligned}$$

(86)

$$\begin{aligned}{} & {} {\tilde{f}}_{yy}={\dot{x}}\{(3h^{2}-1){\dot{x}}^{4}-(3h^{4}-8h^{2}+1) {\dot{x}}^{2}{\dot{y}}^{2}-h^{2}(h^{2}-3){\dot{y}}^{4}\}. \end{aligned}$$

(87)

$$\begin{aligned}{} & {} {\tilde{g}}_{xx}=-2{\dot{x}}^{4}+(h^{2}-5){\dot{x}}^{2}{\dot{y}}^{2}-(2h^{4}-3h^{2}+3){\dot{y}}^{4}, \ \end{aligned}$$

(88)

$$\begin{aligned}{} & {} {\tilde{g}}_{xy}=-2{\dot{x}}^{6}+(h^{2}-5){\dot{x}}^{4}{\dot{y}}^{2} +(4h^{2}-5)(h^{2}+1){\dot{x}}^{2}{\dot{y}}^{4}+2(h^{4}-1){\dot{y}}^{6}, \end{aligned}$$

(89)

$$\begin{aligned}{} & {} {\tilde{g}}_{yx}=2{\dot{x}}^{4}+4{\dot{x}}^{2}{\dot{y}}^{2}+(h^{4}-2h^{2}+3){\dot{y}}^{4}, \end{aligned}$$

(90)

$$\begin{aligned}{} & {} {\tilde{g}}_{yy}={\dot{x}}^{4}+2h^{2}{\dot{x}}^{2}{\dot{y}}^{2}+h^{2}{\dot{y}}^{4}. \end{aligned}$$

(91)

The coefficients of the Berwald connection $G^{i}_{jk}$ are given by

$$\begin{aligned}{} & {} G^{1}_{11}=\beta \xi {\dot{y}}^{3}({\tilde{F}}_{xxx}+2\beta ^{3}h {\tilde{G}}_{xxx}), \end{aligned}$$

(92)

$$\begin{aligned}{} & {} G^{1}_{12}=-\frac{\beta \xi }{1-h^{2}}({\tilde{F}}_{xxy}-\beta ^{3}h{\tilde{G}}_{xxy}), \end{aligned}$$

(93)

$$\begin{aligned}{} & {} G^{1}_{22}=\beta \xi ({\tilde{F}}_{xyy}+2\beta ^{3}h{\tilde{G}}_{xyy}), \end{aligned}$$

(94)

$$\begin{aligned}{} & {} G^{2}_{11}=\frac{\beta \xi }{(1-h^{2})^{2}}({\tilde{F}}_{yxx}+2\beta h{\tilde{G}}_{yxx}), \end{aligned}$$

(95)

$$\begin{aligned}{} & {} G^{2}_{12}=-\beta \xi {\dot{y}}^{3}({\tilde{F}}_{yxy}+\beta ^{3}{\tilde{G}}_{yxy}), \end{aligned}$$

(96)

$$\begin{aligned}{} & {} G^{2}_{22}=-\frac{\beta \xi }{1-h^{2}}({\tilde{F}}_{yyy}+2\beta ^{3}h{\tilde{G}}_{yyy}). \end{aligned}$$

(97)

Here, the coefficients in the Berwald connection are defined by

$$\begin{aligned}{} & {} \xi =\frac{h'}{2\beta ^{4}({\dot{x}}^{2}+{\dot{y}}^{2})^{3}}, \end{aligned}$$

(98)

$$\begin{aligned}{} & {} {\tilde{F}}_{xxx}=-{\dot{y}}^{2}\{(9h^{2}+3){\dot{x}}^{4}-6(h^{4}-h^{2}-1) {\dot{x}}^{2}{\dot{y}}^{2}+(2h^{4}-3h^{2}+3){\dot{y}}^{4}\}, \end{aligned}$$

(99)

$$\begin{aligned}{} & {} {\tilde{F}}_{xxy}={\dot{x}}^{3}\{2{\dot{x}}^{6}-3(h^{2}-3){\dot{x}}^{4} {\dot{y}}^{2}+6(2h^{4}-3h^{2}+2){\dot{x}}^{2}{\dot{y}}^{4} \nonumber \\{} & {} \hspace{4.0cm}-(8h^{6}-20h^{4}+15h^{2}-5){\dot{y}}^{6}\}, \end{aligned}$$

(100)

$$\begin{aligned}{} & {} {\tilde{F}}_{xyy}=-{\dot{y}}^{3}\{5(3h^{2}+1){\dot{x}}^{6} -6(2h^{2}+1)(h^{2}-2){\dot{x}}^{4}{\dot{y}}^{2} \nonumber \\{} & {} \hspace{3cm}-3(2h^{2}-3)(h^{2}+1){\dot{x}}^{2}{\dot{y}}^{4}-2(h^{4}-1){\dot{y}}^{6}\}, \end{aligned}$$

(101)

$$\begin{aligned}{} & {} {\tilde{F}}_{yxx}={\dot{x}}\{2(1+h^{2}){\dot{x}}^{8}-3(h^{2}-3) (h^{2}+1){\dot{x}}^{6}{\dot{y}}^{2} \nonumber \\{} & {} \hspace{1.5cm}+3(h^{6}-4h^{4}+h^{2}+6){\dot{x}}^{4}{\dot{y}}^{4} -(2h^{8}+2h^{6}-7h^{4}+16h^{2}-17){\dot{x}}^{2}{\dot{y}}^{6} \nonumber \\{} & {} \hspace{2cm}+3(2h^{8}-7h^{6}+8h^{4}-5h^{2}+2){\dot{y}}^{8}\}, \end{aligned}$$

(102)

$$\begin{aligned}{} & {} {\tilde{F}}_{yxy}=3(1+h^{2}){\dot{x}}^{6}-3(h^{4}+3h^{2}-2){\dot{x}}^{4}{\dot{y}}^{2} \nonumber \\{} & {} \hspace{2cm}+3(2h^{4}-5h^{2}+1){\dot{x}}^{2}{\dot{y}}^{4}+h^{2}(h^{2}-3){\dot{y}}^{6}, \end{aligned}$$

(103)

$$\begin{aligned}{} & {} {\tilde{F}}_{yyy}=-{\dot{x}}^{3}\{(3h^{2}-1){\dot{x}}^{6}-3h^{2} (3h^{2}-5){\dot{x}}^{4}{\dot{y}}^{2} \nonumber \\{} & {} \hspace{2cm}+3(2h^{6}-4h^{4}+3h^{2}+1){\dot{x}}^{2}{\dot{y}}^{4} -(2h^{6}-5h^{4}+3h^{2}-2){\dot{y}}^{6}. \end{aligned}$$

(104)

$$\begin{aligned}{} & {} {\tilde{G}}_{xxx}={\dot{x}}({\dot{x}}-3{\dot{y}}^{2}), \end{aligned}$$

(105)

$$\begin{aligned}{} & {} {\tilde{G}}_{xxy}=2{\dot{x}}^{6}+3(h^{2}+1){\dot{x}}^{4}{\dot{y}}^{2} -6(h^{2}-2){\dot{x}}^{2}{\dot{y}}^{4}-(h^{2}-3){\dot{y}}^{6}, \end{aligned}$$

(106)

$$\begin{aligned}{} & {} {\tilde{G}}_{xyy}={\dot{x}}^{3}{\dot{y}}(3{\dot{x}}^{2}-{\dot{y}}^{2}), \end{aligned}$$

(107)

$$\begin{aligned}{} & {} {\tilde{G}}_{yxx}=-2{\dot{x}}^{8}+2(h^{2}-4){\dot{x}}^{6}{\dot{y}}^{2} +3(h^{4}-3){\dot{x}}^{4}{\dot{y}}^{4} \nonumber \\{} & {} \hspace{2.0cm}-(3h^{6}-8h^{4}+h^{2}+6){\dot{x}}^{2}{\dot{y}}^ {6}+(h^{6}-3h^{4}+5h^{2}-3){\dot{y}}^{8}, \end{aligned}$$

(108)

$$\begin{aligned}{} & {} {\tilde{G}}_{yxy}=8h{\dot{x}}^{3}, \end{aligned}$$

(109)

$$\begin{aligned}{} & {} {\tilde{G}}_{yyy}={\dot{x}}^{6}+3(2h^{2}-1){\dot{x}}^{4} {\dot{y}}^{2}+3h^{2}{\dot{x}}^{2}{\dot{y}}^{4}+h^{2}{\dot{y}}^{6}. \end{aligned}$$

(110)