Time-Extremal Navigation in Arbitrary Winds on Conformally Flat Riemannian Manifolds

Abstract

This paper aims at solving Zermelo’s navigation problem on conformally flat Riemannian manifolds admitting a ship’s variable self-speed, under the action of arbitrary winds including space and time dependence for both perturbation and ship’s speed. Our approach is a variational one under application of the Euler–Lagrange equations with reference to the initial studies of this problem. First of all, we distinguish the navigation cases in non-critical, i.e. weak or strong, and critical winds, which are then unified into an arbitrary wind. After having considered the second variation of a given functional, we obtain the conditions for both time-minimal, i.e. the typical solutions to Zermelo’s problem, and time-maximal extremals. The anomalous paths are also emphasized. Moreover, some classification results are presented with respect to the kinds of perturbation considered separately and under an arbitrary wind. This study is illustrated at its end by a two-dimensional example including a prolate ellipsoid in the presence of a rotational vector field, wherein the solution types are being compared.

Appendices

Appendix A

We sketch below a technical roadmap for the extension of time-extremal navigation to arbitrary Riemannian manifolds as a background space in the presence of arbitrary winds.

Let (M, h) be a Riemannian manifold of dimension n,  where $h=h_{ij}\hbox {d}x^{i}\otimes \hbox {d}x^{j}$ is a Riemannian metric determined by the components $h_{ij}(x)=h(\frac{\partial }{\partial x^{i}},\frac{\partial }{ \partial z^{j}})$ in the local coordinates $(x^{1},\ldots ,x^{n})$ of $x\in M.$ The length of a tangent vector $v\in T_{x}M$ is denoted by $\left| v\right| _{h}=\sqrt{h(v,v)}=\sqrt{h_{ij}v^{i}v^{j}}$. Substituting $u=v {-}W$ into $f=|u|_{h}=\sqrt{h(u,u)}$, it results that $|u|_{h}=\sqrt{ |v|_{h}^{2}-2h(v,W)+|W|_{h}^{2}}$. Thus, the last relation can be rewritten as

$$\begin{aligned} |v|_{h}^{2}-2h(v,W)-\lambda =0, \end{aligned}$$

(51)

with $\lambda :=f^{2}-|W|_{h}^{2}.$ Since the resulting velocity v is the tangent vector to the trajectory $(x^{1}(t),\ldots ,x^{n}(t))$ and $v^{i}~=~{\dot{x}}^{i}(t)=\frac{\hbox {d}x^{i}}{\hbox {d}t},$ $\hbox {d}s=\left| v\right| _{h}\hbox {d}t,$ where $s=s(t)$ is the arc length function. Moreover, using the notation $L_{d}(x,\hbox {d}x,t):=\hbox {d}t$, it yields the equivalent form of Eq. (51), that is,

$$\begin{aligned} \lambda L_{d}^{2}(x,\hbox {d}x,t)+2W_{i}\hbox {d}x^{i}L_{d}(x,\hbox {d}x,t)-(\hbox {d}s)^{2}=0, \end{aligned}$$

(52)

where $W_{i}:=h_{ij}W^{j}$. This corresponds to Eq. (5), for arbitrary Riemannian metric h. Following the same way and arguments like in Sect. 3, Eq. (52) leads to the $C^{\infty }$ homogeneous Lagrangian $L(x,{\dot{x}},t):=\frac{1 }{\hbox {d}t}L_{d}(x,\hbox {d}x,t)$ in arbitrary variables x, ${\dot{x}}:=\frac{\hbox {d}x}{\hbox {d}t}$ and t. It is a positive root of the equation

$$\begin{aligned} \lambda L^{2}+2h(v,W)L-|v|_{h}^{2}=0, \end{aligned}$$

(53)

and it satisfies $L(x(t),{\dot{x}}(t),t)=1$ along the curves given by the equations of motion (3). The explicit equations for optimal navigation with arbitrary Riemannian metric h, which generalize Eq. (23), can be obtained by developing the Euler–Lagrange equations (11) corresponding to the Lagrangian L from Eq. (53). For example, for space-dependent perturbations of arbitrary force $|W|_{h}$ and unit self-speed ($|u|_{h}=1$), the time-extremal solutions to the navigation problem on (M, h) are determined by the differential system consisting of the general optimality condition

$$\begin{aligned} \frac{\hbox {d}u^{i}}{\hbox {d}t}=\left[ W_{j|k}u^{j}u^{k}u^{i}-h^{ij}W_{k|j}-\varGamma _{jk}^{i}(W^{j}+u^{j})\right] u^{k}, \qquad i=1,\ldots ,n, \end{aligned}$$

and Eq. (3), if $1+W_{k}u^{k}\ne 0,$ where $\varGamma _{jk}^{i}$ are the Christoffel symbols of h and $W_{j|k}:=\frac{\partial W_{j}}{\partial x^{k}}-W_{l}\varGamma _{jk}^{l}$.

Appendix B: Proof of Proposition 4.2

We make use of Proposition 11.2.1 from [44, p.287], that is,

Proposition B.1

[44] Suppose $(Q_{ij})$ is a non-singular $n\times n$ complex matrix with inverse $(Q^{ij}),$ $C_{i},$ $i=1,\ldots ,n$ are complex numbers, $C^{i}:=Q^{ij}C_{j}$, $C^{2}:=C^{i}C_{i}$ and $G_{ij}:=Q_{ij}+ \varepsilon C_{i}C_{j},$ with $\varepsilon$ a constant. Then

• 1. $\det (G_{ij})=(1+\varepsilon C^{2})\det (Q_{ij})$,

• 2. whenever $1+\varepsilon C^{2}\ne 0,$ the matrix $(G_{ij})$ is invertible and in this case its inverse is $G^{ij}=Q^{ij}-\frac{\varepsilon }{1+\varepsilon C^{2}}C^{i}C^{j}.$

In order to justify our claim, first of all, we need to solve the following equation

$$\begin{aligned} \det \left( H_{ij}(x(t),{\dot{x}}(t),t)-r\delta _{ij}\right) =0. \end{aligned}$$

Denoting $E_{ij}:=H_{ij}(x(t),{\dot{x}}(t),t)-r\delta _{ij},$ we apply Proposition B.1 in a recursive algorithm in three steps. We write $E_{ij}$ in the form

$$\begin{aligned} E_{ij}{=}(\frac{\rho }{f}{-}r)\left( \delta _{ij}{+}\frac{\rho ^{2}}{f(\rho -rf)} W^{i}W^{j}{-}\frac{\rho ^{2}}{f(\rho -rf)}{\dot{x}}^{i}{\dot{x}}^{j}{+}\frac{pf\rho ^{2}}{(\lambda {+}p)(\rho -rf)}\alpha _{i}\alpha _{j}\right) . \end{aligned}$$

Step 1. We set $Q_{ij}:=\delta _{ij},$ $\varepsilon :=\frac{ \rho ^{2}}{f(\rho -rf)}$ and $C_{i}:=W^{i}.$ We obtain $Q^{ij}=\delta _{ij}$, $C^{i}=W^{i}$, $C^{2}=S^{2}|W|_{h}^{2}$ and $1+\varepsilon C^{2}=\frac{(\lambda +p)(\rho -rf)+\rho |W|_{h}^{2}}{(\lambda +p)(\rho -rf)}.$ So, the matrix $G_{ij}=\delta _{ij}+\frac{\rho ^{2}}{f(\rho -rf)}W^{i}W^{j}$ is invertible, with $G^{ij}=\delta _{ij}-\frac{\rho }{S^{2}X}W^{i}W^{j}$ and the determinant $\det \left( \delta _{ij}+\frac{\rho ^{2}}{f(\rho -rf)}W^{i}W^{j}\right) = \frac{X}{(\lambda +p)(\rho -rf)},$ where $X:=(\lambda +p)(\rho -rf)+\rho |W|_{h}^{2}.$

Step 2. Now, we consider $Q_{ij}:=\delta _{ij}+\frac{\rho ^{2}}{f(\rho -rf)} W^{i}W^{j},$ $\varepsilon :=\frac{\rho ^{2}}{f(\rho -rf)}$ and $C_{i}:={\dot{x}}^{i}$. Having applied Proposition B.1 again, it gives $Q^{ij}=\delta _{ij}-\frac{\rho }{S^{2}X}W^{i}W^{j},$ $C^{i}={\dot{x}}^{i}- \frac{\rho p}{X}W^{i},$ $C^{2}=\frac{S^{2}(|v|_{h}^{2}X-\rho p^{2})}{X}$ and $1+\varepsilon C^{2}=\frac{X(\lambda +p)(\rho -rf)-\rho (|v|_{h}^{2}X-\rho p^{2})}{X(\lambda +p)(\rho -rf)}\ne 0.$ These imply the existence of the inverse of $G_{ij}=\delta _{ij}+\frac{\rho ^{2}}{f(\rho -rf)}W^{i}W^{j}-\frac{\rho ^{2}}{f(\rho -rf)}{\dot{x}}^{i}{\dot{x}} ^{j}$. Namely, $G^{ij}=\delta _{ij}-\frac{\rho }{S^{2}X}W^{i}W^{j}-\frac{\rho X}{S^{2}Y} \left( {\dot{x}}^{i}-\frac{\rho p}{X}W^{i}\right) \left( {\dot{x}}^{j}-\frac{ \rho p}{X}W^{j}\right)$ and the determinant

$$\begin{aligned} \det \left( \delta _{ij}+\frac{\rho ^{2}}{f(\rho -rf)}W^{i}W^{j}-\frac{\rho ^{2}}{f(\rho -rf)}{\dot{x}}^{i}{\dot{x}}^{j}\right) =\frac{Y}{(\lambda +p)^{2}(\rho -rf)^{2}}, \end{aligned}$$

where $Y:=X(\lambda ~+~p) (\rho -rf)-\rho (|v|_{h}^{2}X-\rho p^{2}).$

Step 3. Finally, we put $Q_{ij}:=\delta _{ij}+\frac{\rho ^{2}}{f(\rho -rf)} W^{i}W^{j}-\frac{\rho ^{2}}{f(\rho -rf)}{\dot{x}}^{i}{\dot{x}}^{j}+\frac{pf\rho ^{2}}{(\lambda +p)(\rho -rf)}\alpha _{i}\alpha _{j}$, $\varepsilon :=\frac{ pf\rho ^{2}}{(\lambda +p)(\rho -rf)}$ and $C_{i}:=\alpha _{i}.$ We obtain $Q^{ij}=\delta _{ij}-\frac{\rho }{S^{2}X}W^{i}W^{j}-\frac{\rho X}{S^{2}Y} \left( {\dot{x}}^{i}-\frac{\rho p}{X}W^{i}\right) \left( {\dot{x}}^{j}-\frac{ \rho p}{X}W^{j}\right)$. By computation, it reads $1+\varepsilon C^{2}=1+\varepsilon Q^{ij}C_{i}C_{j}=\frac{r}{Y}(r-\frac{\rho f|v|_{h}^{2}}{(\lambda +p)^{2}}).$ So,

$$\begin{aligned} \begin{aligned} \det&\left( \delta _{ij}+\frac{\rho ^{2}}{f(\rho -rf)}W^{i}W^{j}-\frac{\rho ^{2}}{f(\rho -rf)}{\dot{x}}^{i}{\dot{x}}^{j}+\frac{pf\rho ^{2}}{(\lambda +p)(\rho -rf)}\alpha _{i}\alpha _{j}\right) \\&=(1+\varepsilon C^{2})\frac{Y}{(\lambda +p)^{2}(\rho -rf)^{2}}\\&=\frac{1}{(\lambda +p)^{2}(\rho -rf)^{2}}r\left( r-\frac{\rho f|v|_{h}^{2}}{(\lambda +p)^{2}}\right) . \end{aligned} \end{aligned}$$

Thus, we get

$$\begin{aligned} \begin{aligned} \det E_{ij}=&\left( \frac{\rho }{f}-r\right) ^{n}\det \left( \delta _{ij}+\frac{\rho ^{2}}{f(\rho -rf)}W^{i}W^{j}-\frac{\rho ^{2}}{f(\rho -rf)}{\dot{x}}^{i}{\dot{x}}^{j}+\frac{pf\rho ^{2}}{(\lambda +p)(\rho -rf)}\alpha _{i}\alpha _{j}\right) \\ =&\frac{1}{(\lambda +p)^{2}f^{2}}r\left( \frac{\rho }{f}-r\right) ^{n-2}\left( r-\frac{ \rho f|v|_{h}^{2}}{(\lambda +p)^{2}}\right) . \end{aligned} \end{aligned}$$

Our claim immediately results therefrom. $\square$