1 Introduction

A sub-Riemannian structure on a manifold M is given by a smoothly varying distribution D on M and a smoothly varying positively definite metric g on the distribution. The triple (M, D, g) is called a sub-Riemannian manifold, which has been applied in control theory, quantum physics, C-R geometry, and the other areas [1–4, 9, 15, 19]. Some efforts have been made to generalize sub-Riemannian manifold. One of them leads to the following question: what kind of geometrical features the mentioned triple will have if we change the positively definite metric to an indefinite nondegenerate metric? It is natural to start with the Lorentzian metric of index 1. In this case, the triple: manifold, distribution, and Lorentzian metric on the distribution is called a sub-Lorentzian manifold by analogy with a Lorentzian manifold. For the details concerning the sub-Lorentzian geometry, the reader is referred to [11]. To our knowledge, there are only a few works devoted to this subject (see [10–14, 16, 18]). In [10], Chang, Markina, and Vasiliev have systematically studied the geodesics in an anti-de Sitter space with a sub-Lorentzian metric and a sub-Riemannian metric, respectively. In [13], Grochowski computed reachable sets starting from a point in the Heisenberg sub-Lorentzian manifold on \(\mathbb {R}^{3}\). It was shown in [18] that the Heisenberg group \(\mathbb {H}\) with a Lorentzian metric on \(\mathbb {R}^{3}\) possesses the uniqueness of Hamiltonian geodesics of time-like or space-like type.

The Engel group was first named by Cartan [8] in 1901. It is a prolongation of a three-dimensional contact manifold and is a Goursat manifold. In [5–7], A. Ardentov and Yu. L. Sachkov computed minimizers on the sub-Riemannian Engel group. In the present article, we study the Engel group furnished with a sub-Lorentzian metric. This is an interesting example of sub-Lorentzian manifolds because the Engel group is the simplest manifold with nontrivial abnormal extremal trajectories, and the vector distribution of the Engel group is not 2− generating, its growth vector is (2, 3, 4). We first study some properties of horizontal curves in the Engel group. Second, we use the Hamiltonian formalism and Pontryagin maximum principle to write the equations for geodesics. Furthermore, we give a complete description of the Hamiltonian geodesics in the Engel group.

Apart from the introduction, this paper contains three sections. Section 2 contains some preliminaries as well as definitions of sub-Lorentzian manifolds, the Engel group. In Section 3, we study some properties of horizontal curves in the Engel group. In Section 4, we prove that the time-like normal geodesics are locally maximal in the Engel group, and explicitly calculate the non-space-like Hamiltonian geodesics.

2 Preliminaries

A sub-Lorentzian manifold is a triple (M, D, g), where M is a smooth n-dimensional manifold, D is a smooth distribution on M, and g is a smoothly varying Lorentzian metric on D. For each point p ∈ M, a vector v ∈ D p is said to be horizontal. An absolutely continuous curve γ(t) is said to be horizontal if its derivative \(\gamma ^{\prime }\)(t) exists almost everywhere and lies in D γ(t).

A vector v ∈ D p is said to be time-like if g(v, v) < 0; space-like if g(v, v) > 0 or v = 0; null (light-like) if g(v, v) = 0 and v ≠ 0; and non-space-like if g(v, v) ≤ 0. A curve γ(t) is said to be time-like if its tangent vector \(\dot \gamma (t)\) is time-like a.e.; space-like if \(\dot \gamma (t)\) is space-like a.e.; null if \(\dot \gamma (t)\) is null a.e.; non-space-like if \(\dot \gamma (t)\) is non-space-like a.e.

By a time orientation of (M, D, g), we mean a continuous time-like vector field on M. From now on, we assume that (M, D, g) is time-oriented. If X is a time orientation on (M, D, g), then a non-space-like vector v ∈ D p is said to be future directed if g(v, X(p)) < 0, and past directed if g(v, X(p)) > 0. Throughout this paper, “f.d.” stands for “future directed,” “t.” for “time-like,” and “nspc.” for “non-space-like.”

Let v, w ∈ D be two non-space-like vectors, we have the following reverse Schwartz inequality (see page 144 in [17]):

$$|g(v,w)|\geq\|v\|\cdot\|w\|, $$

where \(\|v\|=\sqrt {|g(v,v)|}\). The equality holds if and only if v and w are linearly dependent.

We introduce the space H γ(t) of horizontal nspc. curves:

$$\begin{array}{@{}rcl@{}} H_{\gamma(t)}=&&\{\gamma:[0,1]\rightarrow M|\ \gamma(t) \text{ is absolutely continuous },\ \ g(\dot{\gamma}(t),\dot{\gamma}(t))\leq0, \\ &&\,\,\dot{\gamma}(t)\in D_{\gamma(t)} \text{ for almost all }t \in [0,1]\}. \end{array} $$
(2.1)

The sub-Lorentzian length of a horizontal nspc. curve γ(t) is defined as follows:

$$l(\gamma) = {{\int}_{0}^{1}} \|\gamma^{\prime}(t)\|dt, $$

where \(\|\gamma ^{\prime }(t)\|=\sqrt {|g(\gamma ^{\prime }(t),\gamma ^{\prime }(t))|}\). We use the length to define the sub-Lorentzian distance d U (q 1, q 2) with respect to a set \(U\subset M\) between two points q 1, q 2 ∈ U:

$$d_{U}(q_{1},q_{2}) = \left\{\begin{array}{ll} \sup\{l(\gamma), \gamma\in H_{U}(q_{1},q_{2})\} & \text{ if } H_{U}(q_{1},q_{2})\neq \emptyset \\ 0 &\text{ if } H_{U}(q_{1},q_{2}) = \emptyset, \end{array}\right. $$

where H U (q 1, q 2) is the set of all nspc.f.d curves contained in U and joining q 1 and q 2.

A nspc. curve is said to be a maximizer if it realizes the distance between its endpoints. We also use the name U-geodesic for a curve in U whose each suitably short sub-arc is a U-maximizer.

A distribution \(D\subset TM\) is called bracket generating if any local frame {X i }1 ≤ i ≤ r for D, together with all of its iterated Lie brackets [X i , X j ],[X i ,[X j , X k ]], ⋯ span the tangent bundle TM. Bracket generating distributions are sometimes also called completely nonholonomic distributions, or distributions satisfying H\(\ddot {o}\)rmander’s condition.

Theorem 2.1

(Chow) Fix a point q ∈ M. If the distribution \(D\subset TM\) is bracket generating, then the set of points that can be connected to q by a horizontal curve is the component of M containing q.

By Chow’s Theorem, we know that if D is bracket generating and M is connected, then any two points of M can be joined by a horizontal curve.

Now, we describe the Engel group E. We consider the Engel group E with coordinates \(q=(x_{1},x_{2},y,z)\in \mathbb {R}^{4}\). The group law is denoted by ⊙ and defined as follows:

$$\begin{array}{@{}rcl@{}} &&(x_{1},x_{2},y,z)\odot(x_{1}^{\prime},x_{2}^{\prime},y^{\prime},z^{\prime})\\ &&=\left( x_{1}+x_{1}^{\prime},x_{2}+x_{2}^{\prime},y+y^{\prime}+\frac{x_{1}x_{2}^{\prime}-x_{1}^{\prime}x_{2}}{2},z+z^{\prime}+\frac{x_{2}x_{2}^{\prime}}{2}(x_{2}+x_{2}^{\prime})+x_{1}y^{\prime}+\frac{x_{1}x_{2}^{\prime}}{2}(x_{1}+x_{1}^{\prime})\right). \end{array} $$

A vector field X is said to be left invariant if it satisfies d L q X(e) = X(q), where L q denotes the left translation \(p\rightarrow L_{q}(p) = q\odot p\) and e is the identity of E. This definition implies that any left invariant vector field on E is a linear combination of the following vector fields:

$$\begin{array}{@{}rcl@{}} X_{1}&=&\frac{\partial}{\partial x_{1}}-\frac {x_{2}}{2} \frac{\partial}{\partial y};\ \ X_{2}=\frac{\partial}{\partial x_{2}}+\frac {x_{1}} {2}\frac{\partial}{\partial y}+\frac{{x_{1}^{2}}+{x_{2}^{2}}}{2}\frac{\partial}{\partial z};\\ X_{3}&=&\frac{\partial}{\partial y}+x_{1}\frac{\partial}{\partial z};\ \ \ \ X_{4}=\frac{\partial}{\partial z}. \end{array} $$
(2.2)

The distribution D = span{X 1, X 2} of E satisfies the bracket generating condition, since X 3 = [X 1, X 2], X 4 = [X 1, X 3]. The Engel group is a nilpotent Lie group, since [X 1, X 4] = [X 2, X 3] = [X 2, X 4] = 0. We define a smooth Lorentzian metric \(\tilde {g}\) on E such that \(\tilde {g}(X_{i},X_{j}) = (-1)^{\delta _{1i}}\delta _{ij}\), i, j = 1,⋯ ,4, where δ i j is the Kronecker symbol. It is not difficult to compute the coefficients of \(\tilde {g}\) under the local coordinates \((x_{1},x_{2},y,z)\in \mathbb {R}^{4}\). The coefficients can be expressed as

$$(\tilde{g}_{ij}) = \left( \begin{array}{cccc} -1+\frac{{x_{2}^{2}}}{4}+\frac{{x_{1}^{2}}{x_{2}^{2}}}{4} & -\frac{x_{1}x_{2}}{4}+\frac{x_{1}{x_{2}^{3}}}{4} & \frac{x_{2}}{2}+\frac{x_{2}{x_{1}^{2}}}{2} & -\frac{x_{1}x_{2}}{2} \\ -\frac{x_{1}x_{2}}{4}+\frac{x_{1}{x_{2}^{3}}}{4} & 1+\frac{{x_{1}^{2}}}{4}+\frac{{x_{2}^{4}}}{4} & -\frac{x_{1}}{2}+\frac{x_{1}{x_{2}^{2}}}{2} & -\frac{{x_{2}^{2}}}{2} \\ \frac{x_{2}}{2}+\frac{x_{2}{x_{1}^{2}}}{2} & -\frac{x_{1}}{2}+\frac{x_{1}{x_{2}^{2}}}{2} & 1+{x_{1}^{2}} & -x_{1} \\ -\frac{x_{1}x_{2}}{2} & -\frac{{x_{2}^{2}}}{2} & -x_{1} & 1 \end{array}\right) $$
(2.3)

When we restrict \(\tilde {g}\) to D, we can get a smooth sub-Lorentzian metric \(g=\tilde {g}_{D}\), which satisfies

$$ g(X_{1},X_{1}) = -1,\ \ g(X_{2},X_{2}) = 1, \ \ g(X_{1},X_{2}) = 0. $$
(2.4)

On the other hand, any sub-Lorentzian metric on D can be extended to a (usually not unique) Lorentzian metric on E. In this paper, we assume that X 1 is the time orientation.

3 Horizontal Curves

Chow’s theorem states that any two points can be connected by a horizontal curve, but we have no information about the character of horizontal curves. In this section, we will investigate some properties of horizontal curves.

An absolutely continuous curve γ(s) : [0, 1] → E is said to be horizontal if the tangent vector \(\dot {\gamma }(s)\) can be expressed linearly by the horizontal directions X 1, X 2; hence, we have the following lemma.

Lemma 3.1

A curve γ(s) = (x 1 (s),x 2 (s),y(s),z(s)) is horizontal with respect to the distribution D, if and only if

$$\begin{array}{@{}rcl@{}} \frac{x_{2}\dot{x_{1}}}{2}-\frac{x_{1}\dot{x_{2}}}{2}+\dot{y}&=&0,\\ -\frac{{x_{1}^{2}}+{x_{2}^{2}}}{2}\dot x_{2}+\dot{z}&=&0. \end{array} $$
(3.1)

Proof

The distribution D is the annihilator of the one-forms:

$$\omega_{1}=\frac{x_{2}}{2}dx_{1}-\frac{x_{1}}{2}dx_{2}+dy,\ \ \ \omega_{2}=-\frac{{x_{1}^{2}}+{x_{2}^{2}}}{2}dx_{2}+dz $$

so γ(s) is horizontal if and only if (3.1) holds. □

By the same method, we can easily calculate the left invariant coordinates u 1(s) and u 2(s) of the horizontal curve γ(s):

$$ u_{1}=\dot{x_{1}},\ \ u_{2}=\dot{x_{2}}. $$
(3.2)

The square of the velocity vector for the horizontal curve is as follows:

$$ g(\dot\gamma,\dot\gamma) = -{u_{1}^{2}}+{u_{2}^{2}}=-\dot{x}_{1}^{2}+\dot{x}_{2}^{2}. $$
(3.3)

So whether a horizontal curve is time-like (or nspc.) is determined by the sign of \(-\dot {x}_{1}^{2}+\dot {x}_{2}^{2}\).

Next, we present a left invariant property of horizontal curves in a Lie group with sub-Lorentzian metric. That is to say, the causal character (time-like, space-like, light-like, or non-space-like) of horizontal curves will not change under left translations. Hence, it is also true for the Engel group.

Let us consider a left invariant sub-Lorentzian structure on a Lie group G: \(\mathcal {D}=\text {span}(X_{1},X_{2},\cdots ,X_{k})\subset TG\), \(g(X_{i},X_{j}) = (-1)^{\delta _{1i}}\delta _{ij}\), with a time orientation X 1. The vector fields X i are assumed to be left invariant, i.e.,

$$L_{x\ast}X_{i}(q) = X_{i}(x\cdot q),\ \ x,q\in G, \ \ i=1,\cdots, k. $$

Proposition 3.2

Left translations preserve the causal character of horizontal curves of a left invariant sub-Lorentzian structure on a Lie group G, and the property of future-directness is also preserved.

Proof

Let c(t) be a causal horizontal curve, and

$$ \dot{c}(t) = \sum\limits_{i=1}^{k}u_{i}(t)X_{i}(c(t)). $$

Then, the left translation γ(t) = x ⊙ c(t) has the same causal character, since

$$\begin{array}{@{}rcl@{}} \dot{\gamma}(t)&=&L_{x\ast}\dot{c}(t) = L_{x\ast}\left( \sum\limits_{i=1}^{k}u_{i}(t)X_{i}(c(t))\right) =\sum\limits_{i=1}^{k}u_{i}(t)L_{x\ast}(X_{i}(c(t)))\\ &=&\sum\limits_{i=1}^{k}u_{i}(t)X_{i}(x\odot c(t)) = \sum\limits_{i=1}^{k}u_{i}(t)X_{i}(\gamma(t)). \end{array} $$

Therefore,

$$\begin{array}{@{}rcl@{}} &&g(\dot{c}(t),\dot{c}(t)) = \sum\limits_{i=1}^{k}(-1)^{\delta_{i1}}{u_{i}^{2}}=g(\dot{\gamma}(t),\dot{\gamma}(t)),\\ &&g(\dot{c}(t),X_{1}) = -u_{1}=g(\dot{\gamma}(t),X_{1}). \end{array} $$

□

By Chow’s Theorem, we know that any two points on the Engel group can be connected by a horizontal curve. But we do not know its causal character. Next, we will present some particular examples to show its complexity.

Example 1

Let \(\dot {x}_{2}=0\). Then, \(x_{2}={x_{2}^{0}}\) is constant. The horizontal condition (3.1) becomes

$$\frac{x_{2}}{2}\dot{x_{1}}+\dot{y}=0, $$
(3.4)
$$\dot{z}=0. $$
(3.5)

Then, therefore, the square of the velocity vector

$$ -{u_{1}^{2}}+{u_{2}^{2}}=-\dot{x}_{1}^{2}\leq0. $$
(3.6)

It follows that, the curves satisfying (3.4) and (3.5) are all non-space-like curves. Furthermore, we obtain,

$$ y(s) = -\frac{1}{2}{x_{2}^{0}}x_{1}(s)+\frac{1}{2}{x_{2}^{0}}{x_{1}^{0}}+y^{0},\quad z(s) = z^{0}. $$
(3.7)

Therefore, all nonconstant horizontal curves \(c(s) = \left (x_{1}(s),{x_{2}^{0}},-\frac {x_{1}(s){x_{2}^{0}}}{2}+\frac {{x_{1}^{0}}{x_{2}^{0}}}{2}+y^{0},z^{0}\right )\) are time-like. These curves are straight lines. If \(\dot x_{1}=0\), c(s) degenerate into some points, so there are no null curves in this family.

Example 2

Let \(\dot {x}_{2}\neq 0\). We choose x 2 as a parameter, then the horizontal condition (3.1) becomes

$$\frac{x_{2}}{2}\dot{x_{1}}-\frac{x_{1}}{2}+\dot{y}=0, $$
(3.8)
$$-\frac{{x_{1}^{2}}+{x_{2}^{2}}}{2}+\dot{z}=0. $$
(3.9)

And the square of the velocity vector

$$ -{u_{1}^{2}}+{u_{2}^{2}}=-\dot{x}_{1}^{2}+1. $$
(3.10)

We consider three different cases.

  1. (a)

    If \(\dot {x}_{1}=0\), then \(x_{1}={x_{1}^{0}}\) is constant, (3.8) and (3.9) become

    $$-\frac{{x_{1}^{0}}}{2}+\dot{y}=0, $$
    (3.11)
    $$-\frac{\left( {x_{1}^{0}}\right)^{2}+{x_{2}^{2}}}{2}+\dot{z}=0. $$
    (3.12)

    In this case, \(|\dot {c}(s)|^{2}=1\), so the curves satisfying (3.11) and (3.12) are all space-like. Furthermore, we obtain,

    $$ y(s) = \frac{{x_{1}^{0}}}{2}x_{2}+y^{0},\quad z(s) = \frac{1}{6}{x_{2}^{3}}+\frac{\left( {x_{1}^{0}}\right)^{2}}{2}x_{2}+z_{0}. $$
    (3.13)

    Therefore, all nonconstant horizontal curves c(s) =\(\left (\hspace *{-3pt}{x_{1}^{0}}\hspace *{-1pt},\hspace *{-1pt}x_{2}\hspace *{-.5pt},\hspace *{-1pt}\frac {{x_{1}^{0}}}{2}x_{2}\hspace *{-1pt}+\hspace *{-1pt}y^{0}\hspace *{-1pt},\hspace *{-1pt}\frac {1}{6}{x_{2}^{3}}\hspace *{-1pt}+\hspace *{-1pt}\frac {\left ({x_{1}^{0}}\right )^{2}}{2}x_{2}\hspace *{-1pt}+\hspace *{-1pt}z_{0}\hspace *{-3pt}\right )\) are space-like. There are no null or time-like horizontal curves in this family.

  2. (b)

    If \(\dot {y}=0\), Eqs. 3.8 and 3.9 become

    $$x_{2}\dot{x_{1}}-x_{1}=0, $$
    (3.14)
    $$-\frac{{x_{1}^{2}}+{x_{2}^{2}}}{2}+\dot{z}=0. $$
    (3.15)

    From Eq. 3.14, we get

    $$\frac{1}{x_{2}}=\frac{\dot{x}_{1}}{x_{1}}, $$

    integrating with respect to x 2, we calculate x 1 = ι x 2, where \(\iota =\frac {{x_{1}^{0}}}{{x_{2}^{0}}}\), i.e., \(x_{1}=\frac {{x_{1}^{0}}}{{x_{2}^{0}}}x_{2}\), substituting x 1 in Eq. 3.15, we obtain

    $$ z=\frac{1}{6}\left( 1+\iota^{2}\right){x_{2}^{3}}+z^{0}. $$
    (3.16)

    Therefore, all nonconstant horizontal curves

    $$ c(s) = \left( \iota x_{2},x_{2},y^{0},\frac{1}{6}\left( 1+\iota^{2}\right){x_{2}^{3}}+z^{0}\right) $$
    (3.17)

    are time-like when |ι| > 1. If |ι| < 1 (= 1), they are space-like (null).

  3. (c)

    If \(\dot {z}=0\), the horizontal condition becomes:

    $$\frac{x_{2}}{2}\dot{x_{1}}-\frac{x_{1}}{2}+\dot{y}=0, $$
    (3.18)
    $$-\frac{{x_{1}^{2}}+{x_{2}^{2}}}{2}=0. $$
    (3.19)

    So x 1 = x 2 = 0, y = y 0. The curves degenerate into some points. There are no causal (time-like, space-like, null) horizontal curves in this family.

Thus, any two points \(P_{1}\left ({x_{1}^{0}},{x_{2}^{0}},y^{0},z^{0}\right )\), \(Q_{1}\left (x_{1},{x_{2}^{0}},y^{1},z^{0}\right )\) can be connected by a time-like horizontal curve if \(y^{1}=-\frac {x_{1}{x_{2}^{0}}}{2}+\frac {{x_{1}^{0}}{x_{2}^{0}}}{2}+y^{0}\). Especially, any two points \(\left ({x_{1}^{0}},0,y^{0},z^{0}\right )\), \(\left (x_{1},0,y^{0},z^{0}\right )\) can be connected by a time-like horizontal straight line.

Any two points \(P_{1}\left ({x_{1}^{0}},{x_{2}^{0}},y^{0},z^{0}\right )\), \(Q_{2}\left ({x_{1}^{0}},x_{2},y^{1},z^{1}\right )\) can be connected by a space-like horizontal curve if \(y^{1}=\frac {{x_{1}^{0}}}{2}x_{2}+y^{0}\), \(z^{1}=\frac {1}{6}{x_{2}^{3}}+\frac {\left ({x_{1}^{0}}\right )^{2}}{2}x_{2}+z_{0}\).

Any two points \(P_{1}\left ({x_{1}^{0}},{x_{2}^{0}},y^{0},z^{0}\right )\), \(Q_{3}\left (x_{1},x_{2},y^{0},z^{1}\right )\) can be connected by a time-like (space-like, null) horizontal curve if x 1 = ι x 2, \(z^{1}= \frac {1}{6}\left (1+\iota ^{2}\right ){x_{2}^{3}}+z^{0},\) and \(\mid \iota \mid =\left |\frac {{x_{1}^{0}}}{{x_{2}^{0}}}\right |>1(<1,\ =1)\).

4 Sub-Lorentzian Geodesics

In the Lorentzian geometry, there are no curves of minimal length because two arbitrary points can be connected by a piecewise light-like curve whose length is always 0. For example, let \(\mathbb {R}^{2}\) be the two-dimensional Minkowski space, \(\hat {p}=(\hat {x},\hat {y})\) is any one point in this space. We want to find a light-like curve going from the origin to \(\hat {p}\). First, we choose a curve γ 1(t):(x(t), y(t)) = (t, t) which connects the origin and the point \(\left (\frac {\hat x+\hat y}{2},\frac {\hat x+\hat y}{2}\right )\); then, we choose the second curve \(\gamma _{2}(t):(x(t), y(t)) = (t,-t+\hat x+\hat y)\) which joins \(\left (\frac {\hat x+\hat y}{2},\frac {\hat x+\hat y}{2}\right )\) and \(\hat {p}\). It is easy to check that the curve γ(t) consisting of γ 1 and γ 2 is a light-like curve. It goes from the origin to the point \(\hat {p}\), and the length is 0. However, there do exist time-like curves with maximal length which are time-like geodesics [17]. For this reason, we will study the optimality of time-like geodesics and compute the longest curve among all horizontal time-like ones on the sub-Lorentzian Engel group. The computation will be given by extremizing the action integral \({S}=\frac 1 2\int \left (-{u_{1}^{2}}+{u_{2}^{2}}\right )dt\) under constraint (3.1). By Proposition 3.2, horizontal time-like curves are left invariant, so we can assume that the initial point is origin, i.e., x 1(0) = x 2(0) = y(0) = z(0) = 0, and time-like initial velocity is \(-{u_{1}^{2}}(0)+{u_{2}^{2}}(0) = -1\).

Let ξ = (ξ 1, ξ 2, ξ 3, ξ 4) be the vector of costate variables, so the Hamiltonian function of Pontryagin’s maximum principle is

$$ H(\xi_{0},\xi,q,u) = \xi_{0}\frac{-{u^{2}_{1}}+{u^{2}_{2}}}{2}+\xi_{1}u_{1}+\xi_{2}u_{2}+\xi_{3}\frac{x_{1}u_{2}-x_{2}u_{1}}{2}+\xi_{4}\frac{{x_{1}^{2}}+{x_{2}^{2}}}{2}u_{2}. $$
(4.1)

where ξ 0 is a constant which equals to 0 or −1. Also, we get the Hamiltonian system:

$$\begin{array}{@{}rcl@{}} &&\dot x_{1}=H_{\xi_{1}}=u_{1}, \quad\dot x_{2}\,=\,H_{\xi_{2}}=u_{2}, \quad \dot y\,=\,H_{\xi_{3}}=\frac{x_{1}u_{2}-x_{2}u_{1}}{2}, \quad\dot z=H_{\xi_{4}}=\frac{{x_{1}^{2}}+{x_{2}^{2}}}{2}u_{2},\\ &&\dot \xi_{1}=-H_{x_{1}}=-\frac{\xi_{3}u_{2}}{2}-\xi_{4}x_{1}u_{2},\quad \dot\xi_{2}=-H_{x_{2}}=\frac{\xi_{3}u_{1}}{2}-\xi_{4}x_{2}u_{2},\quad \dot\xi_{3}=\dot\xi_{4}=0, \end{array} $$
(4.2)

and the maximum condition:

$$ H(\xi_{0},\xi(t),q(t),u(t)) = \max\limits_{\tilde{u}\in\mathbb{R}^{2}}H(\xi_{0},\xi(t),\tilde{q}(t),\tilde{u}), \xi_{0}\leq 0, $$
(4.3)

where u(t) is the optimal control, and (ξ 0, ξ(t)) ≠ 0.

4.1 Abnormal Extremal Trajectories

We shall investigate the abnormal case ξ 0 = 0. From the maximum condition (4.3), we obtain

$$H_{u_{1}}=\xi_{1}-\frac{\xi_{3}x_{2}}{2}=0, $$
(4.4)
$$H_{u_{2}}=\xi_{2}+\frac{\xi_{3}x_{1}}{2}+\frac{\xi_{4}({x_{1}^{2}}+{x_{2}^{2}})}{2}=0. $$
(4.5)

Differentiating Eqs. 4.4 and 4.5, we obtain

$$0=\dot\xi_{1}-\frac{\xi_{3}\dot x_{2}}{2}=\dot\xi_{1}-\frac{\xi_{3}u_{2}}{2}=-u_{2}(\xi_{3}+\xi_{4}x_{1}), $$
(4.6)
$$0=\dot\xi_{2}+\frac{\xi_{3}\dot x_{1}}{2}+\xi_{4}(x_{1}\dot x_{1}+x_{2}\dot x_{2}) = u_{1}(\xi_{3}+\xi_{4}x_{1}). $$
(4.7)

For a time-like curve, we assume that \(-{u_{1}^{2}}+{u_{2}^{2}}=-1\), so ξ 3+ξ 4 x 1 = 0. If ξ 4 = 0, then, ξ 3 = 0, and therefore, ξ = 0. It is a contradiction with the nontriviality of the costate variables; hence, ξ 4 ≠ 0. In this case, \(x_{1}=\frac {-\xi _{3}}{\xi _{4}}\) is a constant, and u 1 = 0, u 2 = ±i, so there is no time-like abnormal extremal in the Engel group E.

For a space-like curve, we assume that \(-{u_{1}^{2}}+{u_{2}^{2}}=1\), by using the same method, we get that u 1 = 0, u 2 = ±1, so the space-like abnormal extremal trajectories are given by the following expression:

$$ \gamma(s) = \left( 0, \pm s,0,\pm \frac{s^{3}}{6}\right). $$
(4.8)

For a null curve, suppose that \(-{u_{1}^{2}}+{u_{2}^{2}}=0\), we can easily get that u 1 = 0, u 2 = 0, so the null abnormal extremal trajectories are trivial curves.

4.2 Normal Geodesics

4.2.1 Normal Hamiltonian System

Now, we look at the normal case ξ 0 = −1. It follows from the maximum condition (4.3) that \(H_{u_{1}}=H_{u_{2}}=0\). Hence,

$$ u_{1}=-\left( \xi_{1}-\frac{x_{2}\xi_{3}}{2}\right),\quad u_{2}=\xi_{2}+\frac{\xi_{3}x_{1}}{2}+\frac{\xi_{4}({x_{1}^{2}}+{x_{2}^{2}})}{2}. $$
(4.9)

Let ζ i = (ξ, X i ), i = 1,2, be the Hamiltonian corresponding to the basis vector fields X 1, X 2 in the cotangent space \(T_{q}^{*}E\). They are linear on the fibers of the cotangent space T ∗ E, and

$$ \zeta_{1}=\xi_{1}-\frac{x_{2}}{2}\xi_{3},\quad \zeta_{2}=\xi_{2}+\frac{x_{1}}{2}\xi_{3}+\frac{{x_{1}^{2}}+{x_{2}^{2}}}{2}\xi_{4}. $$
(4.10)

So u 1 = −ζ 1 and u 2 = ζ 2.

The Hamiltonian system in the normal case becomes:

$$ \left\{ \begin{array}{ll} \dot x_{1} =\frac{\partial H}{\partial \xi_{1}}=-(\xi_{1}-\frac{x_{2}}{2}\xi_{3}) = -\zeta_{1},\\ \dot x_{2}=\frac{\partial H}{\partial \xi_{2}}=\left( \xi_{2}+\frac{x_{1}}{2}\xi_{3}+\frac{{x_{1}^{2}}+{x_{2}^{2}}}{2}\xi_{4}\right) = \zeta_{2}, \\ \dot y=\frac{\partial H}{\partial \xi_{3}}=\zeta_{1}\frac{x_{2}}{2}+\zeta_{2}\frac{x_{1}}{2}=\frac 1 2(x_{1}\zeta_{2}+x_{2}\zeta_{1}),\\ \dot z=\frac{\partial H}{\partial \xi_{4}}=\frac{{x_{1}^{2}}+{x_{2}^{2}}}{2}\zeta_{2},\\ \dot \xi_{1}=-\frac{\partial H}{\partial x_{1}}=-\zeta_{2}\left( \frac{\xi_{3}}{2}+x_{1}\xi_{4}\right),\\ \dot \xi_{2}=-\frac{\partial H}{\partial x_{2}}=-\frac 1 2\xi_{3}\zeta_{1}-x_{2}\xi_{4}\zeta_{2},\\ \dot\xi_{3}=-\frac{\partial H}{\partial y}=0,\\ \dot\xi_{4}=-\frac{\partial H}{\partial z}=0. \end{array} \right. $$
(4.11)

Definition 4.1

A normal geodesic in the sub-Lorentzian manifold (E, D, g) is a curve \(\gamma :[a,b]\rightarrow E\) that admits a lift \({\Gamma }:[a,b]\rightarrow T^{*}M\), which is a solution of the Hamiltonian equations (4.11). In this case, we say that Γ is a normal lift of γ.

Associate with the expression of H, a sub-Lorentzian geodesic is time-like if H < 0; space-like if H > 0; light-like if H = 0.

Remark 4.1

In fact, abnormal extremal trajectories (4.8) are also normal geodesics, since we can choose the costate variables as \(\widetilde {\xi }=(0, \pm 1,0,0)\); it is easy to check that \({\Gamma }(t) = (\gamma ,\widetilde {\xi })\) satisfies Hamiltonian equation (4.11). This example also confirms that normal geodesics and abnormal trajectories are sometimes not mutually exclusive.

Lemma 4.2

The causal character of normal sub-Lorentzian geodesics does not depend on time.

Proof

The Hamiltonian H is an integral of the Hamiltonian system, i.e., \(\dot H(s) = 0\); this implies that the causality character does not change for all \(t\in [0,\infty )\). □

Remark 4.3

If γ(t) is a nspc. normal geodesic on the Engel group, then the orientation will not change along the curve. In fact, if γ(t) is time-like, and it is future directed at t = 0, then, we have \(-{u_{1}^{2}}(t)+{u_{2}^{2}}(t) = -1, u_{1}(0)>0\). We only need to show that u 1(t) will not be equal to 0 along the curve γ(t). Actually, if there is a t 1 > 0, such that u 1(t 1) = 0, then we have \({u_{2}^{2}}(t_{1}) = -1\); it is impossible. So, u 1(t) will not change the sign (since u 1(t) = −ζ 1(t) is continuous), and γ(t) is future directed along the curve. It is also true for the other cases.

Differentiating ζ i ,

$$\dot \zeta_{1}=\dot\xi_{1}-\frac{\xi_{3}}{2}\dot x_{2}=-\zeta_{2}(\xi_{3}+x_{1}\xi_{4}), $$
(4.12)
$$\dot\zeta_{2}=\dot\xi_{2}+\frac{1}{2}\dot{x}_{1}\xi_{3}+(x_{1}\dot{x}_{1}+x_{2}\dot{x}_{2})\xi_{4} =-\zeta_{1}(\xi_{3}+x_{1}\xi_{4}). $$
(4.13)

Let

$$ \beta(s) = -(\xi_{3}+x_{1}\xi_{4}), \ \ \dot\beta=\xi_{4}\zeta_{1}, $$
(4.14)

then, we have

$$ \dot\zeta_{1}=\beta\zeta_{2},\ \ \dot\zeta_{2}=\beta\zeta_{1},\ \ \dot\beta=\xi_{4}\zeta_{1}. $$
(4.15)

4.2.2 Maximality of Short Arcs of Geodesics

Definition 4.2

Let φ be a smooth function on M, and U an open subset in M. The horizontal gradient ∇ H φ of φ is a smooth horizontal vector field on U such that for each p ∈ U and v ∈ H, ∂ v φ(p) = g(∇ H φ(p), v).

Locally, we can write

$$\nabla_{H}\varphi=-(\partial_{X_{1}}\varphi)X_{1}+\sum\limits_{i=2}^{r}(\partial_{X_{i}}\varphi)X_{i}. $$

Now, we give a proof that the time-like normal geodesics are locally maximizing curves on the Engel group.

Proposition 4.4

If γ is a t.f.d. (t.p.d.) normal geodesic on the Engel group, then every sufficiently short sub-arc of γ is a maximizer.

Proof

Assume that \(\gamma :(a,b)\rightarrow E\) is parameterized by arc-length, \(\dot \gamma (t) = {u_{1}^{0}}(t)X_{1}(\gamma (t))+{u_{2}^{0}}(t)X_{2}(\gamma (t)), X_{1}\) is the time orientation, and \(\tilde {\Gamma }(t) = (\gamma (t),\lambda (t))\) is the normal lift of γ. So, we have \(H(\gamma (t),\lambda (t)) = -\frac {1}{2},\ t\in (a,b)\). For any c ∈ (a, b), 𝜖 > 0, let \(J_{c}=(c-\epsilon ,c+\epsilon )\subset (a,b)\) be a neighborhood of c. We will prove that \(\gamma |_{J_{c}}\) is maximal for any c ∈ (a, b) and small 𝜖 > 0. Since the sub-Lorentzian metric is left invariant, so we can assume that γ(c) = 0, λ(c) = λ 0. Consider an three-dimensional hypersurface S passing through the origin 0, and satisfying λ 0(T 0(S)) = 0. Let \(\bar {\lambda }\) be a smooth one-form on an open neighborhood Ω of 0, such that \(\bar {\lambda }(0) = \lambda _{0}\), and \(\forall p\in S\cap {\Omega },\bar {\lambda }(p)(T_{p}S) = 0\), \(H(p,\bar {\lambda }(p)) = -\frac {1}{2}\). Let Γ p = (γ p , λ p ) be the solution of \(\dot {\Gamma }(t) = \vec {H}({\Gamma }(t)),\ {\Gamma }(c) = (p,\bar \lambda (p))\). Then, clearly \({\Gamma }_{0}=\tilde {\Gamma }\). Since \(\dot \gamma (0)\not \in T_{0}S,\) by the Implicit Function Theorem, there exits a diffeomorphism:

$$\nu:(c-\epsilon,c+\epsilon)\times W\rightarrow U\subset E, $$
$$(t,p)\rightarrow \gamma_{p}(t), $$

where W is a neighborhood of 0 in S, \(U\subset {\Omega }\) is a neighborhood of 0 in E. Define a smooth function \(V:U\rightarrow \mathbb {R}\) as follows:

$$V(x) = t,\quad if\ x=\gamma_{p}(t), $$

we will show that ∥∇ H V∥ = 1. For this purpose, let Y 1 be the vector field on U given by

$$Y_{1}(x) = \dot{\gamma}_{p}(t) = u_{1}(p,t)X_{1}(\gamma_{p}(t))+u_{2}(p,t)X_{2}(\gamma_{p}(t)), \quad if \ x=\gamma_{p}(t), $$

where u 1(p, t), u 2(p, t) are smooth functions on W × (c−𝜖, c+𝜖), and \(u_{1}(0,t) = {u_{1}^{0}}(t), u_{2}(0,t) = {u_{2}^{0}}(t)\). Since \(H(p,\bar {\lambda }(p)) = -\frac {1}{2}\), by the construction of Γ p (t), we have \(H(\gamma _{p}(t),\lambda _{p}(t)) = -\frac {1}{2}\), and \(-{u_{1}^{2}}+{u_{2}^{2}}=-1\). It is easy to check that Y 1 = u 1 X 1+u 2 X 2, Y 2 = u 2 X 1+u 1 X 2 is also an orthonormal basis of D, so \(\partial _{Y_{1}}V=1,\ \partial _{Y_{2}}V=0\). Therefore, \(\nabla _{H}V=-Y_{1}, \|\nabla _{H}V\|=\sqrt {|g(-Y_{1},-Y_{1})|}=\sqrt {|-{u_{1}^{2}}+{u_{2}^{2}}|}=1\). Choose t 1, t 2 in the domain of γ. If γ(t) is a t.f.d. geodesic, then \(|{u_{1}^{0}}|>|{u_{2}^{0}}|\), and \({u_{1}^{0}}>0\). Since \(u_{1}(0,t) = {u_{1}^{0}},\) and u 1(p, t) is a smooth function, so there exists a neighborhood \(W_{1}\times (c-\epsilon _{1},c+\epsilon _{1})\subset W\times (c-\epsilon ,c+\epsilon )\) such that u 1(p, t) > 0. Thus, ∇ H V = −Y 1 is past directed. On the other hand, since \(-{u_{1}^{2}}+{u_{2}^{2}}=-1\), we have |u 1| > |u 2|. Let \(\eta :[0,\alpha ]\rightarrow U\) be a t.f.d. curve with η(0) = γ(t 1), η(α) = γ(t 2), and \(\dot \eta =v_{1}X_{1}+v_{2}X_{2},\) then |v 1| > |v 2|, v 1 > 0, so \(g(\dot \eta ,\nabla _{H}V) = u_{1}v_{1}-u_{2}v_{2}>0,\) and

$$\begin{array}{@{}rcl@{}} L(\gamma|_{[t_{1},t_{2}]})&=&t_{2}-t_{1}=V(\gamma(t_{2}))-V(\gamma(t_{1})) = {\int}_{0}^{\alpha}\frac{dV(\eta(s))}{ds}ds\\ &=&{\int}_{0}^{\alpha}g(\dot\eta,\nabla_{H}V)ds\geq{\int}_{0}^{\alpha}\|\dot\eta(s)\|ds=L(\eta|_{[0,\alpha]}). \end{array} $$

By the reverse Schwartz inequality, L(γ) = L(η) holds if and only if η can be reparameterized as a trajectory of −∇ H V. If γ(t) is a t.p.d. geodesic, then \(|{u_{1}^{0}}|>|{u_{2}^{0}}|\), and \({u_{1}^{0}}<0\). By the same method, we choose a neighborhood \(W_{2}\times (c-\epsilon _{2},c+\epsilon _{2})\subset W\times (c-\epsilon ,c+\epsilon )\) such that u 1(p, t) < 0. Thus, ∇ H V = −Y 1 is future directed. Let \(\rho :[0,\alpha ]\rightarrow U\) be a t.p.d. curve with ρ(0) = γ(t 1), ρ(α) = γ(t 2), and \(\dot \rho =\mu _{1}X_{1}+\mu _{2}X_{2},\) then |μ 1| > |μ 2|, μ 1 < 0, so \(g(\dot \rho ,\nabla _{H}V) = u_{1}\mu _{1}-u_{2}\mu _{2}>0,\) and

$$\begin{array}{@{}rcl@{}} L(\gamma|_{[t_{1},t_{2}]})&=&t_{2}-t_{1}=V(\gamma(t_{2}))-V(\gamma(t_{1})) = {\int}_{0}^{\alpha}\frac{dV(\eta(s))}{ds}ds\\ &=&{\int}_{0}^{\alpha}g(\dot\rho,\nabla_{H}V)ds\geq{\int}_{0}^{\alpha}\|\dot\rho(s)\|ds=L(\rho|_{[0,\alpha]}). \end{array} $$

By the reverse Schwartz inequality, L(γ) = L(ρ) holds if and only if ρ can be reparameterized as a trajectory of −∇ H V. In conclusion, the t.f.d (t.p.d.) normal geodesics are locally maximizers. This ends the proof. □

4.2.3 Light-Like Geodesics

Next, we compute the expressions of light-like geodesics and time-like geodesics on the Engel group. Firstly, we study the case of light-like sub-Lorentzian geodesics.

By the definition, we have \(H=\frac {1}{2}\left (-{\zeta _{1}^{2}}+{\zeta _{2}^{2}}\right ) = 0\), thus ζ 2 = ±ζ 1. If ζ 2 = ζ 1, then light-like trajectories satisfy the ODE:

$$\dot{\gamma}=-\zeta_{1}(X_{1}-X_{2}), $$

i.e., they are reparameterizations of the one-parametric subgroup of the field X 1−X 2. We assume \(\dot {\gamma }=X_{1}-X_{2}\), so

$$\dot{x}_{1}=1,\ \dot{x}_{2}=-1,\ \dot{y}=-\frac{1}{2}(x_{1}+x_{2}),\ \dot{z}=-\frac{1}{2}\left( {x_{1}^{2}}+{x_{2}^{2}}\right), $$

thus,

$$x_{1}=t,\ x_{2}=-t,\ y=0,\ z=-\frac{1}{3}t^{3}. $$

If h 2 = −h 1, similarly, we obtain

$$x_{1}=t,\ x_{2}=t,\ y=0,\ z=\frac{1}{3}t^{3}. $$

In conclusion, we get the following theorem:

Theorem 4.5

Light-like horizontal geodesics starting from the origin are reparameterizations of the curves:

$$x_{1}=t,\ x_{2}=\pm t,\ y=0,\ z=\pm\frac{1}{3}t^{3}, $$

i.e., they are reparameterizations of the one-parameter subgroups corresponding to the vector fields X 1 ± X 2.

4.2.4 Time-Like Geodesics

Secondly, we study time-like sub-Lorentzian geodesics on the Engel group.

We consider the case of ξ 4 = 0 at first. This case is also of interest since it reproduces the earlier known results for the Heisenberg group [18]. In this case, β = −(ξ 3+x 1 ξ 4) = −ξ 3 is a constant. Equations 4.15 become

$$ \dot\zeta_{1}=-\xi_{3}\zeta_{2},\ \ \dot\zeta_{2}=-\xi_{3}\zeta_{1}, $$
(4.16)

where ξ 3 is a constant. There are two separate cases:

Case 1

If ξ 3 = 0, we have ζ 1 and ζ 2 are constants, i.e., ζ 1(s) = ζ 1(0) = ξ 1(0) and ζ 2(s) = ζ 2(0) = ξ 2(0). According to Eq. 4.11, ξ 1 and ξ 2 are constants. On the other hand, by integrating \(\dot x_{1}=-\zeta _{1}\) and \(\dot x_{2}=\zeta _{2}\), we get

$$ x_{1}(s) = -\xi_{1}s \quad \text{ and }\quad x_{2}(s) = \xi_{2}s. $$
(4.17)

Since \(\dot y=\frac 1 2(x_{1}\zeta _{2}+x_{2}\zeta _{1}) = 0\), then y(s) = 0. Also

$$\dot z=\frac{{x_{1}^{2}}+{x_{2}^{2}}}{2}\zeta_{2}=\frac{{\xi_{1}^{2}}+{\xi_{2}^{2}}}{2}\xi_{2}s^{2}, $$

so

$$z(s) = \frac{{\xi_{1}^{2}}+{\xi_{2}^{2}}}{6}\xi_{2}s^{3}=\frac{{x_{1}^{2}}(s)x_{2}(s)+{x_{2}^{3}}(s)}{6}. $$

Theorem 4.6

In the case of ξ 3 = ξ 4 = 0, there is a unique time-like horizontal geodesic joining the origin to a point (x 1 ,x 2, y, z), if and only if y = 0, z is the following function of x 1, x 2:

$$ z=\frac{{x_{1}^{2}}x_{2}+{x_{2}^{3}}}{6}. $$
(4.18)

The expression of the geodesic is

$$ x_{1}(s) = -\xi_{1}s,\ \ x_{2}(s) = \xi_{2}s,\ \ y(s) = 0,\ \ z(s) = \frac{{\xi_{1}^{2}}+{\xi_{2}^{2}}}{6}\xi_{2}s^{3}, $$
(4.19)

where ξ 1, ξ 2 are constants. The arc-length is given by

$$ l=\sqrt{{x_{1}^{2}}-{x_{2}^{2}}}. $$
(4.20)

Its projection to the (x 1 ,x 2 ) plane is a straight line.

Case 2

If ξ 3 ≠ 0, from Eq. 4.16, we have

$$\begin{array}{@{}rcl@{}} &&\zeta_{1}(s) = {\xi_{1}^{0}}\cosh(\xi_{3}s)-{\xi_{2}^{0}}\sinh(\xi_{3}s), \end{array} $$
(4.21)
$$\begin{array}{@{}rcl@{}} &&\zeta_{2}(s) = -{\xi_{1}^{0}}\sinh(\xi_{3}s)+{\xi_{2}^{0}}\cosh(\xi_{3}s), \end{array} $$
(4.22)

where \({\xi _{1}^{0}}=\xi _{1}(0)\), \({\xi _{2}^{0}}=\xi _{2}(0)\). So,

$$\begin{array}{@{}rcl@{}} &&x_{1}=-{{\int}_{0}^{s}}\zeta_{1}(t)dt=-\frac{{\xi_{1}^{0}}}{\xi_{3}}\sinh(\xi_{3}s)+\frac{{\xi_{2}^{0}}}{\xi_{3}}\left( \cosh(\xi_{3}s)-1\right), \end{array} $$
(4.23)
$$\begin{array}{@{}rcl@{}} &&x_{2}={{\int}_{0}^{t}}\zeta_{2}(t)dt=-\frac{{\xi_{1}^{0}}}{\xi_{3}}\left( \cosh(\xi_{3}s)-1\right)+\frac{{\xi_{2}^{0}}}{\xi_{3}}\sinh(\xi_{3}s). \end{array} $$
(4.24)

Substituting them into the expression of \(\dot y, \dot z\) in Eq. 4.11, and integrating, we get

Theorem 4.7

In the case of ξ 3 ≠ 0, ξ 4 = 0, the time-like horizontal geodesics starting from the origin are given by the following:

$$\begin{array}{@{}rcl@{}} &&x_{1}(s) = -A_{1}\sinh(\xi_{3}s)+A_{2}\left( \cosh(\xi_{3}s)-1\right), \end{array} $$
(4.25)
$$\begin{array}{@{}rcl@{}} &&x_{2}(s) = -A_{1}\left( \cosh(\xi_{3}s)-1\right)+A_{2}\sinh(\xi_{3}s), \end{array} $$
(4.26)
$$\begin{array}{@{}rcl@{}} &&y(s) = \frac{1}{2}\left( {A_{2}^{2}}-{A_{1}^{2}}\right)\left( \xi_{3}s-\sinh(\xi_{3}s)\right), \end{array} $$
(4.27)
$$\begin{array}{@{}rcl@{}} &&z(s) = A_{2}\left( {A_{1}^{2}}+{A_{2}^{2}}\right)\cosh^{2}(\xi_{3}s)\sinh(\xi_{3}s) -\frac{2}{3}{A_{2}^{3}}\sinh^{3}(\xi_{3}s)-\frac{1}{3}A_{1}\left( {A_{1}^{2}}+3{A_{2}^{2}}\right)\cosh^{3}(\xi_{3}s)\\ &&\quad +\frac{1}{2}A_{1}\left( {A_{1}^{2}}+3{A_{2}^{2}}\right)\cosh^{^{2}}(\xi_{3}s)-\frac{1}{2}A_{2}\left( 3{A_{1}^{2}}+{A_{2}^{2}}\right)\sinh(\xi_{3}s)\cosh(\xi_{3}s)-\frac{1}{2}A_{2}\left( 3{A_{1}^{2}}+{A_{2}^{2}}\right){s}\\ &&\quad -\frac{1}{6}A_{1}\left( {A_{1}^{2}}+3{A_{2}^{2}}\right). \end{array} $$
(4.28)

where \({\xi _{1}^{0}}=\xi _{1}(0), {\xi _{2}^{0}}=\xi _{2}(0)\) is the initial value, ξ 3 , \(A_{1}=\frac {{\xi _{1}^{0}}}{\xi _{3}}\) , \(A_{2}=\frac {{\xi _{2}^{0}}}{\xi _{3}}\) are constants.

Projections of geodesics to the plane (x 1, x 2) are hyperbolas, for \(\xi (0) = \left (\sqrt {2},1,1,0\right )\), \(\xi (0) = \left (\frac {\sqrt {5}}{2},\frac {1}{2},1,0\right )\) and \(\xi (0) = \left (\frac {\sqrt {5}}{2},\frac {1}{2},-1,0\right ),\) they are shown in Fig. 1.

Fig. 1
figure 1

Projections of geodesics to the plane (x 1, x 2) when ξ 3 ≠ 0, ξ 4 = 0

Full size image

From this theorem, we obtain a description of the reachable set by geodesics ξ 3 ≠ 0, ξ 4 = 0 starting from the origin.

Corollary 4.8

In the case of ξ 3 ≠ 0, ξ 4 = 0, let (x 1 ,x 2 ,y,z) be a point on a time-like geodesic, then we have

$$-1<\frac{4y}{-{x_{1}^{2}}+{x_{2}^{2}}}<1. $$

Proof

By Eqs. 4.25 and 4.26, we get

$$ -{x_{1}^{2}}+{x_{2}^{2}}=4\left( {A_{2}^{2}}-{A_{1}^{2}}\right)\sinh^{2}\left( \frac{\xi_{3}}{2}\right), $$
(4.29)

substituting Eq. 4.29 into Eq. 4.27, we obtain the following equation:

$$ y=\frac{\left( -{x_{1}^{2}}+{x_{2}^{2}}\right)(\xi_{3}-\sinh(\xi_{3}))}{8\sinh^{2}\left( \frac{\xi_{3}}{2}\right)}, $$
(4.30)

if we set \(\tau =\frac {\xi _{3}}{2}\), then

$$ y=\frac{\left( -{x_{1}^{2}}+{x_{2}^{2}}\right)}{4}\left( \frac{\tau}{\sinh^{2}(\tau)}-{\coth(\tau)}\right), $$
(4.31)

or

$$ \frac{4y}{\left( -{x_{1}^{2}}+{x_{2}^{2}}\right)}=\frac{\tau}{\sinh^{2}(\tau)}-{\coth(\tau)}. $$
(4.32)

It is easy to check that the right-hand side of Eq. 4.32 is a decreasing function in \((-\infty ,\) \(+\infty ),\) and its range is (−1, 1). That is to say, the points on the time-like geodesics should satisfy

$$-1<\frac{4y}{-{x_{1}^{2}}+{x_{2}^{2}}}<1. $$

This ends the proof. □

Next, we consider the case ξ 4 ≠ 0. Recall that

$$ \dot\zeta_{1}=\beta\zeta_{2},\quad \dot\zeta_{2}=\beta\zeta_{1},{\kern3pt} \text{where}{\kern3pt} \beta(s) = -(\xi_{3}+x_{1}\xi_{4}),{\kern3pt} \dot\beta=\xi_{4}\zeta_{1}. $$
(4.33)

Combining the expressions for \(\dot \beta \) and \(\dot \zeta _{2}\) to get

$$ \xi_{4}\dot\zeta_{2}=\beta\xi_{4}\zeta_{1}=\beta\dot\beta. $$
(4.34)

Integrating both sides, we have

$$ \xi_{4}\zeta_{2}=\frac{\beta^{2}}{2}+C_{1},{\kern3pt} \text{where}{\kern3pt} C_{1}=\xi_{4}\zeta_{2}(0)-\frac{\beta^{2}(0)}{2}=\xi_{4}{\xi_{2}^{0}}-\frac{{\xi_{3}^{2}}}{2}. $$
(4.35)

This yields

$$ x_{1}(s) = -\frac{\beta(s)+\xi_{3}}{\xi_{4}}, $$
(4.36)

and

$$ \zeta_{2}(s) = \frac{1}{\xi_{4}}\left( \frac{\beta^{2}(s)}{2}+C_{1}\right). $$
(4.37)

Since \(\dot x_{2}=\zeta _{2},\) we deduce

$$ x_{2}(s) = {{\int}_{0}^{s}}\zeta_{2}(t)dt=\frac{1}{\xi_{4}}{{\int}_{0}^{s}}\left( \frac{\beta^{2}(t)}{2}+C_{1}\right)dt. $$
(4.38)

To compute y(s) in term of β(s), we note that

$$ \dot y=\frac 1 2(x_{1}\zeta_{2}+x_{2}\zeta_{1}) = \frac12(x_{1}\dot x_{2}-x_{2}\dot x_{1}), $$
(4.39)

then integration by parts yields

$$\begin{array}{@{}rcl@{}} y(s)&=&\frac12{{\int}_{0}^{s}}(x_{1}\dot x_{2}-x_{2}\dot x_{1})dt={{\int}_{0}^{s}}x_{1}\zeta_{2}dt-\frac12x_{1}x_{2}\\ &=&-\frac{1}{{\xi_{4}^{2}}}{{\int}_{0}^{s}}\left( {\beta(t)+\xi_{3}}\right)\left( \frac{\beta^{2}(t)}{2}+C_{1}\right)dt-\frac12x_{1}x_{2}. \end{array} $$
(4.40)

Finally, since \(\dot z=\frac {{x_{1}^{2}}+{x_{2}^{2}}}{2}\zeta _{2},\)

$$\begin{array}{@{}rcl@{}} z(s)&=&{{\int}_{0}^{s}}\frac{{x_{1}^{2}}+{x_{2}^{2}}}{2}\zeta_{2}dt=\frac12{{\int}_{0}^{s}}{x_{1}^{2}}\zeta_{2}dt+\frac16{x_{2}^{3}}\\ &=&\frac{1}{2{\xi_{4}^{3}}}{{\int}_{0}^{s}}\left( {\beta(t)+\xi_{3}}\right)^{2}\left( \frac{\beta^{2}(t)}{2}+C_{1}\right)dt+\frac16{x_{2}^{3}}. \end{array} $$
(4.41)

Once we find β, we can find the geodesic (x 1(s), x 2(s), y(s), z(s)) explicitly.

Since \(\dot \beta (s) = \xi _{4}\zeta _{1}\), \(\dot \beta (0) = \xi _{4}\zeta _{1}(0) = \xi _{4}{\xi _{1}^{0}}\), we have

$$ \ddot\beta(s) = \xi_{4}\dot\zeta_{1}=\xi_{4}\beta(s)\zeta_{2}=\beta(s)(\xi_{4}\zeta_{2}) = \beta(s)\left( \frac{\beta^{2}(s)}{2}+C_{1}\right). $$
(4.42)

Multiplying both sides by \(2\dot \beta (s)\) and integrating, we have

$$ \dot\beta^{2}(s) = \frac{\beta^{4}(s)}{4}+C_{1}\beta^{2}(s)+C_{2}=\left( \frac{\beta^{2}(s)}{2}+C_{1}\right)^{2}+C_{2}-{C_{1}^{2}}, $$
(4.43)

where C 2 is a constant, and

$$ C_{2}=\dot\beta^{2}(0)-\frac{\beta^{4}(0)}{4}-C_{1}\beta^{2}(0) = ({\xi_{1}^{0}})^{2}{\xi_{4}^{2}}+\frac{{\xi_{3}^{4}}}{4}-{\xi_{2}^{0}}{\xi_{3}^{2}}\xi_{4}. $$
(4.44)

Then,

$$ C_{2}-{C_{1}^{2}}=({\xi_{1}^{0}})^{2}{\xi_{4}^{2}}+\frac{{\xi_{3}^{4}}}{4}-{\xi_{2}^{0}}{\xi_{3}^{2}}\xi_{4}-\left( {\xi_{2}^{0}}\xi_{4}-\frac{{\xi_{3}^{2}}}{2}\right)^{2}={\xi_{4}^{2}}(({\xi_{1}^{0}})^{2}-({\xi_{2}^{0}})^{2}) = {\xi_{4}^{2}}, $$
(4.45)

since \(({\xi _{1}^{0}})^{2}-({\xi _{2}^{0}})^{2}=1\).

Assume \(\dot \beta (s)>0\), we have

$$ \frac{d\beta(s)}{ds}=\sqrt{\left( \frac{\beta^{2}(s)}{2}+C_{1}\right)^{2}+{\xi_{4}^{2}}}. $$
(4.46)

Hence,

$$ ds=\frac{d\beta}{\sqrt{\left( \frac{\beta^{2}(s)}{2}+C_{1}\right)^{2}+{\xi_{4}^{2}}}}. $$
(4.47)

Let ρ 2=C 1+ξ 4 i, \(\bar {\rho }^{2}=C_{1}-\xi _{4}i\) and \(u=\frac {\beta }{\sqrt 2}\), integrating (4.47) from 0 to s, we obtain

$$ s={\int}_{\frac{\beta(0)}{\sqrt 2}}^{\frac{\beta(s)}{\sqrt 2}}\frac{\sqrt 2du}{\sqrt{(u^{2}+\rho^{2})(u^{2}+\bar\rho^{2})}}. $$
(4.48)

Set

$$\begin{array}{@{}rcl@{}} k^{2}&=&-\frac{(\rho-\bar\rho)^{2}}{4\rho\bar\rho}=\frac{\sqrt{{C_{1}^{2}}+{\xi_{4}^{2}}}-C_{1}}{2\sqrt{{C_{1}^{2}}+{\xi_{4}^{2}}}},\\ \mathfrak{g}&=&\frac{1}{2\sqrt{\rho\bar\rho}}=\frac{1}{2\left( {C_{1}^{2}}+{\xi_{4}^{2}}\right)^{\frac{1}{4}}}.\ \end{array} $$

Since

$$ {\int}_{y}^{\infty}\frac{dt}{\sqrt{(t^{2}+\rho^{2})(t^{2}+\bar\rho)}}=\mathfrak{g}\cdot cn^{-1}\ (cos\ \varphi,k) = \mathfrak{g}F(\varphi,k), $$
(4.49)

where c n −1 (y, k) is a Jacobi’s Inverse Elliptic Functions, and

$$ \varphi=\cos^{-1}\ \left( \frac{y^{2}-\rho\bar\rho}{y^{2}+\rho\bar\rho}\right),\ \quad F(\varphi,k) = {\int}_{0}^{\varphi}\frac{dt}{\sqrt{1-k^{2}sin^{2}\ t}}. $$

Hence,

$$ {\int}_{\frac{\beta(0)}{\sqrt 2}}^{\frac{\beta(s)}{\sqrt 2}}\frac{\sqrt {2}du}{\sqrt{(u^{2}+\rho^{2})(u^{2}+\bar\rho^{2})}}={\int}_{\frac{\beta(0)}{\sqrt {2}}}^{\infty}\frac{\sqrt {2}du}{\sqrt{(u^{2}+\rho^{2})(u^{2}+\bar\rho^{2})}}-{\int}_{\frac{\beta(s)}{\sqrt {2}}}^{\infty}\frac{\sqrt {2}du}{\sqrt{(u^{2}+\rho^{2})(u^{2}+\bar\rho^{2})}}. $$

According to Eq. 4.49, we have

$$ {\int}_{\frac{\beta(0)}{\sqrt {2}}}^{\infty}\frac{\sqrt {2}du}{\sqrt{(u^{2}+\rho^{2})(u^{2}+\bar\rho^{2})}}=\sqrt{2}\mathfrak{g}F(\varphi_{1},k) = constant, $$
(4.50)

where

$$\varphi_{1}=cos^{-1}\ \left( \frac{{\xi_{3}^{2}}-2\sqrt{C_{1}+{\xi_{4}^{2}}}}{{\xi_{3}^{2}}+2\sqrt{C_{1}+{\xi_{4}^{2}}}}\right). $$

Since

$$ {\int}_{\frac{\beta(s)}{\sqrt 2}}^{\infty}\frac{\sqrt 2du}{\sqrt{(u^{2}+\rho^{2})(u^{2}+\bar\rho^{2})}}=\sqrt2\mathfrak{g}\cdot cn^{-1}\ \left( \frac{\beta^{2}(s)-2\rho\bar\rho}{\beta^{2}(s)+2\rho\bar\rho}\right). $$
(4.51)

Hence,

$$ cn^{-1}\ \left( \frac{\beta^{2}(s)-2\rho\bar\rho}{\beta^{2}(s)+2\rho\bar\rho}\right) = F(\varphi_{1},k)-\frac{s}{\sqrt{2}\mathfrak{g}}, $$
(4.52)

let F = F(φ 1, k), we obtain

$$ \beta^{2}(s) = \frac{2\rho\bar\rho\left( 1+cn\ \left( F -\frac{s}{\sqrt{2}\mathfrak{g}},k\right)\right)}{\left( 1-cn\ \left( F-\frac{s}{\sqrt{2}\mathfrak{g}},k\right)\right)}= \frac{2\rho\bar\rho\left( 1+cn\ \left( 2\tilde{s},k\right)\right)}{\left( 1-cn\ \left( 2\tilde{s},k\right)\right)}, $$
(4.53)

where \(2\tilde {s}=F-\frac {s}{\sqrt {2}\mathfrak {g}}\).

Since

$$ \frac{1-cn\ (2s)}{1+cn\ (2s)}=tn^{2}\ (s)dn^{2}\ (s), $$
(4.54)

hence,

$$ \beta(s) = \frac{\sqrt{2\rho\bar\rho}}{tn\ (\tilde s,k)dn\ (\tilde s,k)}=\sqrt{2\rho\bar\rho}cs\ (\tilde s,k)nd\ (\tilde s,k). $$
(4.55)

For the case of \(\dot {\beta }(s)<0,\) we can calculate by the same method, and get the same result. But the expression of the parameter \(\tilde {s}\) in Eqs. 4.53 and 4.55 should be changed to

$$\frac{1}{2}\left( F+\frac{s}{\sqrt{2}\mathfrak{g}}\right). $$

Thus, the sign of \(\dot {\beta }(s)\) will not affect the expression of the geodesics.

Therefore, integrating Eqs. 4.36, 4.38, 4.40, and 4.41, we get a complete description of the Hamiltonian time-like geodesics in the Engel group.

Theorem 4.9

In the case of ξ 4 ≠ 0, time-like geodesics starting from the origin are given by the following:

$$\begin{array}{@{}rcl@{}} x_{1}(s)&=&-\frac{1}{\xi_{4}}(\beta(s)+\xi_{3}), \end{array} $$
(4.56)
$$\begin{array}{@{}rcl@{}} x_{2}(s)&=&\frac{1}{2\xi_{4}}(B_{2}(s)+2C_{1}), \end{array} $$
(4.57)
$$\begin{array}{@{}rcl@{}} y(s)&=&-\frac{1}{2{\xi_{4}^{2}}}(B_{3}(s)+2C_{1}B_{1}(s)+\xi_{3}B_{2}(s)+2C_{1}\xi_{3}s)-\frac{1}{2}x_{1}(s)x_{2}(s), \end{array} $$
(4.58)
$$\begin{array}{@{}rcl@{}} z(s)&=&\frac{1}{4{\xi_{4}^{3}}}(B_{4}(s)+2C_{1}B_{2}(s)+2\xi_{3}B_{3}(s)+4C_{1}\xi_{3}B_{1}(s)+{\xi_{3}^{2}}B_{2}(s)+2C_{1}{\xi_{3}^{2}})+\frac{1}{6}{x_{2}^{3}}(s),\\ \end{array} $$
(4.59)

where \(C_{1}=\xi _{4}{\xi _{2}^{0}}-\frac {{\xi _{3}^{2}}}{2}\) , \(B_{i}(s) = {{\int }^{s}_{0}}\beta ^{i}(t)dt,\ i=1,\ldots ,4,\) and the expressions of B i (s) are presented in Appendix.

Projections of geodesics to the plane (x 1, x 2) with ξ(0) = (1, 0, 1, 1), \(\xi (0) = \left (\frac {\sqrt {5}}{2},1/2,2,1\right )\) and \(\xi (0) = \left (\frac {\sqrt {5}}{2},1/2,1,1\right )\) are shown in Fig. 2.

Fig. 2
figure 2

Projections of geodesics to the plane (x 1, x 2) when ξ 3 ≠ 0, ξ 4 ≠ 0

Full size image