On the Integration of the Regularized Equations of the Two-Body Problem

M. S. Yarov-Yarovoi

In the present work a general method is proposed for regularizing canonical equations and the corresponding Hamilton–Jacobi equation. The method is applied to the two-body problem. In this same problem, for the rectangular coordinates of the radius vector, extended regularized equations are obtained, having the form of nonhomogeneous linear differential equations with constant coefficients. The solution of these equations is found for all types of orbits. The results of this work will be used for constructing a perturbation theory for the regularized equations of problems of celestial mechanics and astrodynamics.

1. Along with the well-known device of Poincaré (for example, [1], p. 58) for constructing regularized canonical equations, one may use the following theorem [2].

Theorem 1.1. In order to find the complete integral

\[ V = V(q^1,\ldots,q^n;\alpha_1,\ldots,\alpha_n)+\alpha_{n+1} \tag{1.1} \]

of the first-order partial differential equation

\[ p+H(t,q^1,\ldots,q^n;p_1,\ldots,p_n)=0 \tag{1.2} \]

\[ (p=\partial V/\partial t,\quad p_i=\partial V/\partial q^i;\quad i=1,\ldots,n) \]

it is sufficient to find a complete integral \(W\) of the form

\[ W=-\alpha_0\tau+W_1(t,q^1,\ldots,q^n;\alpha_0,\alpha_1,\ldots,\alpha_n)+\alpha_{n+1} \tag{1.3} \]

of the partial differential equation

\[ p_0+pI+IH=0 \tag{1.4} \]

\[ (p_0=\partial W/\partial\tau,\quad p=\partial W/\partial t,\quad p_i=\partial W/\partial q^i,\quad i=1,\ldots,n), \]

where the function

\[ I=I(t,q^1,\ldots,q^n,p,p_1,\ldots,p_n) \tag{1.5} \]

does not vanish for any values of its variables in their domain of variation, and to set in this complete integral the constant \(\alpha_0=0\).

Proof. The characteristic equations (canonical equations) for equation (1.4) have the obvious first integral

\[ pI+IH=\alpha_0. \tag{1.6} \]

Therefore, if in the left-hand side we substitute the complete integral \(W\) of the form (1.3) and put \(\alpha_0=0\), then we arrive at the relation

\[ V=\left. W\right|_{\alpha_0=0}, \tag{1.7} \]

and this is precisely what is asserted in our theorem.

To find the complete integral \(W\) of equation (1.4), one may apply Cauchy’s method of characteristics [3]. The characteristic equations have the canonical form

\[ \tau'=\frac{\partial H^*}{\partial p_0},\quad p_0'=-\frac{\partial H^*}{\partial \tau}, \]

\[ t'=\frac{\partial H^*}{\partial p},\quad p'=-\frac{\partial H^*}{\partial t}, \tag{1.8} \]

\[ (q^i)'=\frac{\partial H^*}{\partial p_i},\quad (p_i)'=-\frac{\partial H^*}{\partial q^i} \quad (i=1,\ldots,n) \]

with characteristic function \(H^*\) of the form

\[ H^*=p_0+pI+IH. \tag{1.9} \]

Since \(\partial H^*/\partial p_0=1\), \(\partial H^*/\partial \tau=0\), in the system (1.8) \(\tau\) is an independent variable, and this system has the first integral (1.6), expressing the constancy of the momentum \(p_0\) of the auxiliary variable \(\tau\). Therefore the system (1.8) reduces to a system of canonical equations of order \((2n+2)\)

\[ t'=\frac{\partial H_1}{\partial p},\quad p'=-\frac{\partial H_1}{\partial t}, \]

\[ (q^i)'=\frac{\partial H}{\partial p_i},\quad (p_i)'=-\frac{\partial H^1}{\partial q^i} \tag{1.10} \]

with characteristic function

\[ H_1=Ip+IH. \tag{1.11} \]

This system also has the first integral (1.6). Consider the original canonical system

\[ (q^i)'=\frac{\partial H}{\partial p_i},\quad (p_i)'=-\frac{\partial H}{\partial q^i}. \tag{1.12} \]

Theorem 1.2. The general solution of the system (1.12) is obtained from the general solution of the system (1.10), if in the latter one sets \(\alpha_0=0\) and eliminates \(\tau\).

The proof of this theorem is obvious.

At first glance it seems that such an introduction of a new auxiliary variable \(\tau\) raises the order of the system by two units, and therefore the solution of the problem is complicated. However, this is not so. By making use of the arbitrariness in the choice of the function \(I\), one can arrange that the function \(H_1\) and its partial derivatives have no analytic singularities of the kind that, generally speaking, the function \(H\) and its partial derivatives possess. In the case of regularization, the analytic properties of the general solution of the system (1.10) will be simpler than those of the general solution of the system (1.12). For these reasons \(\tau\) is called a regularizing variable, the function \(I\) a regularizing multiplier, and equation (1.4) the regularizing equation corresponding to equation (1.1).

Remark 1.1. By the same method one can introduce one more, or several more, regularizing variables. The corresponding regularizing equation may, for example, have the form

\[ p_k+p_{k-1}I_k+p_{k-2}I_{k-1}I_k+\ldots+p_0 I_k\ldots I_1+ \]
\[ +pI_k\ldots I_0+I_k\ldots I_0H=0. \tag{1.13} \]

2. The integration of a regularizing equation in partial derivatives can sometimes be carried out by the method of separation of variables [2].

Theorem 2.1. If the partial differential equation

\[ \Phi(q^1,\ldots,q^n;\,p_1,\ldots,p_n)=0 \qquad (p_i=\partial V/\partial q^i,\ i=1,\ldots,n) \tag{2.1} \]

can be solved with respect to each of the momenta

\[ p_i=\Phi_i(q^1,\ldots,q^n;\,p_1,\ldots,p_{i-1},p_{i+1},\ldots,p_n) \tag{2.2} \]

and the initial conditions \(q_0^i,\ p_i^0\) \((i=1,\ldots,n)\) for the Cauchy problem satisfy the equation itself

\[ \Phi(q_0^1,\ldots,q_0^n;\,p_1^0,\ldots,p_n^0)=0, \tag{2.3} \]

then a necessary and sufficient condition that equation (2.1) be integrated by the method of separation of variables, i.e. that equation (2.1) have a complete integral of the form

\[ V=V^1(q^1)+\ldots+V^n(q^n), \tag{2.4} \]

is the identity

\[ \Phi(q^1,\ldots,q^n;\,\Phi_1^0,\ldots,\Phi_n^0)=0 \tag{2.5} \]

with respect to \(q^1,\ldots,q^n;\ q_0^1,\ldots,q_0^n;\ p_1^0,\ldots,p_n^0\),

if

\[ \Phi_i^0= \]

\[ =\Phi_i(q_0^1,\ldots,q_0^{\,i-1},q^i,q_0^{\,i+1},\ldots,q_0^n;\, p_1^0,\ldots,p_{i-1}^0,p_{i+1}^0,\ldots,p_n^0). \tag{2.6} \]

Proof. If a function \(V\) of the form (2.4) is substituted in equation (2.2), then on the left there will be only the function \(q^i\), and on the right a function \(q^i\) and all the remaining variables \(q^j\) \((j\ne i)\). Since \(\operatorname{Det}\|\partial^2V/\partial q^k\partial\alpha_l\|\ne0\), the arbitrary constants of integration \(\alpha_l\) entering into \(V^j\) can be chosen so that

\[ p_j(q_0^j)= \left.\frac{\partial V}{\partial q^j}\right|_{q^j=q_0^j} = \left.\frac{dV^j}{dq^j}\right|_{q^j=q_0^j} =p_j^0. \tag{2.7} \]

But then, in the right-hand side of equation (2.2), all the variables \(q^j,\ p_j\) \((j\ne i)\) may be set equal to the initial values \(q_0^j,\ p_j^0\). This gives

\[ V^i=\int \Phi_i^0\,dq^i. \tag{2.8} \]

The function \(V=V^1+\ldots+V^n\) satisfies equation (2.1) if the identity (2.5) is fulfilled. The theorem is proved.

It is essential that Theorem 2.1 makes it possible to find a complete integral with separated variables without integrating the canonical equations themselves. The main ideas of this theorem were already announced by us in [2].

3. Let us apply these results to the construction and integration of the regularizing equation of the two-body problem.

If, instead of rectangular coordinates, spherical coordinates are introduced by the formulas

\[ x=r\cos\varphi\cos\lambda,\quad y=r\cos\varphi\sin\lambda,\quad z=r\sin\varphi, \tag{3.1} \]

then the Hamilton–Jacobi equation for the two-body problem has the form [4]

\[ p+\frac{1}{2}\left(p_r^2+\frac{1}{r^2}p_\varphi^2+\frac{1}{r^2\cos^2\varphi}p_\lambda^2\right)-\frac{\mu}{r}=0 \tag{3.2} \]

\[ (p=\partial V/\partial t,\quad p_r=\partial V/\partial r,\quad p_\varphi=\partial V/\partial\varphi,\quad p_\lambda=\partial V/\partial\lambda). \]

Here \(\mu=f(m_1+m_2)\), \(f\) is the gravitational constant, \(m_i\) are the masses of the bodies, and \(r\) is the distance separating them.

The left-hand side of this equation has an analytic singularity at \(r=0\). The singularity in parentheses is removed by the choice of a coordinate system (for example, rectangular coordinates). Therefore it is sufficient to remove the singularity of type \(1/r\) and to choose

\[ l=\frac{r}{\mu}. \tag{3.3} \]

Then the corresponding regularizing equation has the form

\[ p_0+\frac{r}{\mu}p+\frac{1}{2}\left(\frac{r}{\mu}p_r^2+\frac{1}{\mu r}p_\varphi^2+\frac{1}{\mu r\cos^2\varphi}p_\lambda^2\right)-1=0. \tag{3.4} \]

We shall seek a complete integral of this equation in the form

\[ W=W_0(\tau)+W_1(t)+W_2(r)+W_3(\varphi)+W_4(\lambda) \tag{3.5} \]

by the method described in the preceding subsection. Omitting lengthy calculations, by Theorem 2.1 one may find

\[ W=-\alpha_4-\alpha_0\tau-\alpha_1 t-\alpha_2\lambda+ \]

\[ +\int \sqrt{\frac{2\mu(1+\alpha_0)}{r}+2\alpha_1-\frac{\alpha_3}{r^2}}\,dr +\int \sqrt{\alpha_3-\frac{\alpha_2^2}{\cos^2\varphi}}\,d\varphi . \tag{3.6} \]

The arbitrary constants of integration \(\alpha_i\) are expressed in terms of the initial values of the variables and momenta as follows:

\[ \alpha_0=-p_0^0,\quad \alpha_1=-p^0,\quad \alpha_2=-p_\lambda^0, \]

\[ \alpha_3=(p_\varphi^0)^2+\frac{1}{\cos^2\varphi_0}(p_\lambda^0)^2 =-r_0^2(p_r^0)^2-2r_0^2p^0-2\mu r_0p_0^0+2\mu r_0. \tag{3.7} \]

The complete integral \(V\) of the original equation (3.2) is obtained from expression (3.6) for \(\alpha_0=0\).

The expressions for the coordinates and time in terms of the regularizing variable \(\tau\) may be found from the equations [1]

\[ \frac{\partial W}{\partial\alpha_0}=\beta_0,\quad \frac{\partial W}{\partial\alpha_1}=\beta_1,\quad \frac{\partial W}{\partial\alpha_2}=\beta_2,\quad \frac{\partial W}{\partial\alpha_3}=\beta_3, \tag{3.8} \]

where in the left-hand sides, after differentiating \(W\), one must put \(\alpha_0=0\).

If \(W\) from (3.6) is substituted into (3.8) and \(\alpha_0=0\) is put, then

\[ -\tau+\mu\int\frac{dr}{\sqrt{2\alpha_1r^2+2\mu r-\alpha_3}}=\beta_0, \]

\[ -t+\int\frac{r\,dr}{\sqrt{2\alpha_1r^2+2\mu r-\alpha_3}}=\beta_1, \tag{3.9} \]

\[ -\lambda-a_2\int \frac{d\varphi}{\cos\varphi\sqrt{a_3\cos^2\varphi-a_2^2}}=\beta_2, \tag{3.9} \]

\[ -\frac12\int \frac{dr}{r\sqrt{2a_1r^2+2\mu r-a_3}} +\frac12\int \frac{\cos\varphi\,d\varphi}{\sqrt{a_3\cos^2\varphi-a_2^2}}=\beta_3. \]

The first equation gives the relation between the radius vector and \(\tau\). The remaining equations have the usual form [4], since the regularizing variable \(\tau\) itself does not enter into them.

1. Let a material point with coordinates \(x, y, z\) move in a force field with potential \(U(x,y,z,t)\). Then the characteristic function of the problem has the form

\[ H=\frac12(p_x^2+p_y^2+p_z^2)-U(x,y,z,t), \tag{4.1} \]

and the regularizing characteristic function is

\[ H_1=\overline{I}_p+IH. \tag{4.2} \]

If, for this case, we write the canonical system (1.10) and take (4.2) into account, we obtain

\[ t'=I,\qquad x'=Ip_x,\qquad y'=Ip_y,\qquad z'=Ip_z, \]

\[ p_x'=I\frac{\partial U}{\partial x},\qquad p_y'=I\frac{\partial U}{\partial y},\qquad p_z'=I\frac{\partial U}{\partial z}, \tag{4.3} \]

since we need only those trajectories on which \(p+H=0\).

We shall assume that \(I\) does not depend on the momenta, and we shall eliminate the momenta \(p_x,p_y,p_z\) from equations (4.3),

\[ t'=I,\qquad \left(\frac{x'}{I}\right)'=I\frac{\partial U}{\partial x},\qquad \left(\frac{y'}{I}\right)'=I\frac{\partial U}{\partial y},\qquad \left(\frac{z'}{I}\right)'=I\frac{\partial U}{\partial z} \tag{4.4} \]

or

\[ t'=I,\qquad x''=\frac{I'}{I}x'+I^2\frac{\partial U}{\partial x}, \tag{4.5} \]

\[ y''=\frac{I'}{I}y'+I^2\frac{\partial U}{\partial y},\qquad z''=\frac{I'}{I}z'+I^2\frac{\partial U}{\partial z}. \]

These are the regularized equations for the motion of a material point in space.

1. For the two-body problem \(U=\mu/r\), and let \(I=1/U=r/\mu\). Then equations (4.5) take the form

\[ t'=\frac{r}{\mu},\qquad x''=\frac{r'}{r}x'-\frac{1}{\mu}\frac{x}{r}, \]

\[ y''=\frac{r'}{r}y'-\frac{1}{\mu}\frac{y}{r},\qquad z''=\frac{r'}{r}z'-\frac{1}{\mu}\frac{z}{r}. \tag{5.1} \]

Theorem 5.1. The system of equations (5.1) can be reduced to the form

\[ t'=\frac{r}{\mu},\qquad x''=\frac{2h}{\mu^2}x-\frac{f_1}{\mu^2}, \]

\[ y''=\frac{2h}{\mu^2}\,y-\frac{f_2}{\mu^2},\qquad z''=\frac{2h}{\mu^2}\,z-\frac{f_3}{\mu^2}, \tag{5.2} \]

where \(h\) is the constant of the energy integral, and \(f_1, f_2, f_3\) are the constants of the Laplace integrals for system (5.1). In this case the equation for the radius vector takes the form

\[ r''=\frac{2h}{\mu^2}\,r+\frac{1}{\mu}. \tag{5.3} \]

Theorem 5.1 in fact establishes that the dependence of the rectangular coordinates and of the radius vector on the regularizing variable \(\tau\) can be found from linear inhomogeneous differential equations with constant coefficients. The variables \(x, y, z\) in system (5.2) and \(r\) in system (5.3) are separated, and therefore each of these equations is integrated independently. The dependence of \(t\) on \(\tau\) reduces to the performance of an elementary quadrature.

Proof. Let us obtain the energy integral for system (5.1). To do this, multiply both sides of the second equation by \(x'\), the third equation by \(y'\), and the fourth equation by \(z'\), and add. As a result we obtain

\[ \frac{1}{2}\,(x'^2+y'^2+z'^2)'-\frac{r'}{r}\,(x'^2+y'^2+z'^2)=-\frac{1}{\mu}\,r'. \]

The integrating factor here is \(\mu^2/r^2\). The energy integral has the form

\[ \frac{1}{2}\,(x'^2+y'^2+z'^2)=\frac{r}{\mu}+h\,\frac{r^2}{\mu^2}. \tag{5.4} \]

It is immediately clear from the energy integral that as \(r\to 0\) the regularized velocity also tends to zero (whereas the ordinary velocity tends to infinity).

The area integrals are obtained by eliminating, from two equations out of the last three, the second terms, which have the factor \(1/\mu r\). They have the form

\[ yz'-zy'=\frac{c_1}{\mu}\,r,\qquad zx'-xz'=\frac{c_2}{\mu}\,r,\qquad xy'-yx'=\frac{c_3}{\mu}\,r. \tag{5.5} \]

Here \(c_1,c_2,c_3\) are arbitrary constants of integration.

Let us now obtain the equation for \(r\) (5.3). Differentiating twice the identity \(r^2=x^2+y^2+z^2\), we obtain

\[ rr''+r'^2=xx''+yy''+zz''+x'^2+y'^2+z'^2. \]

Substituting here \(x'',y'',z''\) from system (5.1), and \(x'^2+y'^2+z'^2\) from the energy integral (5.4), we obtain

\[ rr''+r'^2=r'^2-\frac{r}{\mu}+2\,\frac{r}{\mu}+2h\,\frac{r^2}{\mu^2}, \]

which, after collecting like terms and cancelling by \(r\), gives (5.3).

In order to derive the Laplace integrals, we set up an equation for the auxiliary variable

\[ r^*=rr'=\mu r'. \tag{5.6} \]

Differentiating both sides of (5.3) with respect to \(\tau\), we obtain

\[ r'''=\frac{2h}{\mu^2}\,r'. \]

From (5.3),

\[ \frac{2h}{\mu^2}=\frac{r''}{r}-\frac{1}{\mu r}. \]

Therefore

\[ r'''=\frac{r'}{r}\,r''-\frac{1}{\mu r}\,r'. \]

Expressing \(r'\), \(r''\), \(r'''\) in terms of \(r^*\) and its derivatives with respect to \(\tau\), we obtain

\[ r^{*''}=\frac{r'}{r}\,r^{*'}-\frac{1}{\mu r}\,r^* . \tag{5.7} \]

This equation for \(r^*\) has the same form as the equations for \(x,y,z\) (5.1). To obtain one of Laplace’s first integrals, multiply both sides of the second equation of system (5.1) by \(-r^*\) and equation (5.7) by \(x\). As a result we obtain

\[ (xr^{*'}-r^*x')'=\frac{r'}{r}\,(xr^{*'}-r^*x') . \]

Hence

\[ xr^{*'}-r^*x'=\frac{f_1}{\mu}\,r,\qquad yr^{*'}-r^*y'=\frac{f_2}{\mu}\,r,\qquad zr^{*'}-r^*z'=\frac{f_3}{\mu}\,r . \tag{5.8} \]

Here \(f_1,f_2,f_3\) are arbitrary constants of integration. The other two Laplace integrals (5.8) are obtained analogously.

The Laplace integrals make it possible essentially to simplify the right-hand sides of the last three equations of system (5.1) and to bring them to the form (5.2), i.e. to prove Theorem 5.1 completely.

From equation (5.3) we find the expression for

\[ r^{*'}=\frac{2h}{\mu}\,r+1 . \tag{5.9} \]

Using (5.6), (5.9), we find

\[ \frac{2h}{\mu}\,rx+x-\mu r'x'=\frac{f_1}{\mu}\,r,\qquad \frac{2h}{\mu}\,ry+y-\mu r'y'=\frac{f_2}{\mu}\,r, \]

\[ \frac{2h}{\mu}\,rz+z-\mu r'z'=\frac{f_3}{\mu}\,r . \]

Hence

\[ \frac{r'}{r}\,x'-\frac{1}{\mu}\,\frac{x}{r} =\frac{2h}{\mu^2}\,x-\frac{f_1}{\mu^2},\qquad \frac{r'}{r}\,y'-\frac{1}{\mu}\,\frac{y}{r} =\frac{2h}{\mu^2}\,y-\frac{f_2}{\mu^2}, \]

\[ \frac{r'}{r}\,z'-\frac{1}{\mu}\,\frac{z}{r} =\frac{2h}{\mu^2}\,z-\frac{f_3}{\mu^2}, \]

i.e. equations (5.2) will be valid. The theorem is proved.

Corollary 5.1. The components of the regularized velocity \(x',y',z'\) satisfy a system of linear homogeneous differential equations with constant coefficients

\[ (x')''=\frac{2h}{\mu^2}\,x',\qquad (y')''=\frac{2h}{\mu^2}\,y',\qquad (z')''=\frac{2h}{\mu^2}\,z', \tag{5.10} \]

in which the variables are separated.

Equations (5.10) are obtained by differentiating with respect to \(\tau\) the last three equations of system (5.2).

Thus, as a result of the regularization of the equations of motion in the two-body problem, not only the equations for the rectangular coordinates and the radius vector, but also those for the components of the regularized velocity turn out to be separated.

6. Theorem 6.1. If, as the regularizing multiplier for the two-body problem, one chooses \(I=r/\mu\), then the rectangular coordinates, the radius vector, the components of the regularized velocity, and the time are represented in the form of absolutely convergent power series in the regularizing variable for all its values, real or complex, and for all initial values of the coordinates, velocity components, and time, provided only that the initial value of the radius vector is different from zero.

The proof of the theorem will begin with an obvious remark.

Remark 6.1. After finding the general solution of equations (5.2), (5.3), (5.10), in order to obtain the general solution of equations (5.1), i.e., the general solution of the two-body problem, it is necessary and sufficient to express the arbitrary constants of integration—both those introduced by integrating equations (5.2), (5.3), (5.10) and \(h, f_1, f_2, f_3\)—as functions of the seven initial values \(t_0, \xi_0, \eta_0, \zeta_0, \xi'_0, \eta'_0, \zeta'_0\).

This remark is a consequence of the fact that, although in equations (5.2), (5.3), (5.10) the coefficients \(i\) will be constants, they will change their values as functions of the initial data. The latter consideration is especially essential when considering the equations of perturbed motion.

Proof of Theorem 6.1. Unfortunately, in modern mathematics there is no single notation for two linearly independent solutions of the equation

\[ x'' = k_1 x + k_2 \qquad (k_1,\ k_2 \text{ are constants}), \tag{6.1} \]

which have no analytic singularities for \(k_1 = 0\). Therefore, in order not to introduce new notation, we shall integrate equations (5.2), (5.3), (5.9) by means of power series.

Let the Cauchy problem be posed for these equations; i.e., for \(\tau = 0\) let

\[ t = t_0,\quad x = \xi_0,\quad y = \eta_0,\quad z = \zeta_0,\quad x' = \xi'_0,\quad y' = \eta'_0,\quad z' = \zeta'_0 . \tag{6.2} \]

We define

\[ \rho_0 = \sqrt{\xi_0^2 + \eta_0^2 + \zeta_0^2},\qquad \rho'_0 = \frac{1}{\rho_0}\left(\xi_0 \xi'_0 + \eta_0 \eta'_0 + \zeta_0 \zeta'_0\right). \tag{6.3} \]

Remark 6.2. If, instead of the initial values of the components of the regularized velocity \(\xi'_0, \eta'_0, \zeta'_0\), the initial values of the components of the ordinary velocity \(\dot \xi_0, \dot \eta_0, \dot \zeta_0\) are given, then

\[ \xi'_0 = \dot \xi_0 \frac{\rho_0}{\mu},\qquad \eta'_0 = \dot \eta_0 \frac{\rho_0}{\mu},\qquad \zeta'_0 = \dot \zeta_0 \frac{\rho_0}{\mu}, \tag{6.4} \]

where \(\rho_0\) is determined by the first equation (6.3).

Next, in accordance with Remark 6.1, we find

\[ h = \frac{1}{2}\frac{\mu^2}{\rho_0^2} \left(\xi_0'^2 + \eta_0'^2 + \zeta_0'^2\right) - \frac{\mu}{\rho_0},\qquad \rho_0^{*} = \mu \rho'_0, \]

\[ \rho_0^{*\,\prime} = \frac{2h}{\mu}\rho_0 + 1, \tag{6.5} \]

\[ f_1 = \frac{\mu}{\rho_0}\left(\xi_0 \rho_0^{*\,\prime} - \rho_0^{*}\xi'_0\right),\qquad f_2 = \frac{\mu}{\rho_0}\left(\eta_0 \rho_0^{*\,\prime} - \rho_0^{*}\eta'_0\right), \]

\[ f_3 = \frac{\mu}{\rho_0}\left(\zeta_0 \rho_0^{*\,\prime} - \rho_0^{*}\zeta'_0\right). \]

The constants of Laplace integrals \(f_1, f_2, f_3\) can also be found from the formulas

\[ f_1 = \frac{\mu^2 \xi_0}{\rho_0^2}\left(\eta_0'^2 + \zeta_0'^2\right) - \frac{\mu^2}{\rho_0^2}\xi'_0\left(\eta_0\eta'_0 + \zeta_0\zeta'_0\right) - \frac{\mu \xi_0}{\rho_0}, \]

\[ f_2 = \frac{\mu^2 \eta_0}{\rho_0^2}\left(\zeta_0'^2 + \xi_0'^2\right) - \frac{\mu^2}{\rho_0^2}\eta'_0\left(\zeta_0\zeta'_0 + \xi_0\xi'_0\right) - \frac{\mu \eta_0}{\rho_0}, \tag{6.6} \]

\[ f_3 = \frac{\mu^2 \zeta_0}{\rho_0^2}\left(\xi_0'^2 + \eta_0'^2\right) - \frac{\mu^2}{\rho_0^2}\zeta'_0\left(\xi_0\xi'_0 + \eta_0\eta'_0\right) - \frac{\mu \zeta_0}{\rho_0}. \]

After this we compute the initial values of the components of the regularized acceleration \(\xi''_0, \eta''_0, \zeta''_0\).

\[ \xi_0''=\frac{2h}{\mu^2}\,\xi_0-\frac{f_1}{\mu^2},\qquad \eta_0''=\frac{2h}{\mu^2}\,\eta_0-\frac{f_2}{\mu^2},\qquad \zeta_0''=\frac{2h}{\mu^2}\,\zeta_0-\frac{f_3}{\mu^2}. \tag{6.7} \]

or

\[ \xi_0''=\frac{\rho_0'}{\rho_0}\,\xi_0'-\frac{1}{\mu}\,\frac{\xi_0}{\rho_0},\qquad \eta_0''=\frac{\rho_0'}{\rho_0}\,\eta_0'-\frac{1}{\mu}\,\frac{\eta_0}{\rho_0}, \]

\[ \zeta_0''=\frac{\rho_0'}{\rho_0}\,\zeta_0'-\frac{1}{\mu}\,\frac{\zeta_0}{\rho_0}, \tag{6.8} \]

the velocities of the accelerations \(\xi_0'''\), \(\eta_0'''\), \(\zeta_0'''\):

\[ \xi_0'''=\frac{2h}{\mu^2}\,\xi_0',\qquad \eta_0'''=\frac{2h}{\mu^2}\,\eta_0',\qquad \zeta_0'''=\frac{2h}{\mu^2}\,\zeta_0', \tag{6.9} \]

and the second derivative with respect to \(\tau\) of the radius vector \(\rho_0''\)

\[ \rho_0''=\frac{2h}{\mu^2}\,\rho_0+\frac{1}{\mu}. \tag{6.10} \]

Then it is not difficult to see that the solution of equations (5.2), (5.3), (5.9) in the form of series in powers of \(\tau\) is written as

\[ x=\xi_0+\xi_0'' A(\tau,h)+\xi_0' B(\tau,h),\qquad y=\eta_0+\eta_0'' A(\tau,h)+\eta_0' B(\tau,h), \]

\[ z=\zeta_0+\zeta_0'' A(\tau,h)+\zeta_0' B(\tau,h),\qquad x'=\xi_0'+\xi_0''' A(\tau,h)+\xi_0'' B(\tau,h), \]

\[ y'=\eta_0'+\eta_0''' A(\tau,h)+\eta_0'' B(\tau,h),\qquad z'=\zeta_0'+\zeta_0''' A(\tau,h)+\zeta_0'' B(\tau,h), \tag{6.11} \]

\[ r=\rho_0+\rho_0'' A(\tau,h)+\rho_0' B(\tau,h),\qquad t=t_0+\frac{\rho_0}{\mu}\,\tau+\frac{\rho_0''}{\mu}\,C(\tau,h)+\frac{\rho_0'}{\mu}\,A(\tau,h). \]

Here

\[ A(\tau,h)=\sum_{k=1}^{\infty}\frac{1}{(2k)!}\left(\frac{2h}{\mu^2}\right)^{k-1}\tau^{2k},\qquad B(\tau,h)=\sum_{k=0}^{\infty}\frac{1}{(2k+1)!}\left(\frac{2h}{\mu^2}\right)^k \tau^{2k+1}, \tag{6.12} \]

\[ C(\tau,h)=\sum_{k=1}^{\infty}\frac{1}{(2k+1)!}\left(\frac{2h}{\mu^2}\right)^{k-1}\tau^{2k+1} \]

are entire functions of \(\tau,h\). The theorem is proved.

The functions \(A, B, C\) have different analytic representations depending on what sign \(h\) has, or whether it is equal to zero.

6a) Hyperbolic motion \((h>0)\).
If we introduce a new variable

\[ H=\frac{\sqrt{2h}}{\mu}\,\tau, \]

then

\[ A=\frac{\mu^2}{2h}(-1+\operatorname{ch}H),\qquad B=\frac{\mu}{\sqrt{2h}}\operatorname{sh}H,\qquad C=\frac{\mu^3}{2h\sqrt{2h}}(\operatorname{sh}H-H). \tag{6.13} \]

6b) Parabolic motion \((h=0)\)

\[ A=\frac{1}{2}\tau^2,\qquad B=\tau,\qquad C=\frac{1}{6}\tau^3. \tag{6.14} \]

6c) Elliptic motion \((h<0)\)

\[ A=-\frac{\mu^2}{-2h}(1-\cos E),\qquad B=-\frac{\mu}{\sqrt{-2h}}\sin E, \]

\[ C=\frac{\mu^3}{-2h\sqrt{-2h}}(E-\sin E), \tag{6.15} \]

where

\[ E=\frac{\sqrt{-2h}}{\mu}\,\tau . \tag{6.16} \]

7. We now analyze the equation for the radius vector

\[ r=\rho_0+\rho_0'' A(\tau,h)+\rho_0' B(\tau,h). \tag{7.1} \]

Theorem 7.1. The radius vector, defined as a function of \(\tau\) by equation (7.1), cannot take negative values.

Proof. In the case of hyperbolic motion \((h>0)\), substitute the values \(A,B\) from (6.13) into equation (7.1):

\[ r=\rho_0-\frac{\mu^2}{2h}\rho_0''+\frac{\mu^2}{2h}\rho_0''\operatorname{ch} H +\frac{\mu}{\sqrt{2h}}\rho_0'\operatorname{sh} H. \]

But from (6.10)

\[ \rho_0-\frac{\mu^2}{2h}\rho_0''=-\frac{\mu}{2h}. \]

Denote this quantity by \(-a\).
Introduce also the auxiliary quantities \(ae\), \(H_0\), defined by the equations

\[ \frac{\mu^2}{2h}\rho_0''=ae\operatorname{ch} H_0,\qquad \frac{\mu}{\sqrt{2h}}\rho_0'=-ae\operatorname{sh} H_0 \]

or

\[ \mu\rho_0''=e\operatorname{ch} H_0,\qquad -\sqrt{2h}\,\rho_0'=e\operatorname{sh} H_0. \tag{7.2} \]

Hence

\[ e=\sqrt{\mu^2(\rho_0'')^2-2h\rho_0'^2} =\sqrt{\,2h\left(\frac{2h}{\mu^2}\rho_0^2+\frac{2}{\mu}\rho_0-\rho_0'^2\right)+1\,}. \tag{7.3} \]

Now choose a rectangular coordinate system so that the orbit lies in the \(x,y\) plane. Then \(z=0\). Denote by \(\varphi\) the polar angle in the \(x,y\) plane, measured from the direction of the \(x\)-axis. Then the square of the regularized velocity is

\[ V'^2=r'^2+r^2\varphi'^2 \]

and

\[ V_0'^2=\rho_0'^2+\rho_0^2\varphi_0'^2. \tag{7.4} \]

This gives

\[ 2h=\frac{\mu^2}{\rho_0^2}\rho_0'^2+\mu^2\varphi_0'^2-\frac{2\mu}{\rho_0}. \tag{7.5} \]

Therefore the expression in parentheses in (7.3) is equal to

\[ \frac{2h}{\mu^2}\rho_0^2+\frac{2\rho_0}{\mu}-\rho_0'^2 =\rho_0^2\varphi_0'^2=\frac{c^2}{\mu^2}, \]

where

\[ c=\sqrt{c_1^2+c_2^2+c_3^2}, \]

and \(c_1,c_2,c_3\) are the constants of the area integrals (5.5).

Thus,

\[ e=\sqrt{1+2h\,\frac{c^2}{\mu^2}}, \tag{7.6} \]

\[ r=a\,[e\operatorname{ch}(H-H_0)-1]. \tag{7.7} \]

Since from (7.6) \(e\geqslant 1\), it follows that \(r\geqslant 0\).

For hyperbolic motion the theorem is true.

In parabolic motion \(h=0\), and therefore, substituting the values \(A, B\) (6.14) into (7.1), we obtain

\[ r=\rho_0+\frac{1}{2}\rho_0''\tau^2+\rho_0'\tau . \tag{7.8} \]

Equality (6.10) gives

\[ \rho_0''=\frac{1}{\mu}. \]

Therefore

\[ r=\rho_0+\frac{1}{2\mu}\tau^2+\rho_0'\tau =\rho_0-\frac{\mu\rho_0'^2}{2}+\frac{1}{2\mu}(\tau+\mu\rho_0')^2 . \]

But from the energy integral in parabolic motion

\[ \rho_0'^2=2\frac{\rho_0}{\mu}-\rho_0\varphi'^2 =2\frac{\rho_0}{\mu}-\rho_0^2\frac{c^2}{\mu^2}. \]

Therefore

\[ r=\frac{1}{2\mu}\rho_0c^2+\frac{1}{2\mu}(\tau-\tau_0)^2, \tag{7.9} \]

where

\[ \tau_0=-\mu\rho_0'. \tag{7.10} \]

From equation (7.9) it is seen that \(r\geq 0\).

In elliptic motion

\[ r=\rho_0+\frac{\mu^2}{-2h}\rho_0'' -\frac{\mu^2}{-2h}\rho_0''\cos E +\frac{\mu}{\sqrt{-2h}}\rho_0'\sin E . \]

But from (6.10)

\[ \rho_0+\frac{\mu^2}{-2h}\rho_0''=\frac{\mu}{-2h}, \]

which we denote by \(a\). We now define the auxiliary quantities \(ae, E_0\) from the equations

\[ -\frac{\mu^2}{-2h}\rho_0''=-ae\cos E_0, \qquad -\frac{\mu}{\sqrt{-2h}}\rho_0'=-ae\sin E_0 \]

or

\[ \mu\rho_0''=e\cos E_0, \qquad \sqrt{-2h}\,\rho_0'=e\sin E_0 . \tag{7.11} \]

For \(e\) we have, just as in hyperbolic motion,

\[ e=\sqrt{\mu^2(\rho_0'')^2-2h\rho_0'^2} =\sqrt{1+2h\frac{c^2}{\mu^2}} \quad(\text{see (7.6)}) \tag{7.6*} \]

But here \(h<0\), and therefore it must be that \(0\leq e\leq 1\). This gives

\[ r=a[1-e\cos(E-E_0)]\geq 0. \tag{7.12} \]

Theorem 7.1 is proved.

Corollary 7.1. The radius vector \(r\) can vanish only when \(c=0\) and when \(H=H_0\) for hyperbolic motion, \(\tau=\tau_0\) for parabolic motion, and \(E=E_0+2k\pi\) \((k=0,\pm1,\ldots)\) for elliptic motion.

Corollary 7.1 is proved with the aid of equations (7.7), (7.9), (7.12), using the relations (7.6), (7.6*).

Theorem 7.2. The regularizing variable is proportional to the eccentric anomaly in hyperbolic and elliptic motions and to \(\tg v/2\) (\(v\) is the true anomaly) in parabolic motion, when \(c\ne0\).

Proof. From the preceding arguments it follows that as the instant \(\tau=0\) one may choose that instant at which either the direction of the velocity is perpendicular to the radius vector (case 7a), or these directions are parallel (case 7b).

Case 7a. The system of rectangular coordinates can be chosen so that, for \(\tau=0\),

\[ t=t_0,\quad x=\xi_0=\rho_0,\quad y=z=0,\quad x'=0,\quad y'=\eta'_0=V'_0,\quad z'=0. \]

Then formulas (6.8), (6.9) give us

\[ \xi''_0=-1/\mu,\quad \eta''_0=\zeta''_0=0,\quad \xi'''_0=0,\quad \eta'''_0=\frac{2h}{\mu^2}V'_0,\quad \zeta'''_0=0. \tag{7.13} \]

Instead of formulas (6.11) we must have

\[ x=\rho_0-\frac{1}{\mu}A(\tau,h),\quad y=V'_0 B(\tau,h),\quad z=0,\quad x'=-\frac{1}{\mu}B(\tau,h), \tag{7.14} \]

\[ y'=V'_0+\frac{2h}{\mu^2}V'_0 A(\tau,h),\quad z'=0,\quad r=\rho_0+\left(\frac{2h}{\mu^2}\rho_0+\frac{1}{\mu}\right)A(\tau,h), \]

\[ t=t_0+\frac{\rho_0}{\mu}\tau+ \left(\frac{2h}{\mu^3}\rho_0+\frac{1}{\mu^2}\right)C(\tau,h). \]

The integral of areas has the constant

\[ c=\rho_0 V_0=\mu V'_0. \tag{7.15} \]

Therefore

\[ e=\sqrt{1+2hV_0^2} \tag{7.16} \]

or

\[ e=\sqrt{1+2h\left(\frac{2\rho_0}{\mu}+\frac{\rho_0^2}{\mu^2}2h\right)} =1+\frac{\rho_0\,2h}{\mu}. \tag{7.17} \]

For what follows it is convenient to introduce the new dimensionless quantity

\[ \alpha=\frac{\rho_0\,2h}{\mu}. \tag{7.18} \]

This quantity is equal to zero in parabolic motion and is close to zero in nearly parabolic motion. We have

\[ e=1+\alpha,\quad \alpha=e-1. \tag{7.19} \]

Moreover,

\[ V'_0=\sqrt{\frac{2\rho_0}{\mu}+\frac{\rho_0^2}{\mu^2}2h} =\sqrt{\frac{\rho_0}{\mu}}\sqrt{2+\alpha} =\sqrt{\frac{\rho_0}{\mu}}\sqrt{1+e}. \tag{7.20} \]

In hyperbolic motion, in (7.14) one must substitute (6.13), (7.18), (7.20); then

\[ x=\rho_0\left[(1+\alpha^{-1})-\alpha^{-1}\operatorname{ch}H\right],\quad y=\rho_0\sqrt{2+\alpha}\,\alpha^{-1/2}\operatorname{sh}H,\quad z=0, \tag{7.21} \]

\[ x'=\sqrt{\frac{\rho_0}{\mu}}\,\alpha^{-1/2}(-\operatorname{sh}H),\quad y'=\sqrt{\frac{\rho_0}{\mu}}\sqrt{2+\alpha}\operatorname{ch}H,\quad z'=0, \]

\[ r=\rho_0\left[-\alpha^{-1}+(1+\alpha^{-1})\operatorname{ch}H\right],\quad t=t_0+\frac{\rho_0\sqrt{\rho_0}}{\sqrt{\mu}}\,\alpha^{-1/2} \left[-\alpha^{-1}H+(1+\alpha^{-1})\operatorname{sh}H\right]. \]

If in hyperbolic motion we introduce the quantities \(\alpha=\dfrac{\mu}{2h}\), \(e=1+\alpha\), then

\[ x=\alpha(e-\operatorname{ch}H),\quad y=\alpha\sqrt{e^2-1}\operatorname{sh}H,\quad z=0, \]

\[ x'=\sqrt{\frac{\alpha}{\mu}}(-\operatorname{sh}H), \tag{7.22} \]

\[ y'=\sqrt{\frac{\alpha}{\mu}}\sqrt{e^2-1}\operatorname{ch}H,\quad z'=0,\quad r=\alpha(e\operatorname{ch}H-1), \]

\[ t=t_0+\frac{a\sqrt{\bar a}}{\sqrt{\mu}}(e\,\operatorname{sh} H-H). \tag{7.22} \]

Since, on the one hand, \(H\) is proportional to \(\tau\), and on the other hand, as is seen from (7.22), it is the eccentric anomaly in hyperbolic motion, the theorem is true in this case.

In parabolic motion in case 7a, instead of \(\tau\) we introduce the dimensionless quantity

\[ \beta=\frac{\tau}{\sqrt{\mu}\sqrt{\rho_0}}. \tag{7.23} \]

Taking into account (7.14), (6.14), (7.20), (7.23), we obtain

\[ x=\rho_0\left(1-\frac{1}{2}\beta^2\right),\quad y=\rho_0\sqrt{2}\,\beta,\quad z=0,\quad x'=\sqrt{\frac{\rho_0}{\mu}}\,(-\beta), \]

\[ y'=\sqrt{\frac{\rho_0}{\mu}}\,\sqrt{2},\quad z'=0,\quad r=\rho_0\left(1+\frac{1}{2}\beta^2\right), \]

\[ t=t_0+\frac{\rho_0\sqrt{\rho_0}}{\sqrt{\mu}}\left(\beta+\frac{1}{6}\beta^3\right). \tag{7.24} \]

Let us note that in parabolic motion the components of the regularized velocity are polynomials in \(\tau\) or \(\beta\) of degree no higher than the first, the coordinates and radius vector of degree no higher than the second, and the time of degree no higher than the third.

The true anomaly \(\vartheta\) in case 7a is determined from the equations

\[ x=r\cos\vartheta,\quad y=r\sin\vartheta. \]

Hence

\[ \operatorname{tg}\frac{\vartheta}{2} =\frac{1-\cos\vartheta}{\sin\vartheta} =\frac{r-x}{y} =\frac{1}{\sqrt{2}}\beta =\frac{1}{\sqrt{2}}\frac{\tau}{\sqrt{\mu}\sqrt{\rho_0}}. \tag{7.25} \]

This proves Theorem 7.2 in case 7a in parabolic motion.

If in elliptic motion one introduces the quantities

\[ a=-\frac{\mu}{2h},\quad e=1+\frac{\rho_0\,2h}{\mu},\quad E=\frac{\sqrt{-2h}}{\mu}\,\tau, \tag{7.26} \]

then from (7.14), (6.15) we must have

\[ x=a(\cos E-e),\quad y=a\sqrt{1-e^2}\sin E,\quad z=0,\quad x'=\sqrt{\frac{a}{\mu}}\,(-\sin E), \]

\[ y'=\sqrt{\frac{a}{\mu}}\sqrt{1-e^2}\cos E,\quad z'=0,\quad r=a(1-e\cos E), \tag{7.27} \]

\[ t=t_0+\frac{a\sqrt{a}}{\sqrt{\mu}}(E-\sin E), \]

which proves the validity of Theorem 7.2 in case 7a in elliptic motion.

Case 7b. Choose the system of rectangular coordinates so that for \(\tau=0\)

\[ t=t_0,\quad x=\xi_0=-\rho_0,\quad y=z=0,\quad x'=\xi'_0=V'_0,\quad y'=z'=0. \]

If \(x'=\xi'_0=-V'_0\), then we replace \(\tau\) by \(-\tau\).

In this case, from equations (6.8), (6.9),

\[ \xi''_0=\frac{1}{\rho_0}V_0^{\prime\,2}-\frac{1}{\mu},\quad \xi'''_0=\frac{2h}{\mu^2}V'_0, \]

\[ \eta''_0=0,\quad \eta'''_0=0,\quad \zeta''_0=0,\quad \zeta'''_0=0. \]

Equations (6.11) give

\[ x=\rho_0+\left(\frac{1}{\rho_0}V_0^{\prime 2}-\frac{1}{\mu}\right)A(\tau,h)+V_0'B(\tau,h),\quad y=0,\quad z=0, \]

\[ x'=V_0'+\frac{2h}{\mu^2}V_0'A(\tau,h)+\left(\frac{1}{\rho_0}V_0^{\prime 2}-\frac{1}{\mu}\right)B(\tau,h),\quad y'=0,\quad z'=0, \]

\[ r=\rho_0+\left(\frac{2h}{\mu^2}\rho_0+\frac{1}{\mu}\right)A(\tau,h)+V_0'B(\tau,h), \tag{7.28} \]

\[ t=t_0+\frac{\rho_0}{\mu}\tau+\left(\frac{2h}{\mu^3}\rho_0+\frac{1}{\mu^2}\right)C(\tau,h)+\frac{1}{\mu}V_0'A(\tau,h). \]

Let us now introduce the dimensionless quantity

\[ \gamma=\frac{\rho_0\,2h}{\mu}. \tag{7.29} \]

Then

\[ V_0'=\sqrt{\frac{2\rho_0}{\mu}+\frac{\rho_0^2}{\mu^2}\,2h} =\sqrt{\frac{\rho_0}{\mu}}\sqrt{2+\gamma}. \tag{7.30} \]

In the hyperbolic motion \((h>0)\) we have

\[ x=r=\rho_0\left[-\gamma^{-1}+(1+\gamma^{-1})\operatorname{ch}H+\sqrt{2+\gamma}\,\gamma^{-1/2}\operatorname{sh}H\right],\quad y=0,\quad z=0, \]

\[ x'=\frac{\sqrt{\rho_0}}{\sqrt{\mu}} \left[\sqrt{2+\gamma}\operatorname{ch}H+\gamma^{-1/2}(1+\gamma)\operatorname{sh}H\right], \quad y'=0,\quad z'=0, \tag{7.31} \]

\[ t=t_0+\frac{\rho_0\sqrt{\rho_0}}{\sqrt{\mu}} \left\{\gamma^{-1/2}\left[-\gamma^{-1}H+(1+\gamma^{-1})\operatorname{sh}H\right] +\sqrt{2+\gamma}\,\gamma^{-1}(-1+\operatorname{ch}H)\right\}, \]

where \(H=\sqrt{2h}\,\tau/\mu\).

Choose the quantities \(a,\ H_0\) so that

\[ a\operatorname{ch}H_0=\rho_0(1+\gamma^{-1}),\quad a\operatorname{sh}H_0=-\rho_0\sqrt{2+\gamma}\,\gamma^{-1/2}. \tag{7.32} \]

Hence

\[ a=\rho_0\gamma^{-1}. \]

Thus,

\[ x=a[\operatorname{ch}(H-H_0)-1],\quad y=0,\quad z=0, \]

\[ x'=\sqrt{\frac{a}{\mu}}\operatorname{sh}(H-H_0),\quad y'=0,\quad z'=0, \tag{7.33} \]

\[ t=t_0^*+\frac{a\sqrt{a}}{\sqrt{\mu}}\,[\operatorname{sh}(H-H_0)-(H-H_0)], \]

where \(t_0^*\) is the value of \(t\) in the last equation (7.31) for \(H=H_0\) (the moment of collision).

These formulas are obtained from equations (7.22) for \(e=1\), if it is taken into account that for \(e=1\) in equations (7.22) one obtains \(x=-r\).

In the parabolic case of rectilinear motion \((h=0,\ c=0)\) introduce the independent variable

\[ \delta=\frac{\tau}{\sqrt{\mu}\sqrt{\rho_0}}. \tag{7.34} \]

This gives

\[ x=r=\rho_0\left(1+\frac{1}{2}\delta^2+\sqrt{2}\,\delta\right),\quad y=0,\quad z=0, \]

\[ x'=\sqrt{\frac{\rho_0}{\mu}}\left(\sqrt{2}+\delta\right),\quad y'=0,\quad z'=0, \tag{7.35} \]

\[ t=t_0+\frac{\rho_0\sqrt{\rho_0}}{\sqrt{\mu}} \left(\delta+\frac{1}{6}\delta^3+\sqrt{2}\,\frac{1}{2}\delta^2\right). \]

Let \(\dot\delta_0=-\sqrt{2}\). Then from equations (7.35)

\[ \begin{gathered} x=r=-\frac{1}{2}\rho_0(\dot\delta-\dot\delta_0)^2,\qquad y=0,\qquad z=0,\\ x'=\sqrt{\frac{\rho_0}{\mu}}(\dot\delta-\dot\delta_0),\qquad y'=0,\qquad z'=0, \end{gathered} \tag{7.36} \]

\[ t=t_0^{*}+\frac{\rho_0\sqrt{\rho_0}}{\sqrt{\mu}}\frac{1}{6}(\dot\delta-\dot\delta_0)^3 . \]

Here \(t_0^{*}\) is the value of \(t\) in the last equation (7.35) for \(\dot\delta=\dot\delta_0\) (the moment of collision).

Equations (7.24) cannot be used if \(\rho_0\to0\). Denote in them the quantity \(\rho_0\) by \(q\) and introduce a new dimensionless variable \(\dot\delta^{*}\)

\[ \rho_0(\dot\delta^{*})^2=q\beta^2,\qquad \beta=\sqrt{-\frac{\rho_0}{q}}\,\dot\delta^{*}, \tag{7.37} \]

where \(\rho_0\) is some fixed value of the distance. Then, taking into account the notation introduced,

\[ \begin{gathered} x=q-\frac{1}{2}\rho_0(\dot\delta^{*})^2,\qquad y=\sqrt{q\rho_0}\sqrt{2}\,\dot\delta^{*},\qquad r=q+\frac{1}{2}\rho_0(\dot\delta^{*})^2,\qquad z=0,\\ x'=\sqrt{\frac{\rho_0}{\mu}}(-\dot\delta^{*}),\qquad t=t_0+\frac{q\sqrt{-\rho_0}}{\sqrt{\mu}}\dot\delta^{*} +\frac{1}{6}\rho_0\frac{\sqrt{\rho_0}}{\sqrt{\mu}}(\dot\delta^{*})^3,\qquad y'=\sqrt{\frac{q}{\mu}}\sqrt{2}, \tag{7.38} \end{gathered} \]

\[ z'=0. \]

Equations (7.36) are obtained from this for \(q\to0\).

Thus, the variable \(\dot\delta\) in equations (7.35) will no longer have the same simple geometrical meaning as in case 7a the variable \(\beta\) had for parabolic motion.

In the elliptic case, for \(c=0\), introduce two auxiliary quantities \(a,d\):

\[ d=1+\gamma,\qquad a=\frac{\rho_0}{1-d}\qquad (0<d<1). \tag{7.39} \]

This gives

\[ \begin{gathered} x=a(1-d\cos E+\sqrt{1-d^2}\sin E),\qquad y=0,\quad z=0,\\ x'=\sqrt{\frac{a}{\mu}}\left(\sqrt{1-d^2}\cos E+d\sin E\right),\qquad y'=0,\qquad z'=0, \tag{7.40} \end{gathered} \]

\[ t=t_0+\frac{a\sqrt{a}}{\sqrt{\mu}}\,[d\sin E+\sqrt{1-d^2}(-1+\cos E)-E]. \]

If \(E_0\) is chosen so that

\[ \sin E_0=-\sqrt{1-d^2},\qquad \cos E_0=d, \]

then equations (7.37) are easily reduced to the form (7.27), provided only that there one sets \(e=1\) and takes into account that for \(e=1\) in equations (7.27) \(x=-r\). Indeed, in this case from equations (7.40)

\[ x=r=a[1-\cos(E-E_0)],\qquad y=0,\qquad z=0, \]

\[ x'=\sqrt{\frac{a}{\mu}}\sin(E-E_0),\qquad y'=0,\qquad z'=0, \]

\[ t=t_0^{*}+\frac{a\sqrt{a}}{\sqrt{\mu}}[(E-E_0)-\sin(E-E_0)]. \]

Here \(t_0^{*}\) is the value of \(t\) in the last equation (7.37) for \(E=E_0\) (the moment of collision).

Remark 7.1. If, for \(\tau=0\), \(x'=\xi'_0=-V'_0\), then \(\xi'''_0=-2hV'_0/\mu^2\); in equations (7.28), with \(V'_0\) in the first degree, the plus sign must everywhere be replaced by a minus sign, which leads to the fact that in hyperbolic motion in equations (7.31), (7.32) the plus sign must be replaced by a minus sign at the quantity \(\sqrt{2+\gamma}\); in parabolic motion in equations (7.35), instead of \(\sqrt{2}\) one must everywhere put \(-\sqrt{2}\); in equations (7.36) the quantity \(\delta_0=-\sqrt{2}\); in elliptic motion in equations (7.40) the plus sign must everywhere be replaced by a minus sign at the quantity \(\sqrt{1-d^2}\). The angle \(E_0\) must then be determined from the equations

\[ \sin E_0=\sqrt{1-d^2}, \qquad \cos E_0=d. \]

Theorem 7.2 is completely proved.

It follows from this theorem that the expressions of the rectangular coordinates, of the components of the regularized velocity, of the radius vector, and of the time through dimensionless regularizing variables are sufficiently simple, although different for each type of motion.

While preparing this article for publication, I learned from Prof. G. N. Duboshin that, in an analogous problem, but by another method of solution, Kustaanheimo [5] had proposed equations of type (4.7). He obtains equations of type (4.7), but in other coordinates, which are expressed through rectangular ones in a rather complicated way, which is disadvantageous in practice.

The present work was reported at meetings of the Department of Celestial Mechanics and Gravimetry of Moscow State University in June 1963 and April 1964, and also at the conference on the motion of artificial celestial bodies (Riga, May 1964). I express my heartfelt gratitude to G. N. Duboshin, to all members of the department, and to the participants of the conference for the valuable comments made.

References

1. Siegel, K. L. Lectures on Celestial Mechanics. IL, 1959.
2. Yarov-Yarovoi, M. S. PMM, 27, No. 6, 1963, pp. 973–987.
3. Smirnov, V. I. Course of Higher Mathematics, 4. GITTL, 1951, p. 345.
4. Duboshin, G. N. Celestial Mechanics. Basic Problems and Methods. Fizmatgiz, 1963, p. 240.
5. Kustaanheimo, P. Ann. Univ. Turkuens. A. I., 73, 1964, 1 Publ., No. 102 of the Astron. Observatory, Helsinki, 1964.

Received by the editors
January 18, 1965.

State Astronomical Institute
named after P. K. Shternberg