ODE of an eccentric circle

How can we simplify the solution in:

DSolve[ {Sqrt[ 1 - (r[t] - T^2/r[t])^2/(2 a)^2]/(r[t] - T^2/r[t])/(2 a) ==
r'[t]/ r[t], r[0] == T}, r, t]


(r,t) are polar coordinates.

EDIT1:

Second order ODE in terms of $\psi, r, \theta$ which respectively are: angle between arc and radius vector and the polar coordinates. ( Constants T > a)

$$\psi^{''}(s)= \dfrac{T^2 \cos \psi}{2 \,a \,r(s)^3}$$

$$\dfrac {d \theta}{ds}=\dfrac{\sin \psi}{ r(s)} = \dfrac{1-T^2/a^2}{2 a}$$

$$\dfrac {d r}{ds}=\cos \psi$$

Boundary Conditions:

$$@ r= T, \psi(0) = \psi^{'}(0) =0 , \psi^{''}(0)= -1/ (2 a T).$$

Had no luck at start, trying..

EDIT 2:

Corrected errors, and now it works:

$$\psi^{''}(s)= \dfrac{-T^2 \cos \psi}{ a \,r^3(s)}$$

Boundary Conditions:

$$@ r= T, \psi(0) = 0, \,\psi^{'}(0) =1/a.$$

T = 4; a = 3; ri = T; smax = 2 Pi a;

CIRC = {SI''[s] == -(T^2/a ) Cos[SI[s]]/R[s]^3, SI'[0] == 1/a,
SI[0] == 0, R'[s] == Cos[SI[s]], R[0] == ri,
TH'[s] == Sin[SI[s]]/R[s], TH[0] == 0};

NDSolve[CIRC, {SI, R, TH}, {s, 0, smax}];

{si[u_], r[u_], th[u_]} = {SI[u], R[u], TH[u]} /. First[%];

CENTRAL =  ParametricPlot[T { Cos[v], Sin[v]}, {v, .0, 2 Pi},
PlotStyle -> {Purple, Thick}]

ECC = ParametricPlot[{r[s] Cos[th[s]], r[s] Sin[th[s]]}, {s, .0,
smax}, PlotLabel -> ECCENTRIC_CIRCLE, PlotStyle -> {Red, Thick},
GridLines -> Automatic, AspectRatio -> Automatic, PlotRange -> All]

Show[ECC, CENTRAL]

• What do you mean by "simplify the solution"? What do you get now; what would you like to get? For what purpose would you like to use the solution? Commented Aug 19, 2015 at 20:44
• The solution is lengthy so did not transfer it here. Even if boundary condition is taken as r[0]== r1, it is still unwieldy. The square root has two signs here, one for the convex and other concave part. Reference is math.stackexchange.com/questions/1398614/…, where we can further show the result applies to two different parts of same curve or circle. Much appreciate any help to express it (r-t) form also. Commented Aug 19, 2015 at 21:14
• Have you tried to use e.g. ParametricNDSolve to check that at least the equation is well-formed and it represents what you expect? Commented Aug 19, 2015 at 21:23
• I tried NDSolve only. Even at r=r1 as BC it wont take off. At r=T ,it is a tougher start. Commented Aug 19, 2015 at 21:36
• It occurs to me just now, I shall raise it to second order and try, it often works. Commented Aug 19, 2015 at 21:47

The code in this question can be written as

eq = Sqrt[1 - (r[t] - T^2/r[t])^2/(2 a)^2]/(r[t] - T^2/r[t])/(2 a) == r'[t]/r[t];
DSolveValue[{eq, r[0] == T}, r[t], t] // FullSimplify
(* InverseFunction[((-Sqrt[-T^4] ArcTan[(2 a^2 + T^2 - #1^2)/
Sqrt[4 a^2 #1^2 - (-T^2 + #1^2)^2]] + T^2 (-2 Log[#1] +
Log[-2 T^4 + 2 (2 a^2 + T^2) #1^2 +
2 Sqrt[-T^4] Sqrt[4 a^2 #1^2 - (-T^2 + #1^2)^2]]))
Sqrt[4 a^2 #1^2 - (-T^2 + #1^2)^2])/(
2 Sqrt[-T^4] #1 Sqrt[(4 a^2 #1^2 - (-T^2 + #1^2)^2)/(a^2 #1^2)]) &][(
t T^3 - 2 a (a T^3 ArcTan[a/T] + Sqrt[a^2 T^2] Sqrt[-T^4] (-2 Log[T] +
Log[4 (a^2 T^2 + Sqrt[a^2 T^2] Sqrt[-T^4])])))/(4 a T^3)] *)


which, as noted by the OP, is quite complicated.

Alternatively, this problem can be solved using Integrate, because the equation is first order and t does not appear explicitly. Define

exp = eq[[1]] r[t] /. {r[t] -> a r0, T -> a T}
(* (r0 Sqrt[1 - (a r0 - (a T^2)/r0)^2/(4 a^2)])/(2 (a r0 - (a T^2)/r0)) *)


which replaces r[t] by r0 and rescales r0 and t by a. Then, note the turning points of this expression.

Solve[exp == 0, r0]
(* {{r0 -> -1 - Sqrt[1 + T^2]}, {r0 -> 1 - Sqrt[1 + T^2]},
{r0 -> -1 + Sqrt[1 + T^2]}, {r0 -> 1 + Sqrt[1 + T^2]}} *)


Since the minimum value of r0 is given in the question as T, which we assume to be positive, the turning point of interest is the last one, 1 + Sqrt[1 + T^2], beyond which the integral will become complex. Now, perform the integral in this region to obtain t as a function of r.

t = Integrate[1/exp, {r0, T, r}, Assumptions -> 0 < T < r < 1 + Sqrt[1 + T^2]]
(* -2 I a (2 Log[r] + Log[2 - 2 I T] + Log[-4 + 4 I T] -
Log[2 - r^2 + T^2 - I Sqrt[-r^4 - T^4 + 2 r^2 (2 + T^2)]] -
Log[2 T^4 - 2 r^2 (2 + T^2) + 2 I T^2 Sqrt[-r^4 - T^4 + 2 r^2 (2 + T^2)]]) *)


Although complicated, it is far less so that the DSolve expression. Also, rescaling t by a eliminates a from the expression, leaving (the rescaled) T as the only free parameter. As an example of the form of this solution, plot it with T -> 1.

ParametricPlot[{(t/a) /. {T -> 1}, r}, {r, 1, 4},
AspectRatio -> 1/GoldenRatio, AxesLabel -> {"t/a", "r/a"}]


The curve stops at the turning point, r/a = 1 + Sqrt[2].