Clear[energy, radiusInNonDimensionalizedCoordinates, theta, phasePoint]

(* passing points in phase space (p, x) = {., .} *)

energy[m_, ω_, p0_] :=(p0[[1]])^2/(2m) + m ω^2 (p0[[2]])^2/2 ;
radiusInNonDimensionalizedCoordinates[m_, ω_, p0_] := Sqrt[2 energy[m, ω, p0]/m] ;

theta[m_, ω_, p0_] :=ArcTan[ ω m p0[[2]]/p0[[1]]] ; (* x/p *)

phasePoint[t_, m_, ω_, p0_] := 
radiusInNonDimensionalizedCoordinates[m, ω, p0]{
m    Cos[ ω t + theta[m, ω, p0] ] ,
(1/ω)  Sin[ ω t + theta[m, ω, p0] ] 
} ;


FullSimplify[  TrigExpand[  phasePoint[t, m, ω, {(Subscript[p, 0]), (Subscript[x, 0])}] ],(Subscript[p, 0])> 0 && (Subscript[x, 0]) > 0 && m > 0 &&  ω > 0] // TraditionalForm

 
Manipulate[
DynamicModule[{ differenceOfPoints, averagePositionBetweenPoints, radiusOfCircleAtAveragePos, c},

differenceOfPoints = plOuter - plInner ; 
averagePositionBetweenPoints = (plOuter  + plInner)/2 ;
radiusOfCircleAtAveragePos = 0.8 Norm[differenceOfPoints]/2 ;
c[v_] = averagePositionBetweenPoints + radiusOfCircleAtAveragePos {Cos[v], Sin[v]} ;

Show[{
ParametricPlot[
{
phasePoint[u 2 Pi/ω, m, ω, plInner],
phasePoint[u 2 Pi/ω, m, ω, plOuter],
c[2 Pi u],
phasePoint[t1, m, ω, c[2 Pi u]],
phasePoint[t2, m, ω, c[2 Pi u]],
phasePoint[t3, m, ω, c[2 Pi u]]
},
{u,0,1},
 PlotRange -> 4,
AxesLabel->{p, x}
 ] (* plot *)
,
Graphics[{
Text["((Subscript[p, 0,])(Subscript[x, 0]))", {plInner[[1]] + 0.4,plInner[[2]] + 0.4}],
Text["((Subscript[p, 1,])(Subscript[x, 1]))", {plOuter[[1]] + 0.4,plOuter[[2]] + 0.4}]
}]
} ] (* show *)

]
,
{{t1, 5.5}, 0, 20},
{{t2, 8.65}, 0, 20},
{{t3, 6.65}, 0, 20},
{{m,1}, 0.1, 10},
{{ω , Pi/2}, Pi/10, 10 Pi},
{{plInner,{1,1}},Locator},
{{plOuter,{2,2}},Locator}

] (* manip *)

DynamicModule[{m=1,pl={0.96`,2.5200000000000005`},ω=1.4338228870983816`},

Module[
{th$},th$=theta[m,ω,pl];
Show[
{
ParametricPlot[
phasePoint[t,m,ω,pl],
{t,0,20},
PlotRange->4,
AxesLabel->{p, x}
],
Graphics[{
PointSize[Large](*, Pink*),Point[pl],
Text["(\!\(\*SubscriptBox[\(p\), \(\(0\)\(,\)\(\\ \)\)]\)(Subscript[x, 0]))", {pl[[1]] + 0.3,pl[[2]] + 0.3}]
}]
}
]
]
]