Questão 1

Quit

V=V0 (((x^2)/(a^2))-((x^4)/(4 (a^4))))

Plot[{V/.{V0->1,a->1},1,(3/4)},{x,-2.5,2.5}]

D[V,x]
Solve[%==0,x]
V/.%

(V+((m (v^2))/2))/.{V0->1,a->1,m->1}
ContourPlot[%,{x,-2.3,2.3},{v,-2,2},Contours->Table[i,{i,-3,3,0.1}],ImageSize->600,PlotPoints->30]

Solve[V+((m (v^2))/2)==e,v]
%//FullSimplify[#,{m>0,V0>0,a>0,e>0}]&

Solve[V+((m (v^2))/2)==V0,v]
%//FullSimplify[#,{m>0,V0>0,a>0,e>0}]&

Integrate[(1/(((Sqrt[(V0/m)]) (2 (a^2)-(x^2)))/((Sqrt[2]) (a^2)))),x]//FullSimplify[#,{m>0,V0>0,a>0,e>0}]&
Solve[%==t,x]//FullSimplify[#,{m>0,V0>0,a>0,e>0,t∈Reals}]&

Plot[(Sqrt[2]) a Tanh[((t (Sqrt[(V0/m)]))/a)]/.{V0->1,a->1,m->1},{t,-10,10}]

Solve[V+((m (v^2))/2)==(3/4) V0,v]
%//FullSimplify[#,{m>0,V0>0,a>0,e>0}]&

Integrate[(1/((Sqrt[((V0 (3 (a^4)-4 (a^2) (x^2)+(x^4)))/m)])/((Sqrt[2]) (a^2)))),x]//FullSimplify[#,{m>0,V0>0,a>0,e>0}]&

Solve[V+((m (0^2))/2)==(3/4) V0,x]

(1/(Sqrt[((m (a^2))/V0)])) ((4 a)/((Sqrt[((V0 (3 (a^4)-4 (a^2) (x^2)+(x^4)))/m)])/((Sqrt[2]) (a^2))))/.{x->a ξ} //FullSimplify[#,{m>0,V0>0,a>0,e>0}]&
Integrate[%,{ξ,0,1}]
NIntegrate[%,{ξ,0,1}]

Questão 2

Quit

(r/z)==Tan[α]
z==r Cot[α]

T=(m/2) (((r'[t])^2)+((r[t])^2) ((θ'[t])^2)+ ((Cot[α])^2) ((r'[t])^2))
V=m g r[t] Cot[α]
L=T-V //FullSimplify

D[L,θ[t]]
pθ==D[L,θ'[t]]
Solve[%,θ'[t]]

D[L,r'[t]]
D[%,t]
D[L,r[t]]
eq=%-%%/.{θ'[t]->((pθ)/(m ((r[t])^2)))} //FullSimplify

eq/.r->Function[t,r0]
Solve[%==0,pθ]//FullSimplify
Solve[%%==0,r0]//FullSimplify

ω=((pθ)/(m (r0^2)))/.{pθ->(Sqrt[g]) m (r0^(3/2)) (Sqrt[Cot[α]])}//FullSimplify
((pθ)/(m (r0^2)))/.{r0->((((pθ)^(2/3)) ((Tan[α])^(1/3)))/((g^(1/3)) (m^(2/3))))}//FullSimplify

eq/.r->Function[t,r0+ϵ δr[t]]
Series[%,{ϵ,0,1}]//FullSimplify
%/.{pθ->(Sqrt[g]) m (r0^(3/2)) (Sqrt[Cot[α]])}//FullSimplify
%%/.{r0->((((pθ)^(2/3)) ((Tan[α])^(1/3)))/((g^(1/3)) (m^(2/3))))}//FullSimplify

Ω=(Sqrt[((3 g Cot[α])/(r0 ((Csc[α])^2)))])//FullSimplify[#,{0<α<(π/2),r0>0,g>0,m>0}]&

(Sqrt[(((3 (g^(4/3)) (m^(5/3)))/(((pθ)^(2/3)) ((Tan[α])^(4/3))))/(m ((Csc[α])^2)))])//FullSimplify[#,{0<α<(π/2),r0>0,g>0,m>0}]&

(Ω/ω)//FullSimplify[#,{0<α<(π/2),r0>0,g>0,m>0}]&

(Ω/ω)/.{α->(π/3)}//FullSimplify[#,{0<α<(π/2),r0>0,g>0,m>0}]&

eq1=eq/.{α->(π/3),m->1,g->1,pθ->1}//FullSimplify[#,{0<α<(π/2),r0>0,g>0,m>0}]&
r01=((((pθ)^(2/3)) ((Tan[α])^(1/3)))/((g^(1/3)) (m^(2/3))))/.{α->(π/3),m->1,g->1,pθ->1}
N[r01]

NDSolve[{eq1==0,r[0]==r01,r'[0]==0.1},r,{t,0,30}]
rsol=%[[1,1,2]];
Plot[{rsol[t],r01},{t,0,30},PlotRange->{0,2 r01}]

((2π)/ω)/.{α->(π/3),m->1,g->1,pθ->1}/.r0->r01 //N
((2π)/Ω)/.{α->(π/3),m->1,g->1,pθ->1}/.r0->r01 //N

Plot[{rsol[t],r01},{t,0,2 9.06192130944158`},PlotRange->{0,2 r01}]

Questão 4

Quit

V[r_]=-(1/r) (* r em unidades de k=GM *)
F[r_]=-V'[r]

linhas1=ParametricPlot[{{0.4,y},{0.5,y},{0.6,y}},{y,-8,0}];
linhas2=Plot[{F[0.4],F[0.5],F[0.6]},{r,0,2}];
Plot[F[r],{r,0,2},PlotStyle->Black];
Show[%,linhas1,linhas2,PlotRange->{-10,0}]

F[r0+δr]+F[r0-δr]
Series[%,{δr,0,2}]

Plot[{F[1+δr]+F[1-δr],2F[1]},{δr,0,0.1}]