Existence and uniqueness of integral curves #
Main results #
exists_isMIntegralCurveAt_of_contMDiffAt_boundaryless
: Existence of local integral curves for a $C^1$ vector field. This follows from the existence theorem for solutions to ODEs (exists_forall_hasDerivAt_Ioo_eq_of_contDiffAt
).isMIntegralCurveOn_Ioo_eqOn_of_contMDiff_boundaryless
: Uniqueness of local integral curves for a $C^1$ vector field. This follows from the uniqueness theorem for solutions to ODEs (ODE_solution_unique_of_mem_set_Ioo
). This requires the manifold to be Hausdorff (T2Space
).
Implementation notes #
For the existence and uniqueness theorems, we assume that the image of the integral curve lies in the interior of the manifold. The case where the integral curve may lie on the boundary of the manifold requires special treatment, and we leave it as a TODO.
We state simpler versions of the theorem for boundaryless manifolds as corollaries.
TODO #
- The case where the integral curve may venture to the boundary of the manifold. See Theorem 9.34, Lee. May require submanifolds.
Reference #
Tags #
integral curve, vector field, local existence, uniqueness
Existence of local integral curves for a $C^1$ vector field at interior points of a C^1
manifold.
Alias of exists_isMIntegralCurveAt_of_contMDiffAt
.
Existence of local integral curves for a $C^1$ vector field at interior points of a C^1
manifold.
Existence of local integral curves for a $C^1$ vector field on a C^1
manifold without
boundary.
Alias of exists_isMIntegralCurveAt_of_contMDiffAt_boundaryless
.
Existence of local integral curves for a $C^1$ vector field on a C^1
manifold without
boundary.
Local integral curves are unique.
If a $C^1$ vector field v
admits two local integral curves γ γ' : ℝ → M
at t₀
with
γ t₀ = γ' t₀
, then γ
and γ'
agree on some open interval containing t₀
.
Alias of isMIntegralCurveAt_eventuallyEq_of_contMDiffAt
.
Local integral curves are unique.
If a $C^1$ vector field v
admits two local integral curves γ γ' : ℝ → M
at t₀
with
γ t₀ = γ' t₀
, then γ
and γ'
agree on some open interval containing t₀
.
Alias of isMIntegralCurveAt_eventuallyEq_of_contMDiffAt_boundaryless
.
Integral curves are unique on open intervals.
If a $C^1$ vector field v
admits two integral curves γ γ' : ℝ → M
on some open interval
Ioo a b
, and γ t₀ = γ' t₀
for some t ∈ Ioo a b
, then γ
and γ'
agree on Ioo a b
.
Alias of isMIntegralCurveOn_Ioo_eqOn_of_contMDiff
.
Integral curves are unique on open intervals.
If a $C^1$ vector field v
admits two integral curves γ γ' : ℝ → M
on some open interval
Ioo a b
, and γ t₀ = γ' t₀
for some t ∈ Ioo a b
, then γ
and γ'
agree on Ioo a b
.
Alias of isMIntegralCurveOn_Ioo_eqOn_of_contMDiff_boundaryless
.
Global integral curves are unique.
If a continuously differentiable vector field v
admits two global integral curves
γ γ' : ℝ → M
, and γ t₀ = γ' t₀
for some t₀
, then γ
and γ'
are equal.
Alias of isMIntegralCurve_eq_of_contMDiff
.
Global integral curves are unique.
If a continuously differentiable vector field v
admits two global integral curves
γ γ' : ℝ → M
, and γ t₀ = γ' t₀
for some t₀
, then γ
and γ'
are equal.
For a global integral curve γ
, if it crosses itself at a b : ℝ
, then it is periodic with
period a - b
.
Alias of IsMIntegralCurve.periodic_of_eq
.
For a global integral curve γ
, if it crosses itself at a b : ℝ
, then it is periodic with
period a - b
.
A global integral curve is injective xor periodic with positive period.
Alias of IsMIntegralCurve.periodic_xor_injective
.
A global integral curve is injective xor periodic with positive period.