We use the celebrated heat flow method of Eells and Sampson to the question of deformation of a smooth loop
M\in \mathbf{R}^{2}
M
∈
R
2
on a Finsler manifold
\left( N,h\right)
(
N
,
h
)
to a closed geodesic in
N
N
. This leads to the investigation of the corresponding heat equation which is the parabolic initial value problem
\begin{aligned}\frac{\partial u^{i}}{\partial t}-\frac{\partial ^{2}u^{i}}{\partial x^{2}} &=\Gamma _{hk}^{i}\left( u,\frac{\partial u}{\partial x}\right) \frac{\partial u^{h}}{\partial x}\frac{\partial u^{k}}{\partial x}\text{ in } M\times \lbrack 0,T), \\ u\left( x,0\right) &=f\left( x\right);\ i=1,...,n.\end{aligned}
∂
u
i
∂
t
−
∂
2
u
i
∂
x
2
=
Γ
h
k
i
(
u
,
∂
u
∂
x
)
∂
u
h
∂
x
∂
u
k
∂
x
in
M
×
[
0
,
T
)
,
u
(
x
,
0
)
=
f
(
x
)
;
i
=
1
,
.
.
.
,
n
.
The existence of a global in time solution
u\left( x,t\right)
u
(
x
,
t
)
and its subsequent convergence to a closed geodesic
u_{\infty} \colon M\rightarrow N
u
∞
:
M
→
N
as
t\rightarrow \infty
t
→
∞
, are dealt with. Appropriate concepts arising from the Finslerian nature of the problem are introduced