Let $L:(t,q,\dot q)\mapsto L(t,q,\dot q)$ a Lagrangian defined on $U\times \mathbb{R}^n$, with $U\subset \mathbb{R}^n$.

A Lagrangian variational problem consists in, given two points $x$ and $y$ of $U$, finding the curves $c:[a,b]\to U$ such that $c(a)=x, c(b)=y$ and the quantity $$ \Phi(c(.)) = \int_a^b L(c(t),\dot c(t))dt \tag{1} $$ is minimised.

An extremal of a Lagrangian variational problem is a curve which cancels the differential of $\Phi$ (defined on all curves with fixed extremeties).

Consider a Riemannian metric $g$ on $\mathbb{R}^n$ and $L(q,\dot q)=\sqrt{g_q(\dot q)}$. A geodesics is an extremal of the variational problem associated to the Lagrangian $L^2(q,\dot q)=g_q(\dot q)$.

Question : Show that the geodesics are extremals of $L$ with constant speed.

Proof : The Euler Lagrange equations associated to $L^2$ can be written as $$ 2L\frac{\partial L}{\partial q} -\frac{d}{dt}\left(2\dot L \frac{\partial L}{\partial \dot q}\right)=0 $$ that is $$ \frac{\partial L}{\partial q} -\frac{d}{dt}\left(\frac{\partial L}{\partial \dot q}\right) -2\dot L \frac{\partial L}{\partial \dot q}=0. \qquad (*) $$

Now if $q(.)$ is a gedesics, we can show first that $$\frac{d}{dt}(2L^2(q,\dot q))=\frac{d}{dt}(L^2(q,\dot q))$$ using only the Euler Lagrange equations for $L^2$. Thsi proves that $q(.)$ is with constant speed i.e. $\dot L=0$. Then from (*) we get $\frac{\partial L}{\partial q} -\frac{d}{dt}\left(\frac{\partial L}{\partial \dot q}\right)=0$ and $q(.)$ is an extremal of $L$.

Conversely, if $q(.)$ is an extremal of $L$ with constant speed then $\frac{\partial L^2}{\partial q} -\frac{d}{dt}\left(\frac{\partial L^2}{\partial \dot q}\right)=-2\dot L \frac{\partial L}{\partial \dot q}=0$ since $\dot L = \frac{\partial g_q}{\partial \dot q}\ddot q=0$.

Is it correct ?


1 Answer 1

  1. On one hand, a geodesic is a stationary curve to the arclength functional (1), which is parametrization invariant. Hence a geodesic can have arbitrary parametrization with possibly a non-constant speed, apparently contrary to OP's title (v3).

  2. On the other hand, one may show that the solutions to the Euler-Lagrange (EL) equation associated with the squared Lagrangian $L^2$ are the affinely parametrized geodesics, and hence with constant speed. Briefly this is because $L^2$ has no explicit $t$-dependence, so that the corresponding energy (which happens to be $L^2$) is constant, cf. e.g. this Math.SE post and this Phys.SE post.


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