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I have the following problem:

Let [itex]L(q,\dot{q})=\sum g_{ij}(q)\dot{q}_i\dot{q}_j[/itex]. And [itex]l(q,\dot{q})=\sqrt{L(q,\dot{q})}[/itex]. Define the spaces [itex]\mathbb{X},\, \mathbb{Y}[/itex] of parametrized curves

[tex]\mathbb{X}=\{\gamma\,:\,[0,1]\rightarrow \mathbb{R}^n,\,\gamma \in C^\infty,\,\gamma(0)=q_0,\,\gamma(1)=q_1\},[/tex]

[tex]\mathbb{Y}=\{\gamma\,:\,[0,1]\rightarrow \mathbb{R}^n,\,\gamma \in C^\infty,\,\gamma(0)=q_0,\,\gamma(1)=q_1,\,L(\gamma,\dot{\gamma})=k\},[/tex]

([itex]k[/itex] is a constant and [itex]g\in C^\infty[/itex]). Plus, let's define te actions [itex]A_L:\mathbb{X}\rightarrow\mathbb{R}[/itex] and [itex]A_l:\mathbb{Y}\rightarrow\mathbb{R}[/itex] in the usual way. Prove that the critical points of [itex]A_L[/itex] in [itex]\mathbb{X}[/itex] coincide with the ones of [itex]A_l[/itex] in [itex]\mathbb{Y}[/itex]. Give the geometrical interpretation of the action [itex]A_l[/itex] and of the condition [itex]L(\gamma,\dot{\gamma})=k[/itex] in [itex]\mathbb{Y}[/itex].

I've already shown that the critical points coincide. I also know from a previous exercise that [itex]g_{ij}(q)[/itex] is positive definite, that the Euler-Lagrange equations are the ones for the geodesics in that metric and that [itex]\dot{L}(q(t),\dot{q}(t))=0[/itex] if [itex]q(t)[/itex] is a geodesic.

Is [itex]l=\left\|\dot{q}\right\|_g[/itex] the norm of the velocity vector?

If so, what does it means that [itex]L(\gamma,\dot{\gamma})=k[/itex]?

I little help will be much apretiated.

Let [itex]L(q,\dot{q})=\sum g_{ij}(q)\dot{q}_i\dot{q}_j[/itex]. And [itex]l(q,\dot{q})=\sqrt{L(q,\dot{q})}[/itex]. Define the spaces [itex]\mathbb{X},\, \mathbb{Y}[/itex] of parametrized curves

[tex]\mathbb{X}=\{\gamma\,:\,[0,1]\rightarrow \mathbb{R}^n,\,\gamma \in C^\infty,\,\gamma(0)=q_0,\,\gamma(1)=q_1\},[/tex]

[tex]\mathbb{Y}=\{\gamma\,:\,[0,1]\rightarrow \mathbb{R}^n,\,\gamma \in C^\infty,\,\gamma(0)=q_0,\,\gamma(1)=q_1,\,L(\gamma,\dot{\gamma})=k\},[/tex]

([itex]k[/itex] is a constant and [itex]g\in C^\infty[/itex]). Plus, let's define te actions [itex]A_L:\mathbb{X}\rightarrow\mathbb{R}[/itex] and [itex]A_l:\mathbb{Y}\rightarrow\mathbb{R}[/itex] in the usual way. Prove that the critical points of [itex]A_L[/itex] in [itex]\mathbb{X}[/itex] coincide with the ones of [itex]A_l[/itex] in [itex]\mathbb{Y}[/itex]. Give the geometrical interpretation of the action [itex]A_l[/itex] and of the condition [itex]L(\gamma,\dot{\gamma})=k[/itex] in [itex]\mathbb{Y}[/itex].

I've already shown that the critical points coincide. I also know from a previous exercise that [itex]g_{ij}(q)[/itex] is positive definite, that the Euler-Lagrange equations are the ones for the geodesics in that metric and that [itex]\dot{L}(q(t),\dot{q}(t))=0[/itex] if [itex]q(t)[/itex] is a geodesic.

**The problem is that I don't know how to interpret [itex]A_l[/itex] and [itex]L(\gamma,\dot{\gamma})=k[/itex].**Is [itex]l=\left\|\dot{q}\right\|_g[/itex] the norm of the velocity vector?

If so, what does it means that [itex]L(\gamma,\dot{\gamma})=k[/itex]?

I little help will be much apretiated.

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