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I'm currently reading Topological Solitons by Manton & Sutcliffe and am having a bit of trouble with deriving an equation of motion. Suppose $M$ is a smooth manifold of dimension $D$ and $\mathbf{q}(t)=(q^1(t),...,q^D(t))$ is a smooth trajectory on $M$. Suppose we also have a scalar potential $V:M\rightarrow\mathbb{R}$, a Riemannian metric on $M$ with components $g_{ij}(\mathbf{q})$, and a local one-form with components $a_i(\mathbf{q})$. We can then define the Lagrangian density as $$L=\frac{1}{2}g_{ij}\dot{q}^i\dot{q}^j-a_i\dot{q}^i-V.$$ The Euler-Lagrange equations are $$\frac{d}{dt}\frac{\partial L}{\partial \dot{q}^i}-\frac{\partial L}{\partial q^i}=0$$ which are then written as $$\frac{d}{dt}\Big(g_{ij}\dot{q}^j-a_i\Big)-\frac{\partial}{\partial q^i}\Big(\frac{1}{2}g_{jk}\dot{q}^j\dot{q}^k-a_j\dot{q}^j-V\Big)=0.$$

I don't understand why there isn't a factor of $\frac{1}{2}$ in front of the $g_{ij}\dot{q}^j$ term and was wondering what I've missed here?

Adam Rafferty
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1 Answers1

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It's just product rule. Notice that $$\frac{\partial}{\partial \dot{q}_k}\left(\frac{1}{2}g_{ij}\dot{q}^i\dot{q}^j\right)=\frac{1}{2}(g_{ij}\dot{q}^i \delta_k^j+g_{ij}\dot{q}^j \delta_k^i)=\frac{1}{2}(g_{ik}\dot{q}^i+g_{kj}\dot{q}^j)=g_{ij}\dot{q}^j$$ Because $$\frac{\partial x^i}{\partial x^j}=\delta^i_j$$ Notice that I rewrote the indices in the last step, just the leave it in the form you have written, and also don't forget that you may not differentiate with respect to an index that is involved in the summation, because otherwise you will performing an additional summation that was not intended.

Lourenco Entrudo
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