Showing the invariance of the equations of motion

It is strange to me that for a symmetry which involves $\dot{x}$, there seems to always appear a term with $\dddot{x}$ in the variation of the equations of motion, which doesn't makes much sense. I think that probably the procedure I am following is wrong.

I will show you an example: Consider the simple case of a free particle in one dimension, it's Lagrangian is:

$$L=\frac{1}{2}\dot{x}^2$$

It is obvious that the system conserves energy, so the symmetry that must be valid is $(\delta x=0 ,\delta t=\epsilon)$. I may rewrite these as symmetries that doesn't involve time variations (as the paper by E. L. Hill do):

$$(\delta_{*}x=\epsilon\dot{x},\delta_{*}t=0)$$

Now, I calculate the variation of the equations of motion, with the hope of finding that $\delta(\text{e.o.m})=\text{e.o.m}.$ Such result would mean that the equations of motion are invariant under the symmetry in consideration. So:

$$\delta{(\text{e.o.m})}=\delta(\ddot{x})=\ddot{\eta}$$

In this case $\eta = \dot{x}$ (remember that a variation is of the form $\delta{x}=\epsilon\eta$). So $\ddot{\eta}=\dddot{x}$. Hence:

$$\delta(\text{e.o.m})=\dddot{x}\neq\text{e.o.m}$$

This doesn't make sense because, for the system in consideration, the time translation is a Noetherian symmetry that gives conservation of Energy.

My question is: What is failing in this procedure? Is there a general way of showing that some Symmetry is indeed Noetherian?

• I'm not sure entirely what you're saying here, since you interchange terms like equations of motion (which I presume is e.o.m.) with a quantity like $\ddot{x}$. Do you want to show that the given transformation is a symmetry (in other words, it leaves the Lagrangian invariant up to a total derivative)? Do you also want to find the conserved charge (energy) associated with this transformation? – QuantumFool Sep 27 '16 at 6:07

Short answer: In the reduction (when showing that an equation of motion $EOM=0$ has a symmetry), one is also allowed to use consequences of $EOM=0$. E.g. in OP's example: $$\ddot{x}~=~ 0 \quad \Rightarrow \quad\dddot{x}~=~ 0,$$

See also e.g. this related Phys.SE post.

• I noticed that before, actually that is a previous question I made. Still, by doing that, what is obtained is $\delta(\text{e.o.m})=0$, but it is not clear if $\delta(\text{e.o.m})=\text{e.o.m}$. I think the second is what tells you the equations are invariant, isn't it? – Saavestro Sep 26 '16 at 20:13

I'm not sure precisely what you did to solve this problem (for instance, how is the equation of motion, which is by definition an equation, also the quantity $\ddot{x}$), but here's how I would approach it.

If we vary the Lagrangian according to the transformation you provide:

\begin{align} \delta L & = \delta \left ( \frac{1}{2} \dot{x}^2 \right ) \\ & = \frac{1}{2} \delta \left ( \dot{x}^2 \right ) \\ & = \frac{1}{2} \cdot \frac{\partial \dot{x}^2}{\partial \dot{x}} \delta \dot{x} \\ & = \frac{1}{2} 2 \dot{x} \delta \dot{x} \\ & = \frac{1}{2} 2 \dot{x} \delta \frac{\partial x}{\partial t} \\ & = \frac{1}{2} 2 \dot{x} \frac{\partial}{\partial t} \left ( \delta x \right ) \\ & = \frac{1}{2} 2 \dot{x} \frac{\partial}{\partial t} \left ( \epsilon \dot{x} \right ) \\ & = \dot{x} \frac{\partial}{\partial t} \left ( \epsilon \dot{x} \right ) \\ & = \frac{\epsilon}{2} \left [ \dot{x} \frac{\partial \dot{x}}{\partial t} + \frac{\partial \dot{x}}{\partial t} \dot{x} \right ] \\ & = \epsilon \frac{\partial}{\partial t} \left ( \frac{1}{2} \dot{x}^2 \right ) \end{align}

We now see $L \to L + \epsilon \partial_{t} J$ for $J = \frac{1}{2} \dot{x}^2 = L$. You can show that this always works out. Since the Lagrangian changed by only a total derivative, we have a symmetry, and you can show that the conserved charge is energy.