# Invariants (sorry i didn't know where to put it)

1. Oct 8, 2006

That's the question, if we can interpretate mathematically using some kind of "invariant" theory for this situation:

- Using Newton's second Law for gravity then we find that:

$$\ddot r = -GM/r$$ (1)

M is the mass of the earth, and m is the mass of the particle, amazingly using the equation (1) above $$r(t) \rightarrow r(m,t) = r(t)$$ , then the trajectories of the system of a particle under the gravity is invariant under mass change of the particle (if we make $$m' = m + \delta m$$ as an infinitesimal mass change of the particle the trajectories r(t) are the same)

- Using the metric in GR we also find that the metric:

$$g_{ab}=N(t)dt^{2} - \sum_{i,j}H_{ij}dx^{i}dx^{j}$$

where i,j=x,y,z

then the solutions of the metric are invariant under infinitesimal changes of the function $$N(t)+ \delta N(t)$$

-And the last example, if we define the "charge" by:

$$Q=e|\Phi|^{2}$$ you can check that charge is invariant under

transformation of the Quantum wave function of the form $$\Phi\rightarrow e^{ia(x)}\Phi$$

My question involving math is if we can define some "conserved" quantities or properties of the system from these "peculiar" invariance of the system under m (mass of the particle) N(t) (any function of t) or invariance under "phase" transformation, hopes the problem seems clear.