# Sign of potential term in Lagrangian mechanics

In summary, the two conventions are approximately equivalent up to a constant term when the object remains close to the surface.
I have heard many times that it does not matter where you put the zero to calculate the potential energy and then ##L=T-V##. But mostly what we are doing is taking potential energy negative like in an atom for electron or a mass in gravitational field and then effectively adding it to kinetic energy. What if I take zero at the centre of atom and on the surface of earth, write ##V## positive and subtract if in lagrangian. I try that and EOM is different something like $$F=ma=-mg$$ for a free fall object. If I change it for a charge an electron in atom something similar happens.

Please someone explain, does $$F=-mg \quad \textrm{vs} \quad F=+mg$$ not matter? This is just a single term but if the EOM have multiple terms, this sign would surely make a difference. And in essence to me, it feels like it voids the negative sign in ##L=T-V##, if we can choose any sign for V. That must definitely not be the case, so where am I wrong in my reasoning?

Delta2
Calculate the equations of motion for T - V and T - V + a constant. Any difference?

You seem to have the mistaken idea that the arbitrariness of the zero means you can flip the sign of the potential. That is a mistake.

You can add a constant to the potential, you cannot multiply it by a constant. That includes multiplying it by -1

vanhees71, Delta2, etotheipi and 1 other person
That must definitely not be the case, so where am I wrong in my reasoning?
For GPE you have two conventions. If the object remains close to the surface, then you can write: $$V = mgh$$ where ##h## is the height above the surface.

And, in the general case we write $$V = -\frac{GMm}{r}$$.
The critical point, however, is that in both cases ##V## increases as ##r## or ##h## increases.

As a useful exercise, you could show that for ##r = R + h##, where ##h << R## we have:
$$V = -\frac{GMm}{r} = -\frac{GMm}{R + h} \approx V_0 + mgh$$ where ##V_0## is constant. This shows that the two conventions are approximately equivalent up to a constant term (in the case where the object remains close to the surface).

vanhees71, Dale and etotheipi

## 1. What is a "sign of potential term" in Lagrangian mechanics?

The sign of potential term in Lagrangian mechanics refers to the positive or negative value assigned to the potential energy of a system. This value is determined by the direction in which the force acts on the system and can affect the equations of motion for the system.

## 2. How does the sign of potential term affect the equations of motion in Lagrangian mechanics?

The sign of potential term affects the equations of motion by determining the direction and magnitude of the forces acting on the system. A positive sign indicates a force acting in the positive direction, while a negative sign indicates a force acting in the negative direction. This can impact the acceleration and velocity of the system.

## 3. What is the significance of the sign of potential term in Lagrangian mechanics?

The sign of potential term is significant because it helps to determine the stability and behavior of a system. A positive sign indicates a stable system, while a negative sign indicates an unstable system. This information is crucial in predicting the motion and equilibrium of a system.

## 4. Can the sign of potential term change over time in Lagrangian mechanics?

Yes, the sign of potential term can change over time in Lagrangian mechanics. This can happen if the forces acting on the system change direction or magnitude, or if the system undergoes a phase transition. It is important to consider the time dependence of the sign of potential term when analyzing the behavior of a system.

## 5. How is the sign of potential term related to the concept of potential energy in Lagrangian mechanics?

The sign of potential term is directly related to the concept of potential energy in Lagrangian mechanics. Potential energy is a scalar quantity that represents the energy stored in a system due to its position or configuration. The sign of potential term determines the direction and magnitude of this potential energy, which in turn affects the equations of motion for the system.

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