Can anyone tell me how does conservative forces work? Confused

Question

From vector calculus, I'd learnt that a conservative vector field satisfies $$ \textbf{F} = \boldsymbol{\nabla} g $$ which $\textbf{F}$ is the gradient of some scalar-valued function, and $g$ is the function.

But in physics, I see someone saying "definition of conservative forces is that is there exists a scalar function $V$ for a field $\textbf{F}$ which satisfies this condition" $$ \textbf{F}=- \boldsymbol{\nabla} V$$

Why does a negative sign exist and why does $\textbf{F} = \boldsymbol{\nabla} g$ in the first place? I don't want to take it as it be.

Thanks a lot.

Answer 1

The answer is very simple: if $\vec{F}$ is a conservative vector field, then $\vec{G} = -\vec{F}$ is also a conservative vector field. So mathematically, the choice to include the minus sign or not doesn't change anything. But physically, the minus sign is important because it lets the physical quantities match our physical intuition. When we write $\vec{F} = -\nabla V$, we mean physically that the force $\vec{F}$ arises from a potential energy function $V$. Intuitively, we like to think that such a force pushes from high to low potential energy -- think about how gravity causes an object to fall from high to low gravitational potential energy, or a spring pushes a block from high to low elastic potential energy. Since $\nabla V$ is a vector that points in the direction of the sharpest increase in $V$, then $\vec{F} = -\nabla V$ pushes the object in the direction of the sharpest decrease in $V$.

Of course, this is all a convention -- if you'd like, you can define a "potential schmenergy" function $\tilde{V} = -V$, and then $\vec{F} = + \nabla \tilde{V}$. None of the physics would be changed, except that objects would fall from low to high potential schmenergy, which might go against the grain of your intuition.