33 votes

Why should Conservative forces have their curl equal to zero?(intuition)

"Curl" is a pretty well named mathematical term--it denotes the degree of "rotation" in the vector field. For this reason, if you go all the way around in a vector field, you'll find that the total ...
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32 votes
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What exactly makes a force conservative?

All fundamental forces are conservatives and I would say that this is a postulate. Fundamental physics is constructed in such way that there is a quantity called energy which can be assigned to every ...
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32 votes

How can you conclude that gravity is a conservative force?

Are you looking for a mathematical proof (which has been given by others), or an experimental demonstration? If gravity is not conservative then that means there would two paths up a mountain that ...
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28 votes
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How do non-conservative forces affect Lagrange equations?

More generally, Lagrange equations$^1$ read $$ \frac{d}{dt}\frac{\partial (T-U)}{\partial \dot{q}^j}-\frac{\partial (T-U)}{\partial q^j}~=~Q_j-\frac{\partial{\cal F}}{\partial\dot{q}^j}+\sum_{\ell=1}...
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27 votes
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Conditions for a force to be conservative

Your conclusions are not correct. Here is a simple counter-example. Consider this force $$\vec{F}=k(x\hat{y}-y\hat{x})$$ where $\hat{x}$ and $\hat{y}$ are the unit-vectors in $x$ and $y$-direction, ...
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26 votes

How can you conclude that gravity is a conservative force?

Stokes' theorem tells us that for any vector field, the closed line integral of that field is equal to the surface integral of the curl of that field over any surface bounded by the closed loop. In ...
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  • 113k
19 votes
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Why the work done in a conservative field around a closed circle does not vanish when calculated in cylindrical coordinates?

The problematic line in your reasoning is in assuming that $\nabla \times \vec F=0$ implies that $\vec F$ is conservative, i.e. that $\vec F = \nabla \phi$ for some $\phi$. If this were true, then it ...
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16 votes

What exactly makes a force conservative?

As you know, energy is always conserved. When we talk of a force being non-conservative, it means that the force is operating within a system from which energy is allowed to escape. Perhaps the most ...
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14 votes
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Conservative or non-conservative? Frame dependent?

In classical mechanics, forces are frame invariant. Work is not, in general, because trajectories are not frame invariant. However, definition of conservative forces requires only that in each ...
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14 votes

How can you conclude that gravity is a conservative force?

The force field due to a small element of mass (which we can think of as a point mass) is spherically symmetric and central, which makes it a conservative field. For the case of field due to a point ...
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14 votes
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Is spring force really a conservative force?

The position of the block is the same in both cases, but the position of the block does not wholey define the potential energy in the spring. The length the spring is stretched defines the potential ...
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14 votes
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Relation between potential energy and conservative force

Forces can be conservative or non-conservative. But conservative forces do work where this work is equal to the change in potential energy. Conservative forces are also characterized by the fact that ...
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14 votes

Does the Newtonian gravitational field have momentum analogous to the Poynting vector?

No, in Newtonian mechanics there is no gravitational momentum. Physically, the reason is that the gravitational field doesn't propagate: it responds instantaneously to changes in the matter ...
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14 votes
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Is David Tong incorrect in this remark about classical mechanics in his QM lectures?

Indeed, I think that the statement in Tong's book is quite ambiguous (though it is not definitely false as I discuss below). In principle there is no relation between the possibility of a Hamiltonian ...
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13 votes

Is spring force really a conservative force?

You have to remember that the energy is "stored" in the spring. The block doesn't have the potential energy. The energy of the third system isn't only dependent on the position of the block (...
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11 votes
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Lagrangian Equations of Motion, Conservative Forces

The generalised Lagrange equations are $$ \frac{\mathrm{d}}{\mathrm{d}t}\frac{\partial T}{\partial \dot{q}_j} - \frac{\partial T}{\partial q_j}=Q_j \tag{1} $$ where $T$ is the kinetic energy of the ...
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10 votes

Why should Conservative forces have their curl equal to zero?(intuition)

A force field is called conservative if its work between any points $A$ and $B$ does not depend on the path. This implies that the work over any closed path (circulation) is zero. This also implies ...
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10 votes

Is spring force really a conservative force?

I think that you are confused about what "independent of path" means. For conservative forces the work done between two STATES of the system is independent of path. We also can define a ...
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9 votes
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A false proof of drag force being conservative

The issue is that the formula that connects force and potential gets an extra term when the force depends on velocity ${\bf v}$. The formula reads (see e.g. Ref. 1) $$\tag{1} {\bf F}~=~\frac{d}{dt} \...
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8 votes

Why should Conservative forces have their curl equal to zero?(intuition)

the heart of a force being conservative is that it is integrable, that, if we have a force ${\vec F}$, then it is possible to find a potential $\phi({\vec x})$ such that ${\vec F} = - {\vec \nabla}\...
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8 votes

Why can't the work done by a non-conservative force be zero?

For forces that change along the way, displacement is not the thing to calculate work with. Let $\gamma : [0,1] \rightarrow \mathbb{R}^3$ be the (closed or open) path that the particle the force is ...
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8 votes

All central forces are conservative forces, but are all conservative forces central forces?

If $\phi=-xy$, and ${F}=-\nabla \phi=y \hat{i}+x \hat{j}$ is conservative but not central
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7 votes

Does the Newtonian gravitational field have momentum analogous to the Poynting vector?

The Lagrangian for Newtonian gravitation (ignoring the kinetic energy of the sources, which will not affect this argument) is $$ \mathcal{L} = - \frac{1}{8 \pi G} (\nabla \phi)^2 - \phi \rho $$ where ...
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6 votes

Why can't the work done by a non-conservative force be zero?

Well, we can do a simple counter-example. Let $$ \vec{F}(\vec{x}) = F_0 \cdot \varrho(\vec x) $$ where $\varrho$ is the function that rotates vectors by 90° counter-clockwise (in matrix form $(\...
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6 votes
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What is a potential?

Electric potential and electric potential energy are two different concepts but they are closely related to each other. Consider an electric charge $q_1$ at some point $P$ near charge $q_2$ (assume ...
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6 votes

Is gravitational energy always conserved?

so net work done would be 0 Net work of all forces is 0, and that is why kinetic energy of the body does not change - it remains 0. This is an example of the general theorem $$\text{work of all ...
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6 votes

All central forces are conservative forces, but are all conservative forces central forces?

A constant force is conservative but not central. For example: $\vec F=F \hat x$ You can check that the curl of this force is $0$, hence it is conservative. Its potential energy function in 3D space ...
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  • 53.8k
6 votes

What is a non-conservative system?

Simply put: a conservative system conserves energy, a nonconservative one doesn't. In a conservative system: trajectories follow paths of constant energy - i.e., if you start the system with a given ...
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