# Tag Info

### Lagrange's equation is form invariant under EVERY coordinate transformation. Hamilton's equations are not under EVERY phase space transformation. Why?

You should really think about the variables we use as being like coordinates on some manifold, the configuration space (roughly the same as the phase space, I won't be careful about the distinction). ...
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### Why do we need coordinate-free descriptions?

That's a very good question. While it may seem "natural" that the world is ordered like a vector space (it is the order that we are accustomed to!), it's indeed a completely unnatural requirement for ...
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### Is it "mathematically wrong" to ignore dual spaces, 1-forms, and covariant/contravariant indices in classical mechanics?

As long as you restrict yourself to orthonormal bases, then that's fine. The reason for this is that indices are "raised" or "lowered" via the metric, and in an orthonormal basis ...
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### Vectors as functions?

However, I have read that the modern way to learn these concepts is to think of vectors as multilinear functions of covectors This is actually not quite true, though the distinction is subtle. In the ...
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### Gradient, divergence and curl with covariant derivatives

For the gradient, your mistake is that the components of the gradient vary contravariantly. On top of that, there is a issue with normalisation that I discuss below. I don't know if you are familiar ...

### Co and contravariant: tensors or components?

Yes, the contravariant components make reference to a different geometrical object than the covariant components. The covariant components are components of a vector from the dual space to the vector ...

### Why do we need a metric to define gradient?

The main point of the gradient is to have the following equation: $$df(v) = \langle \nabla f , v \rangle$$ for every vector $v$ of the tangent space where $\langle \cdot , \cdot \rangle$ is the ...

### Why do we need coordinate-free descriptions?

Why do we need coordinate free in the first place? Let me tell you about a related experience I have had teaching students. When I ask them to define the scalar product then the vast majority will ...

### Difference between symmetry and invariance

Both concepts are mathematical in character and they ultimately describe the same characteristics or situations. "Invariance" is a more technical word because it says "what has to be equal to what" ...
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### What is "a general covariant formulation of newtonian mechanics"?

The development of general relativity has led to a lot of misconceptions about the significance of general covariance. It turns out that general covariance is a manifestation of a choice to represent ...
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### Confusions about Covariant and Contravariant vectors

You're dealing with different geometric objects: Tangent vectors, which can be realized as equivalence classes of curves, and cotangent vectors, which can be realized as equivalence classes of real-...
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### Origin of $\sqrt{-g}$ in the integral of action $S$

When you're working in general relativity, the coordinates are basically arbitrary and they do not even need to have dimensions of length. This means that the "usual" volume element,  dx^{0}dx^{1}dx^...
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### Is there differential form notation for Maxwell's equation in curved spacetime?

Differential forms are natural objects that depend solely on the smooth structure, therefore they are valid in any spacetime. Note that the Hodge star $\star$ does, however, depend on a metric tensor....

### Critique on tensor notation

Forget tensors for a moment and just think about our old friend the matrix. Take a square matrix for example. You can multiply it onto a column vector and get back a column vector. Or you can put a ...

### Notion of Co- and Contravariance in Dirac-Notation

In Dirac notation, $\langle \psi |$ is a covariant vector, while $| \psi \rangle$ is a contravariant vector. Of course, this nomenclature is hardly ever used, because it is more convenient to call ...
In the 'strict' sense, you should only apply the summation convention to a pair of indices if one is raised and another is lowered. For example, consider a vector $v$ and a dual vector $f$ (i.e. a ...