24
votes
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). ...
- 4,799
23
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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 ...
- 16.2k
23
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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 ...
- 63.4k
19
votes
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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 ...
- 63.4k
18
votes
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What is the Difference between Lorentz Invariant and Lorentz Covariant?
A Lorentz invariant quantity does not change under a Lorentz transformation. For example, charge is invariant, but energy is not.
The components of a vector transform contravariantly, i.e. $A^\mu \to \...
- 98.7k
18
votes
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Covariant vs contravariant vectors
We don't talk of covariant and contravariant bases. Start with the basis $\{\mathbf e_i\}$. Then a general vector can be written $$\mathbf v = v^i \mathbf e_i$$
Now if you double the length of a basis ...
- 11.4k
16
votes
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What is the reason to believe that the laws of physics are same in all frames of reference?
Is there any supportive evidence which suggests so other than the evidence of common sense and intuition?
Yes, there is quite a substantial body of experimental evidence supporting the two postulates ...
- 86.3k
15
votes
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Covariant formulation of electrodynamics
It's true that the first equation has the same form in all Lorentz frames, but it's not obvious unless you know how $\rho$ and ${\bf J}$ individually transform. For example, the similar-looking ...
- 45.6k
15
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Critique on tensor notation
You are correct: a tensor can be viewed as a linear function in many different ways. We define a $(p,q)$ tensor $T$ as a function that takes $p$ covectors and $q$ vectors and returns a number:
$$T: {V^...
- 27.3k
14
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Momentum operator in QM - scalar or vector?
Momentum is a vector operator. Period.
When restricted to one-dimensional problems, momentum becomes a one-dimensional vector, which coincides with scalars in that space.
- 128k
14
votes
Lagrange's equation is form invariant under EVERY coordinate transformation. Hamilton's equations are not under EVERY phase space transformation. Why?
A more geometric approach to the Hamiltonian formulation is to consider the $(2n+1)$-dimensional contact manifold ${\cal M}$ with coordinates $(q^i,p_j,t)$. The Hamiltonian action functional is
$$S_H[\...
- 188k
13
votes
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Inconsistency with partial derivatives as basis vectors?
Raising and lowering indices in a vector is not a valid operation. Basis vectors are no exception. While $x_\mu=g_{\mu\nu}x^\nu$ is a valid operation, $\hat e^\mu=g^{\mu\nu}\hat e_\nu$ is not. The ...
13
votes
Why does the analogy between electromagnetism and general relativity differ if you consider them as gauge theories or fiber bundles?
There is an additional structure at play here. Depending on which level you're looking at, this is either the fact that the tangent connection $\nabla$ admits the exchange of arguments (if $\nabla_XY$ ...
- 10.3k
13
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Usefulness of Curl and Divergence as Multilinear Maps
This answer takes into account the elaboration on the question which takes place in the comments.
Let's build a tensor space. I won't bother with the definition of a tangent vector as a "...
- 63.4k
13
votes
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Can Einstein-Hilbert action be derived from symmetry considerations?
Well, one can reason as follows:
One wants a diffeomorphism-invariant action, which must be of the form $$ S=\int d^4x\sqrt{-g}L, $$ where $L$ is a scalar in terms of transformation properties.
...
- 10.3k
12
votes
Under what representation do the Christoffel symbols transform?
First, they do not transform in an actual "representation" in the sense of a linear representation of the group of coordinate transformation since their behaviour under a coordinate transformations $x\...
- 118k
12
votes
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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 ...
- 2,229
11
votes
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 ...
- 21.8k
10
votes
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 ...
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10
votes
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 ...
- 1,662
10
votes
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" ...
- 176k
10
votes
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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 ...
- 2,635
10
votes
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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-...
- 13.1k
10
votes
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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^...
- 128k
10
votes
Accepted
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....
- 10.3k
10
votes
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 ...
- 53.3k
10
votes
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 ...
- 15.4k
9
votes
Accepted
Einstein Summation Convention: One as Upper, One as Lower?
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 ...
- 98.7k
9
votes
How to show the spacetime interval is invariant in general?
You cannot derive the invariance of the line element because it is one of the assumptions on which relativity (both flavours) is based. When you say:
I understand how to derive the spacetime interval ...
- 345k
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