mathmatical physics

1. NPTEL – Physics – Mathematical Physics - 1
Lecture 30
Dynamics of particle : usage of tensors
Let us start by giving a simple example of calculating moment of inertia
of a cylinder of radius R and height H.
The components of the inertia tensor are calculated as,
𝐼𝑥𝑥 = ∑ 𝑚𝑖 (𝑦𝑖 + 𝑧𝑖 )
𝑖
2 2
= ∫ 𝑑𝑥𝑑𝑦𝑑𝑧(𝑦2 + 𝑧2)
Where : density of the
cylinder
= 𝑅2 𝐻
𝑀
Using Cylindrical Co-ordinates, 𝑟⃗ = (𝑟𝑐𝑜𝑠, 𝑟𝑠𝑖𝑛, 𝑧
)
𝑑𝑣 = 𝑑𝑥 𝑑𝑦 𝑑𝑧 = 𝑟 𝑑𝑟 𝑑 𝑑𝑧
So,
𝐼𝑥𝑥 =  ∫ 𝑑𝑧 ∫ 𝑑 ∫ 𝑑𝑟 𝑟 (𝑟2𝑠𝑖𝑛2 + 𝑧2)
𝐻⁄2
−𝐻⁄2
2 𝑅
0 0
=  [𝐻 𝑅4
4 12
+ 𝐻
3
𝑅2]
Using  as in above,
𝐼𝑥𝑥 =
3
𝑀𝐻 +
4
𝑀𝑅
Similarly,
1 1
2 2
𝐼𝑦𝑦 = ∑ 𝑀𝑖 (𝑥𝑖 + 𝑦𝑖 ) = 𝐼𝑥𝑥
𝑖
2 2
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2. NPTEL – Physics – Mathematical Physics - 1
𝐼 = ∑ 𝑚 (𝑥 + 𝑦 ) =
𝑧𝑧 𝑖 𝑖 𝑖
𝑖
2 2 𝑀
𝑅 ℎ
2
∫ 𝑑𝑧 ∫ 𝑑 ∫ 𝑑𝑟 (𝑟2𝑐𝑜𝑠2 + 𝑟2𝑠𝑖𝑛2 )
𝐻⁄2
−𝐻⁄2
2 𝑅
0 0
= 𝑀𝑅2
1
2
For the off diagonal elements
𝐼𝑥𝑦 = 𝐼𝑥𝑦 = − ∑ 𝑚𝑖 𝑥𝑖𝑦𝑖 = − ∫ 𝑑𝑧 ∫ 𝑑 ∫ 𝑑𝑟 𝑟3𝑠𝑖𝑛 𝑐𝑜𝑠 = 0
𝑖 −𝐻
𝐻⁄2
⁄2
2 𝑅
0 0
2
(because of ∫ 𝑠𝑖𝑛 𝑐𝑜𝑠 = 0 )
0
Similarly,
𝐼𝑥𝑧 = 𝐼𝑧𝑥 = 𝐼𝑦𝑧 = 𝐼𝑧𝑦 = 0
Thus the only non-zero components are along the diagonal owing to the
symmetry of the object under consideration. Thus the moment of inertia tensor
takes a form,
Velocity and Acceleration- Application in Mechanics
The portion of a particle in 𝑉3 is represented by 𝑥⃑ = (𝑥𝑖(𝑡)), while the
velocity is given by,
𝑣⃑ = 𝑑𝑥⃑
= (𝑥̇ 1, 𝑥̇ 2, 𝑥̇ 3) and the acceleration is given
by, 𝑑𝑡
𝑎⃑ = 𝑑𝑣⃗⃑
= (𝑥̈ 1, 𝑥̈ 2, 𝑥̈ 3) where the number of dots indicate
the 𝑑𝑡
number of time derivatives taken.
In an arbitrary coordinate system, with 𝑔𝑖𝑗 representing the metric tensor, we
have, by definition,
𝑣𝑖 =
𝑑𝑥𝑖
𝑑𝑡
;
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𝐼𝑥𝑥 0 0
⃡𝐼 = ( 0
0
𝐼𝑦𝑦
0
0 )
𝐼𝑧𝑧

3. NPTEL – Physics – Mathematical Physics - 1
𝑎𝑖 = + {
𝑖
}
𝑑𝑣𝑖
𝑑𝑡 𝑗𝑘
𝑑𝑥𝑗 𝑑𝑥
𝑘
𝑑𝑡 𝑑𝑡
𝑑𝑡
Where 𝑑𝑣
is the intrisic derivative and {
𝑖
} are the Christoffel symbols
𝑖
𝑗𝑘
calculated from the metric tensor 𝑔𝑖𝑗 associated with the reference system X. If
is also written as
𝑎𝑖 = + 𝛤
𝑟𝑠𝑣 𝑣
𝑑𝑣𝑖
𝑑𝑡
𝑖 𝑟
𝑠
=
𝑑2𝑥𝑖
𝑑𝑡2 𝑑𝑡 𝑑𝑡
+ 𝛤
𝑟𝑠
𝑖 𝑑𝑥 𝑑
𝑥
𝑟 𝑠
Where the speed and acceleration are invariant,
v = √𝑔𝑖𝑗𝑣𝑖𝑣𝑗 a = √𝑔𝑖𝑗𝑎𝑖𝑎𝑗
yielding the contravariant components of velocity and acceleration. Also 𝛤𝑟𝑠 are
defined as,
𝑖
𝛤
𝑟𝑠
𝑖
= 𝜕 𝑥̅
2 𝑗 𝑖
𝜕2𝑥𝑟𝜕𝑥5 𝜕𝑥̅𝑗
𝜕𝑥
If the mass of the particle m, the Newton’s equation of motion yields,
𝐹 𝑖 = 𝑚 = 𝑚𝑎𝑖
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𝑑𝑣𝑖
𝑑𝑡
The element of work done by the force 𝐹⃗ in producing a displacement
d𝑟⃑ is given by,
dw =𝐹⃗. 𝑑𝑟⃑
since the components of 𝑟⃑ and 𝑑𝑟⃑ are respectively, 𝐹𝑖 and 𝑑𝑥𝑖 , their
scalar product is equal to,
dw = 𝑔𝑖𝑗𝐹𝑖𝑑𝑥𝑗
= 𝐹𝑗 𝐹𝑑𝑥𝑖

4. NPTEL – Physics – Mathematical Physics - 1
Where 𝐹𝑗 =𝑔𝑖𝑗𝐹𝑖 are the covariant components of the vector 𝐹
⃗. The kinetic energy is written as,
𝑇 = 𝑔𝑖 𝑗 𝑣𝑖 𝑣
𝑗
𝑚
2
Suppose you are given the task of computing the components of
the acceleration in cylindrical coordinates. Here (𝑥1, 𝑥2, 𝑥3) = (𝑟, 𝜃, 𝑧).
The components of the metric tensor as computed from the discussion made
earlier,
𝑔11 = 1, 𝑔22 = (𝑥1)2 = 𝑟2, 𝑔33 = 1
𝑔12 = 𝑔13 = 0 = 𝑔23
Also the components of 𝛤 are computed as,
𝛤
22
1
= −𝑥 = −𝑟 𝛤 = =
1 2
12
𝑥1
1 1
𝑟
Thus, 𝑎1 = − 𝑟 ( )
𝑑2𝑟
𝑑𝑡2
𝑑𝜃 2
𝑑𝑡
𝑎2 =
𝑑2𝜃
𝑑𝑡2 𝑟 𝑑𝑡 𝑑𝑡
𝑑2𝑧
𝑑𝑡2
+
2 𝑑𝑟 𝑑𝜃
𝑎3 =
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