Expressing kinetic energy as sum of vector and tensor terms

In summary, a system of N particles described by vector coordinates ##\mathbf{r}_k, k=1,2, \dots, N## and subject to 3N-f constraints can be expressed in terms of generalised coordinates ##q_i, i=1,2, \dots, f## by ##\mathbf{r}_k = \mathbf{r}_k(q_1, q_2, \dots, q_f, t)##. The 'cancellation of dots' rule can be proven, and the kinetic energy of the system can be written as $$T =
  • #1
CAF123
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Homework Statement


A system of N particles described by the vector coordinates ##\mathbf{r}_k, k = 1,2, \dots, N ## subject to 3N - f constraints can be expressed in terms of generalised coordinates ##q_i, i=1,2, \dots, f## by ##\mathbf{r}_k = \mathbf{r}_k(q_1, q_2, \dots, q_f, t)##

a) Prove the 'cancellation of dots' rule
b) Show that the kinetic energy of the system can be written as $$T = \frac{1}{2}\sum_{k=1}^{N} m_k \mathbf{r}_k^2 = M_o + \sum_{i=1}^f M_i \dot{q_i} + \frac{1}{2}\sum_{i=1}^f \sum_{j=1}^f M_{ij} \dot{q_i} \dot{q_j} $$ expressing ##M_o, M_{i}, M_{ij}## in terms of ##\mathbf{r}_k## and ##t##.

Homework Equations


[/B]
$$\mathbf{\dot{r}}_k = \sum_{i=1}^f \frac{\partial \mathbf{r}_k}{\partial q_i} \dot{q_i} + \frac{\partial \mathbf{r}_k}{\partial t}$$

The Attempt at a Solution


a) and b) are both fine I think. I put my work for b) below anyway to show what I have.

The term is $$\frac{1}{2} \sum_{k=1}^N m_k \left( \frac{\partial \mathbf{r}_k}{\partial q_i} \dot{q_i} + \frac{\partial \mathbf{r}_k}{\partial t}\right)^2,$$ using the Einstein summation convention for the ##i## sum. Then I can write this like $$\frac{1}{2} \sum_{k=1}^k m_k \left( \frac{\partial \mathbf{r}_k}{\partial q_i} \frac{\partial \mathbf{r}_k}{\partial q_j} \dot{q_i}\dot{q_j} + \left( \frac{\partial \mathbf{r}_k}{\partial t}\right)^2 + 2 \frac{\partial \mathbf{r}_k}{\partial q_i} \dot{q_i} \frac{\partial \mathbf{r}_k}{\partial t}\right)$$ from which I can read $$M_o = \frac{1}{2} \sum_{k=1}^N m_k \left(\frac{\partial \mathbf{r}_k}{\partial t}\right)^2 \,\,\,\,\, M_i = \sum_{k=1}^N m_k \frac{\partial \mathbf{r}_k}{\partial q_i} \frac{\partial \mathbf{r}_k}{\partial t} \,\,\,\,\,\, M_{ij} = \sum_{k=1}^N m_k \frac{\partial \mathbf{r}_k}{\partial q_i} \frac{\partial \mathbf{r}_k}{\partial q_j} $$

I have edited my previous version of the OP in which I asked for some guidance on how to progress. Really I am confident in the above result now, so if anyone could clarify that would be great, otherwise if the moderators wish to delete this then that would also be fine. Thanks!
 
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  • #2

The term is $$\frac{1}{2} \sum_{k=1}^N m_k \left( \frac{\partial \mathbf{r}_k}{\partial q_i} \dot{q_i} + \frac{\partial \mathbf{r}_k}{\partial t}\right)^2,$$ using the Einstein summation convention for the ##i## sum. Then I can write this like $$\frac{1}{2} \sum_{k=1}^k m_k \left( \frac{\partial \mathbf{r}_k}{\partial q_i} \frac{\partial \mathbf{r}_k}{\partial q_j} \dot{q_i}\dot{q_j} + \left( \frac{\partial \mathbf{r}_k}{\partial t}\right)^2 + 2 \frac{\partial \mathbf{r}_k}{\partial q_i} \dot{q_i} \frac{\partial \mathbf{r}_k}{\partial t}\right)$$ from which I can read $$M_o = \frac{1}{2} \sum_{k=1}^N m_k \left(\frac{\partial \mathbf{r}_k}{\partial t}\right)^2 \,\,\,\,\, M_i = \sum_{k=1}^N m_k \frac{\partial \mathbf{r}_k}{\partial q_i} \frac{\partial \mathbf{r}_k}{\partial t} \,\,\,\,\,\, M_{ij} = \sum_{k=1}^N m_k \frac{\partial \mathbf{r}_k}{\partial q_i} \frac{\partial \mathbf{r}_k}{\partial q_j} $$

Your result for $M_o$ is correct. However, your results for $M_i$ and $M_{ij}$ are incorrect. The correct expressions are: $$M_i = \sum_{k=1}^N m_k \frac{\partial \mathbf{r}_k}{\partial q_i} \cdot \frac{\partial \mathbf{r}_k}{\partial t}$$ and $$M_{ij} = \sum_{k=1}^N m_k \frac{\partial \mathbf{r}_k}{\
 

1. What is kinetic energy?

Kinetic energy is the energy an object possesses due to its motion.

2. Why is kinetic energy important in science?

Kinetic energy is important in science because it is a fundamental concept in physics and helps describe the motion of objects and their interactions with other objects.

3. What are vector and tensor terms?

Vector and tensor terms refer to mathematical representations of quantities that have both magnitude and direction. Vectors have a specific direction in space, while tensors have multiple components that describe the relationship between different directions.

4. How is kinetic energy expressed as a sum of vector and tensor terms?

In order to express kinetic energy as a sum of vector and tensor terms, we can break down the motion of an object into its individual components (such as velocity and acceleration) and use vector and tensor operations to calculate the total kinetic energy.

5. What is the significance of expressing kinetic energy in terms of vectors and tensors?

Expressing kinetic energy in terms of vectors and tensors allows us to better understand the underlying physical principles and relationships involved in an object's motion. It also allows for more accurate and precise calculations and predictions in various scientific fields such as mechanics, fluid dynamics, and thermodynamics.

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