Is the Lagrangian Invariant Under Coordinate Transformations?

In summary, Lagrangians are scalars that are not invariant under coordinate transformations, while Euler-Lagrange equations are invariant under coordinate transformations. This shows that some scalars can vary under coordinate transformations, such as the components of vectors. In conclusion, time is a scalar quantity.
  • #1
shounakbhatta
288
1
Can you please tell me whether I am right or wrong?

Lagrangians are scalars. They are NOT invariant under coordinate transformations[ the simplest example is when you have a gravitational potential(V=mgz) and you translate z by "a"(some number). L=1/2*m*(dz/dt)^2-mgz--->L=1/2*m*(dz/dt)^2-mgz-mga, thus the Lagrangian changed under this coordinate transformation!]. However Euler-Lagrange equations ARE invariant under coordinate transformations. So some scalars do vary under coordinate transformation! Thus components of vectors are scalars, thus time is. Again, try to say the following sentence:"the vector <a> is the first component of the vector <b>".

Does not make any sense?

Thanks
 
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  • #2
for your question! The sentence does not make sense because a vector cannot be the first component of another vector. However, time is a scalar quantity, so it is correct that time is a scalar.
 

FAQ: Is the Lagrangian Invariant Under Coordinate Transformations?

1. What is the Lagrangian in the context of field theory?

The Lagrangian is a mathematical function that describes the dynamics of a physical system. In field theory, it is used to describe the behavior of a scalar field, which is a field that has a single value at each point in space and time.

2. How is the Lagrangian used to derive the equations of motion for a scalar field?

The Lagrangian is used in the Euler-Lagrange equation, which is a set of differential equations that describe the evolution of a system over time. By plugging the Lagrangian into this equation, we can derive the equations of motion for the scalar field.

3. What is a scalar field and how does it differ from a vector field?

A scalar field is a field that has a single value at each point in space and time. This value can be a scalar, such as temperature or density, or it can be a vector, such as velocity or electric field. A vector field, on the other hand, has a direction and magnitude at each point in space and time.

4. How does the concept of a scalar field relate to the Higgs field in particle physics?

The Higgs field is a type of scalar field that is believed to give particles their mass. In the Standard Model of particle physics, the Higgs field is responsible for the Higgs mechanism, which explains how particles acquire mass as they interact with the Higgs field.

5. Can the Lagrangian be used to describe other types of fields besides scalar fields?

Yes, the Lagrangian can also be used to describe other types of fields, such as vector fields and tensor fields. In fact, the Lagrangian can be used to describe the behavior of any physical system, making it a very powerful tool in theoretical physics.

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