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Partial Differential Equation Help watch

1. Hi,

I was going through the derivation of Lagrangian Equation and came across the following two identities. Why are these identities so obvious?

If these identities are true than can we write the following variation of Lagrange's equation:
Attachment 563802563804?

Or am I missing something?

Attached Images

2. For the first identity:

So if you take partial derivatives with respect to the result follows.

The second result is because the time and spatial derivatives can act in either order, the identity has essentially switched the order and done the time derivative on first.

I'm not sure where the final (attached) equations come from. If you write the Lagrangian as L = T - V and consider V to be independent of then you find

so your first attached equation is correct.

But the Euler-Lagrangre equations apply to L, not to T and V individually, so

and so

.
3. (Original post by AlesanaWill)
For the first identity:

So if you take partial derivatives with respect to the result follows.

The second result is because the time and spatial derivatives can act in either order, the identity has essentially switched the order and done the time derivative on first.

I'm not sure where the final (attached) equations come from. If you write the Lagrangian as L = T - V and consider V to be independent of then you find

so your first attached equation is correct.

But the Euler-Lagrangre equations apply to L, not to T and V individually, so

and so

.
Would you mind telling me what am I getting wrong?

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4. The identity

applies to coordinates which are functions of the generalised coordinates .

The kinetic energy T may be a function of as well as and possibly time. So evaluating the total derivative of T gives

Attachment 564076564084.

Even if it's only a function of , when you take partial derivatives with respect to you aren't going to get the same identity as for coordinates.
Attached Images

5. (Original post by AlesanaWill)
The identity

applies to coordinates which are functions of the generalised coordinates .

The kinetic energy T may be a function of as well as and possibly time. So evaluating the total derivative of T gives

Attachment 564076564084.

Even if it's only a function of , when you take partial derivatives with respect to you aren't going to get the same identity as for coordinates.
Thank you so much!!!!
Really appreciate it.

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