Derive Equation of motion using Lagrangian density?

[safekhan]

Homework Statement [/b]

The attempt at a solution[/b]

I have done the first bit but don't know how to show that phi(r,t) is a solution to the equation of motion.

 

[George Jones]

Substitute the given ##\phi## into the left side of your equation; substitute the given ##\phi## into the right side of your equation. After doing this, show that left = right.

Equivalently, but perhaps a little cleaner: take all your terms to the left side; show that substituting ##\phi## into the left side gives zero.

 

[safekhan]

thanks, but how should I treat p.r term in the solution while differentiating with respect to (t,x,y,z)

 

[George Jones]

thanks, but how should I treat p.r term in the solution while differentiating with respect to (t,x,y,z)

What does

$$\mathbf{p} \cdot \mathbf{r}=?$$

 

[paranormal]

To derive the equation of motion using Lagrangian density, we start by first defining the Lagrangian density, denoted by L, as the difference between the kinetic energy T and the potential energy V:

L = T - V

We then use the principle of least action, which states that the physical path taken by a system between two points in time is the one that minimizes the action integral:

S = ∫L dt

Using the Euler-Lagrange equation, we can obtain the equation of motion for a given system:

∂L/∂q - d/dt(∂L/∂(dq/dt)) = 0

where q represents the generalized coordinates of the system.

Substituting in our definition of the Lagrangian density, we get:

∂T/∂q - d/dt(∂T/∂(dq/dt)) - ∂V/∂q + d/dt(∂V/∂(dq/dt)) = 0

Simplifying, we get:

d/dt(∂T/∂(dq/dt)) - ∂T/∂q + ∂V/∂q = 0

This is the equation of motion for the system described by the Lagrangian density.

To show that phi(r,t) is a solution to this equation of motion, we can plug it into the equation and see if it satisfies the equation. In other words, we need to show that:

d/dt(∂T/∂(dphi(r,t)/dt)) - ∂T/∂phi(r,t) + ∂V/∂phi(r,t) = 0

To do this, we can use the definition of the kinetic energy T, which is given by:

T = (1/2)∂phi(r,t)/∂t * ∂phi(r,t)/∂t

Plugging this into the equation of motion, we get:

d/dt(∂T/∂(dphi(r,t)/dt)) - ∂T/∂phi(r,t) + ∂V/∂phi(r,t) = d/dt(∂phi(r,t)/∂t) - ∂V/∂phi(r,t) + ∂V/∂phi(r,t) = d/dt(