N-D spring-mass damper total energy

[ashapi]

Homework Statement

I am trying to derive the total energy for an n-dimensional linear (Hookeian) spring-mass damper system and show that the total energy is either decreasing or constant.

Homework Equations

1D spring-mass damper equation
n-D spring-mass damper equation

The Attempt at a Solution

I started with the following force equation in 1D to describe the system:

(1) m$$\ddot{x}$$ + b$$\dot{x}$$ + kx = 0

Where:
m$$\ddot{x}$$ = F_acc = m$\frac{d^{2}x}{dt^{2}}$
b$$\dot{x}$$ = F_damp (viscous friction) = C$\frac{dx}{dt}$
kx = F_potential (gradient of some potential) = $$\nabla$$P(x)

So the equation becomes:

(2) -$$\nabla$$P(x) - C$\frac{dx}{dt}$ = m$\frac{d^{2}x}{dt^{2}}$

I then modeled the total energy as follows:

(3) E_tot = P(x) + $\frac{1}{2}$m$$\dot{x}$$²

and

(4) $\frac{dE}{dt}$= $$\nabla$$P(x)$$\dot{x}$$ + m$$\dot{x}$$$$\ddot{x}$$

In (4) I substituted m$$\ddot{x}$$ from (1) and got:

$\frac{dE}{dt}$= $$\nabla$$P(x)$$\dot{x}$$ + $$\dot{x}$$(-$$\nabla$$P(x) - C$$\dot{x}$$)

=> -C$$\dot{x}$$²

...which shows that total energy is either decreasing (|$$\dot{x}$$| > 0) or constant ($$\dot{x}$$ = 0)

I then tried to derive the total energy for the n-dimensional case and relate it to an n-D force equation as I did above. To do this I looked at the 2D case and then generalized the formulas for kinetic and potential energy to n-D but I'm not sure if this was correct. Here are my kinetic and potential energy equations for 2D:

 

[ashapi]

My latex seems to bug out sometimes so if it looks weird please click on the source but let me continue...

KE_tot,2D = $\frac{1}{2}$(m₁₁$$\dot{x}$$₁² + m₂₂$$\dot{x}$$₂² + m₁₂$$\dot{x}$$₁$$\dot{x}$$₂)PE_tot = $\frac{1}{2}$(m₁₁$${x}$$₁² + m₂₂$${x}$$₂² + m₁₂$${x}$$₁$${x}$$₂)

I took the generalized forms to be:

KE_tot,nD = $\sum_{i=1}^{n}$ (m_ii)$$\dot{x}$$_i² + $\sum_{j=1}^{n-1}$ $\sum_{k=j+1}^n$ (m_jk)$$\dot{x}$$_j$$\dot{x}$$_k

...and the same for PE with $${x}$$ replacing $$\dot{x}$$.

My plan was to take the derivatives of these and replace the m$$\ddot{x}$$ terms with the force equivalences (as above). Is this right? I also can't think of how to make a generalized form of the force equation for an n-D vector.

Any help would be greatly appreciated.

Thanks,

ashapi