Differential Equations
Here is the problem I have done 1,2, and 3 but don't know how to do the rest
Consider two masses m1 and m2 lying on a frictionless surface. The two masses are connected by three springs with force constants k1 k2 k3 The left most k1 and right most k3 springs are attached to immovable walls
1) what is the total kinetic energy
I have T=(1/2)m1x1dot^2+(1/2)m2x2dot^2
2) what is the total potential energy
I have U=(1/2)k1x1^2+(1/2)k3x2^2+(1/2)k2(x2-x1)^2
3) construct the lagrangian
L=T-U=((1/2)m1x1dot^2+(1/2)m2x2dot^2)-((1/2)k1x1^2+(1/2)k3x2^2+(1/2)k2(x2-x1)^2)
4) use the euler-lagrange equations to define a ststem of 2nd order differential equations that describe the motion
5) write the system of equations in matrix form and find the general solution with the assumption that x=ve^rt
6) what is the physical significance of the eigenvalues and eigenvectors that youve found
7) construct the hamiltonian function
8) Use hamiltons equation to define a system of 1st order equation that describe the motion
9) let m1=m2=k1=k2=k3=1 use the laplace transform to solve the 1st order system with the initial conditions
x1(0)=-1 p1(0)=0 x2(0)=1 p2(0)=0
Here is the problem I have done 1,2, and 3 but don't know how to do the rest
Consider two masses m1 and m2 lying on a frictionless surface. The two masses are connected by three springs with force constants k1 k2 k3 The left most k1 and right most k3 springs are attached to immovable walls
1) what is the total kinetic energy
I have T=(1/2)m1x1dot^2+(1/2)m2x2dot^2
2) what is the total potential energy
I have U=(1/2)k1x1^2+(1/2)k3x2^2+(1/2)k2(x2-x1)^2
3) construct the lagrangian
L=T-U=((1/2)m1x1dot^2+(1/2)m2x2dot^2)-((1/2)k1x1^2+(1/2)k3x2^2+(1/2)k2(x2-x1)^2)
4) use the euler-lagrange equations to define a ststem of 2nd order differential equations that describe the motion
5) write the system of equations in matrix form and find the general solution with the assumption that x=ve^rt
6) what is the physical significance of the eigenvalues and eigenvectors that youve found
7) construct the hamiltonian function
8) Use hamiltons equation to define a system of 1st order equation that describe the motion
9) let m1=m2=k1=k2=k3=1 use the laplace transform to solve the 1st order system with the initial conditions
x1(0)=-1 p1(0)=0 x2(0)=1 p2(0)=0