Find Hamiltonian & EOM for 2 DOF System w/ Lagrangian

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In summary, the Hamiltonian and Hamilton's equations of motion can be found for a system with two degrees of freedom using the given Lagrangian. The equations of motion do not depend on the symmetric part of Bij because it is already taken into account in the lagrangian as it is diagonalized to become the mass.
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Homework Statement



Find the Hamiltonian and Hamilton's equations of motion for a system with two degrees of
freedom with the following Lagrangian

L = 1/2m1[itex]\dot{}xdot[/itex]12 + 1/2m2[itex]\dot{}xdot[/itex]22 + B12[itex]\dot{}xdot[/itex]1x2 + B21[itex]\dot{}xdot[/itex]1x1 - U(x1, x2)

Explain why equations of motion do not depend on the symmetric part of Bij.

Homework Equations





The Attempt at a Solution



No problem finding the Hamiltonian and the e.o.m. The last part is the problem. All I can think of is that the symmetric part is diagonalised to become the mass, since in general for a lagrangian you have L = 1/2 aij(q)[itex]^{}qdot[/itex]2
 
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- U(q). The aij is the symmetric part so if it's already in the lagrangian then its already taken into account.
 

What is a Hamiltonian in a 2 DOF system?

The Hamiltonian in a 2 DOF (degree of freedom) system is a mathematical function that describes the total energy of a system in terms of its position and momentum variables. It is derived from the Lagrangian of the system and is an important quantity in classical mechanics.

How do you find the Hamiltonian for a 2 DOF system?

The Hamiltonian for a 2 DOF system can be found by using the Legendre transformation on the Lagrangian of the system. This involves solving for the generalized momentum variables in terms of the generalized position variables and then substituting them into the Lagrangian. The resulting function is the Hamiltonian.

What is the equation of motion (EOM) for a 2 DOF system?

The equation of motion (EOM) for a 2 DOF system is a set of differential equations that describe how the system's position and momentum variables change over time. These equations can be derived from the Hamiltonian of the system using Hamilton's equations of motion.

How do you solve for the EOM in a 2 DOF system?

To solve for the EOM in a 2 DOF system, you can use various methods such as analytical techniques (e.g. using separation of variables) or numerical methods (e.g. using Euler's method or Runge-Kutta methods). The specific method will depend on the complexity of the system and the desired level of accuracy.

Why is the Lagrangian used to find the Hamiltonian and EOM in a 2 DOF system?

The Lagrangian is used to find the Hamiltonian and EOM in a 2 DOF system because it is a more general and elegant framework for describing the dynamics of a system compared to the traditional Newtonian approach. It takes into account both kinetic and potential energy and allows for the use of generalized coordinates, making it applicable to a wide range of systems.

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