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Hamiltonian & Lagrangian Mechanics


Lagrangian: 

L=T-V

Principle of Least Action: 

\delta\int{L}dt=0

Euler-Lagrange Equation:

\frac{d}{dt}\left[\frac{{\partial}L}{\partial\dot{q}_i}\right]-\frac{{\partial}L}{{\partial}q_i}=0

Generalised Momentum:

p_i=\frac{{\partial}L}{\partial\dot{q}_i}

Hamiltonian:

H=\Sigma_i{\dot{q}_i}p_i-L



The Terms

L - Lagrangian [Units: J]

T - Kinetic Energy [Units: J]

V - Potential Energy [Units: J]

q_i - Generalised Coordinates

\dot{q}_i - Time derivatives of generalised coordinates

p_i - Generalised Momenta

H - Hamiltonian [Units: J]


What Do They Mean?

The Hamiltonian and Lagrangian formalisms which evolved from Newtonian Mechanics are of paramount important in physics and mathematics. They are two different but closely related mathematically elegant pictures which tell us something deep about the mathematical underpinnings of our physical universe.

The Lagrangian is a function with dimensions of energy that summarises the dynamics of a system. The equations of motion can be obtained by substituting L into the Euler-Lagrange equation.

The action \int{L}dt takes different values for different paths. The Principle of Least Action states that the path followed by any real physical system is one for which the action is stationary, that is it does not vary to first order for infinitessimal deformations of the trajectory.

The Euler-Lagrange equation is a differential equation with solutions L for which the action is stationary. It can be used to obtain the equations of motion for a system with Lagrangian L.

The generalised momenta, that is momenta p_i corresponding to coordinates q_i, are defined as the partial differential of the Lagrangian with respect to the time derivative of the coordinate.

The Hamiltonian H has dimensions of energy and is the Legendre transformation of the Lagrangian L.


Further information at Warwick

The principles and methods associated with Hamiltonian and Lagrangian mechanics are explored in the second year module "PX267 Hamiltonian Mechanics" and the third year module "PX440 Mathematical Methods for Physicists III".