softmatter:Lagrangian

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The Lagrangian of a dynamical system is defined as the Kinetic Energy T minus the Potential Energy U, or

.

To acquire equations of motion for a dynamical system, one simply computes the Lagrange Equation,

for each generalized coordinate q_i. When q_i = r_i (i.e. the generalized coordinates are simply the Cartesian coordinates), Lagrange's equations reduce to Newton's second law.

The n generalized coordinates are the coordinates associated with the n degrees of freedom of the system. For example,

• The rotation angles θ₁ and θ₂ in a double pendulum system
• The 3N spatial coordinates for N particles in three-dimensional space
• The 6N spatial plus rotational coordinates for N rigid bodies in three-dimensional space
• The θ and φ of a particle confined to the surface of a sphere