I need help with a Math problem..
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The equations of motion in Lagrangian mechanics are Lagrange's equations, also known as Euler-Lagrange eqations Below, we sketch out the derivation of Lagrange's equation from Newtons laws of motion. See the references for more detailed and more general derivations.
Consider a single particle with mass m and position vector r. The applied force, F, can be expressed as the gradient of a scalar potential energy function V(r, t):
Such a force is independent of third- or higher-order derivatives of r, so Newtons second law forms a set of 3 second-order ordinary differential entities. Therefore, the motion of the particle can be completely described by 6 independent variables, or degrees of freedom. An obvious set of variables is { rj, r′j | j = 1, 2, 3}, the Cartesian components of r and their time derivatives, at a given instant of time (ie. position (x,y,z) and velocity (vx,vy,vz ) ).
More generally, we can work with a set of generalized coordinates, qj, and their time derivatives, the generalized velocities, q′j. The position vector, r, is related to the generalized coordinates by some transformation equation:
For example, for a simple pendulum of length l, a logical choice for a generalized coordinate is the angle of the pendulum from vertical, θ, for which the transformation equation would be
.
The term "generalized coordinates" is really a leftover from the period when Cartesian coordinates were the default coordinate system.
Consider an arbitrary displacement δr of the particle. The work done by the applied force F is δW = F · δr. Using Newton's second law, we write:
Since work is a physical scalar quantity, we should be able to rewrite this equation in terms of the generalized coordinates and velocities. On the left hand side,
Consider a single particle with mass m and position vector r. The applied force, F, can be expressed as the gradient of a scalar potential energy function V(r, t):
Such a force is independent of third- or higher-order derivatives of r, so Newtons second law forms a set of 3 second-order ordinary differential entities. Therefore, the motion of the particle can be completely described by 6 independent variables, or degrees of freedom. An obvious set of variables is { rj, r′j | j = 1, 2, 3}, the Cartesian components of r and their time derivatives, at a given instant of time (ie. position (x,y,z) and velocity (vx,vy,vz ) ).
More generally, we can work with a set of generalized coordinates, qj, and their time derivatives, the generalized velocities, q′j. The position vector, r, is related to the generalized coordinates by some transformation equation:
For example, for a simple pendulum of length l, a logical choice for a generalized coordinate is the angle of the pendulum from vertical, θ, for which the transformation equation would be
The term "generalized coordinates" is really a leftover from the period when Cartesian coordinates were the default coordinate system.
Consider an arbitrary displacement δr of the particle. The work done by the applied force F is δW = F · δr. Using Newton's second law, we write:
Since work is a physical scalar quantity, we should be able to rewrite this equation in terms of the generalized coordinates and velocities. On the left hand side,
The right hand side is more difficult, but after some shuffling we obtain:
where T = 1/2 m r′ 2 is the kinetic energy of the particle. Our equation for the work done becomes
However, this must be true for any set of generalized displacements δqi, so we must have
or each generalized coordinate δqi. We can further simplify this by noting that V is a function solely of r and t, and r is a function of the generalized coordinates and t. Therefore, V is independent of the generalized velocities:
Inserting this into the preceding equation and substituting L = T - V, we obtain Lagrange's equations:
There is one Lagrange equation for each generalized coordinate qi. When qi = ri (i.e. the generalized coordinates are simply the Cartesian coordinates), it is straightforward to check that Lagrange's equations reduce to Newton's second law.
The above derivation can be generalized to a system of N particles. There will be 6N generalized coordinates, related to the position coordinates by 3N transformation equations. In each of the 3N Lagrange equations, T is the total kinetic energy of the system, and V the total potential energy.
