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Classical Mechanics: Difference between revisions
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<math> \frac{d}{dt} \frac{\partial L(q(t), \dot{q}(t), t)}{\partial \dot{q}}=\frac{\partial L(q(t), \dot{q}(t), t)}{\partial q},\quad m * \frac{d^2 q(t)}{dt^2}=-k*q(t)</math></div> | <math> \frac{d}{dt} \frac{\partial L(q(t), \dot{q}(t), t)}{\partial \dot{q}}=\frac{\partial L(q(t), \dot{q}(t), t)}{\partial q},\quad m * \frac{d^2 q(t)}{dt^2}=-k*q(t)</math> | ||
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Computed in the particular example of a Lagrangian given previously. Regarding the position and velocity variables as functions of time again means that given a particular formula for a Lagrangian, the resulting Euler-Lagrange equation has the form of an ordinary differential equation whose solution is just the function <math> q(t) </math>. The partial derivatives of L with respect to position and velocity at each time indicate vectors in the constant-time planes as in the picture, that point in the direction of increase of L. Thus the Euler-Lagrange equation for motion in one dimension can be interpreted in the 3d geometric picture. The change in the velocity component of this vector as one moves up in time along the trajectory (e.g. it decreases) is the instantaneous value of the position component. This time derivative is in the case of when L is time independent, totally determined by the direction of the trajectory in the 3d graphed trajectory space at that time. Given an initial position and velocity at some time, this fixes the rest of the curve for future times, as it is constrained to follow the direction given by the EL equation at each time. In particular, this gives the endpoint at some chosen later time. | Computed in the particular example of a Lagrangian given previously. Regarding the position and velocity variables as functions of time again means that given a particular formula for a Lagrangian, the resulting Euler-Lagrange equation has the form of an ordinary differential equation whose solution is just the function <math> q(t) </math>. The partial derivatives of L with respect to position and velocity at each time indicate vectors in the constant-time planes as in the picture, that point in the direction of increase of L. Thus the Euler-Lagrange equation for motion in one dimension can be interpreted in the 3d geometric picture. The change in the velocity component of this vector as one moves up in time along the trajectory (e.g. it decreases) is the instantaneous value of the position component. This time derivative is in the case of when L is time independent, totally determined by the direction of the trajectory in the 3d graphed trajectory space at that time. Given an initial position and velocity at some time, this fixes the rest of the curve for future times, as it is constrained to follow the direction given by the EL equation at each time. In particular, this gives the endpoint at some chosen later time. | ||