Invariance of lagrangian under point transformation

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A lagrangian $L(q,dot q, t)$ is invariant under the point transformation $$q_i=q_i(s_1,...,s_n,t)$$



To prove this I show that



$$fracddt fracpartial Lpartial dot s_i - fracpartial Lpartial s_i = 0$$



where



$$fracpartial Lpartial s_i = fracpartial Lpartial q_jfracpartial q_jpartial s_i$$



$$fracpartial Lpartial dot s_i = fracpartial Lpartial dot q_jfracpartial dot q_jpartial dot s_i$$



$$fracpartial dot q_jpartial dot s_i= fracpartial q_jpartial s_i$$



which gives
$$fracddt left(fracpartial Lpartial dot q_jfracpartial q_jpartial s_i right) - fracpartial Lpartial q_jfracpartial q_jpartial s_i = 0$$



but in order to get into the form
$$left(fracddt fracpartial Lpartial dot q_j - fracpartial Lpartial q_jright)fracpartial q_jpartial s_i = 0$$



I need to prove that $$fracddt left(fracpartial Lpartial dot q_jfracpartial q_jpartial s_i right) = fracddt left(fracpartial Lpartial dot q_jright)fracpartial q_jpartial s_i $$



What is the logic behind this? Any help would be greatly appreciated!




calculus-of-variations




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  • 1




    This equation is not correct: $$fracpartial Lpartial s_i = fracpartial Lpartial q_jfracpartial q_jpartial s_i$$ It should be $$ fracpartial Lpartial s_i = fracpartial Lpartial q_jfracpartial q_jpartial s_i + fracpartial Lpartial dot q_j fracpartial dot q_jpartial s_i $$
    – md2perpe
    Jul 23 at 20:03














up vote
1
down vote

favorite












A lagrangian $L(q,dot q, t)$ is invariant under the point transformation $$q_i=q_i(s_1,...,s_n,t)$$



To prove this I show that



$$fracddt fracpartial Lpartial dot s_i - fracpartial Lpartial s_i = 0$$



where



$$fracpartial Lpartial s_i = fracpartial Lpartial q_jfracpartial q_jpartial s_i$$



$$fracpartial Lpartial dot s_i = fracpartial Lpartial dot q_jfracpartial dot q_jpartial dot s_i$$



$$fracpartial dot q_jpartial dot s_i= fracpartial q_jpartial s_i$$



which gives
$$fracddt left(fracpartial Lpartial dot q_jfracpartial q_jpartial s_i right) - fracpartial Lpartial q_jfracpartial q_jpartial s_i = 0$$



but in order to get into the form
$$left(fracddt fracpartial Lpartial dot q_j - fracpartial Lpartial q_jright)fracpartial q_jpartial s_i = 0$$



I need to prove that $$fracddt left(fracpartial Lpartial dot q_jfracpartial q_jpartial s_i right) = fracddt left(fracpartial Lpartial dot q_jright)fracpartial q_jpartial s_i $$



What is the logic behind this? Any help would be greatly appreciated!




calculus-of-variations




share|cite|improve this question















  • 1




    This equation is not correct: $$fracpartial Lpartial s_i = fracpartial Lpartial q_jfracpartial q_jpartial s_i$$ It should be $$ fracpartial Lpartial s_i = fracpartial Lpartial q_jfracpartial q_jpartial s_i + fracpartial Lpartial dot q_j fracpartial dot q_jpartial s_i $$
    – md2perpe
    Jul 23 at 20:03












up vote
1
down vote

favorite









up vote
1
down vote

favorite











A lagrangian $L(q,dot q, t)$ is invariant under the point transformation $$q_i=q_i(s_1,...,s_n,t)$$



To prove this I show that



$$fracddt fracpartial Lpartial dot s_i - fracpartial Lpartial s_i = 0$$



where



$$fracpartial Lpartial s_i = fracpartial Lpartial q_jfracpartial q_jpartial s_i$$



$$fracpartial Lpartial dot s_i = fracpartial Lpartial dot q_jfracpartial dot q_jpartial dot s_i$$



$$fracpartial dot q_jpartial dot s_i= fracpartial q_jpartial s_i$$



which gives
$$fracddt left(fracpartial Lpartial dot q_jfracpartial q_jpartial s_i right) - fracpartial Lpartial q_jfracpartial q_jpartial s_i = 0$$



but in order to get into the form
$$left(fracddt fracpartial Lpartial dot q_j - fracpartial Lpartial q_jright)fracpartial q_jpartial s_i = 0$$



I need to prove that $$fracddt left(fracpartial Lpartial dot q_jfracpartial q_jpartial s_i right) = fracddt left(fracpartial Lpartial dot q_jright)fracpartial q_jpartial s_i $$



What is the logic behind this? Any help would be greatly appreciated!




calculus-of-variations




share|cite|improve this question











A lagrangian $L(q,dot q, t)$ is invariant under the point transformation $$q_i=q_i(s_1,...,s_n,t)$$



To prove this I show that



$$fracddt fracpartial Lpartial dot s_i - fracpartial Lpartial s_i = 0$$



where



$$fracpartial Lpartial s_i = fracpartial Lpartial q_jfracpartial q_jpartial s_i$$



$$fracpartial Lpartial dot s_i = fracpartial Lpartial dot q_jfracpartial dot q_jpartial dot s_i$$



$$fracpartial dot q_jpartial dot s_i= fracpartial q_jpartial s_i$$



which gives
$$fracddt left(fracpartial Lpartial dot q_jfracpartial q_jpartial s_i right) - fracpartial Lpartial q_jfracpartial q_jpartial s_i = 0$$



but in order to get into the form
$$left(fracddt fracpartial Lpartial dot q_j - fracpartial Lpartial q_jright)fracpartial q_jpartial s_i = 0$$



I need to prove that $$fracddt left(fracpartial Lpartial dot q_jfracpartial q_jpartial s_i right) = fracddt left(fracpartial Lpartial dot q_jright)fracpartial q_jpartial s_i $$



What is the logic behind this? Any help would be greatly appreciated!





calculus-of-variations





share|cite|improve this question










share|cite|improve this question




share|cite|improve this question









asked Jul 23 at 17:17









DS08

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  • 1




    This equation is not correct: $$fracpartial Lpartial s_i = fracpartial Lpartial q_jfracpartial q_jpartial s_i$$ It should be $$ fracpartial Lpartial s_i = fracpartial Lpartial q_jfracpartial q_jpartial s_i + fracpartial Lpartial dot q_j fracpartial dot q_jpartial s_i $$
    – md2perpe
    Jul 23 at 20:03












  • 1




    This equation is not correct: $$fracpartial Lpartial s_i = fracpartial Lpartial q_jfracpartial q_jpartial s_i$$ It should be $$ fracpartial Lpartial s_i = fracpartial Lpartial q_jfracpartial q_jpartial s_i + fracpartial Lpartial dot q_j fracpartial dot q_jpartial s_i $$
    – md2perpe
    Jul 23 at 20:03







1




1




This equation is not correct: $$fracpartial Lpartial s_i = fracpartial Lpartial q_jfracpartial q_jpartial s_i$$ It should be $$ fracpartial Lpartial s_i = fracpartial Lpartial q_jfracpartial q_jpartial s_i + fracpartial Lpartial dot q_j fracpartial dot q_jpartial s_i $$
– md2perpe
Jul 23 at 20:03




This equation is not correct: $$fracpartial Lpartial s_i = fracpartial Lpartial q_jfracpartial q_jpartial s_i$$ It should be $$ fracpartial Lpartial s_i = fracpartial Lpartial q_jfracpartial q_jpartial s_i + fracpartial Lpartial dot q_j fracpartial dot q_jpartial s_i $$
– md2perpe
Jul 23 at 20:03















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